(This game sounds familiar. Is it original?)

This is a game for 2 players.

Start with a carefully drawn n-by-n grid.

The players take turns completely filling in a total of n squares of the grid. (So, the first player to move fills in ceiling(n/2) squares, and the second player to move fills in floor(n/2) squares.)

After the squares are filled in, then the second player who filled in the squares is the first player to move in the next phase of the game.

The players take turns. On each turn a player draws a straight line from any empty vertex on the edge of the grid (where a grid-line meets the grid's perimeter) to any other empty vertex on any other edge of the grid.

By "empty" vertex, I mean a vertex that has not yet had a line drawn to it or from it in this phase of the game.

No line may pass through any filled-in square. But a line may touch a filled-in square (along an edge or touching at a corner).

(Also, lines may be vertical or horizontal. For this reason, I suggest that the grid be lightly drawn.)

Every time a line passes through a previously drawn line (previously drawn by either player in the second phase of the game) then the player's OPPONENT gets a point for each line crossed by the player's line.

Players move until there are no more possible lines that can be drawn under the rules.

(If a player claims that he/she cannot move any more, then the player's opponent may challenge this assertion and find, if possible, a path the player's line can indeed follow.)

Highest score wins.

Thanks,

Leroy Quet

PS: See the post (to my blog "Amorphous Trapezoid") about games-related poetry at:

http://prism-of-spirals.blogspot.com/2008/12/blog-post_21.html

## Tuesday, December 23, 2008

### Palindromic Card Game

This is a card game for 2 players. (Although this game doesn't technically require cards, using them makes the game easier to play.)

Start with 2n cards, n red and n black cards. (You can play with a standard deck {no jokers}, letting n be 26, and all spades and clubs are black, all hearts and diamonds are red. All that matters in this game is the colors of the cards' suits.)

Deal n cards to each of the players.

Players arrange their n cards in any order in a row, face up.

(One row of cards per player.)

After the cards are arranged, each player then tries to find as many distinct palindromes (symmetric patterns of of redness and blackness) within their opponent's row of cards, where each palindrome starts and ends with a red card. By DISTINCT palindrome I mean that each particular arrangement of reds and blacks counts only once. (Also, different distinct palindromes may share some of the same cards. And a palindrome may consist of exactly one red card.)

For example, if we have an n of 12 and we have the following row:

R B B R R B B B R B B R

Then the palindrome (R B B R) would be counted once, even though it occurs twice in the row.

(The palindrome (BBRBB) would not count at all because it starts and ends with black cards.)

A player gets this many points:

(number of red cards in the player's row) - (number of DISTINCT palindromes found by the player's opponent in the player's row).

(This score will always be 0 or higher.)

Highest score wins.

Thanks,

Leroy Quet

PS: See the post (to my blog "Amorphous Trapezoid") about games-related poetry at:

http://prism-of-spirals.blogspot.com/2008/12/blog-post_21.html

Start with 2n cards, n red and n black cards. (You can play with a standard deck {no jokers}, letting n be 26, and all spades and clubs are black, all hearts and diamonds are red. All that matters in this game is the colors of the cards' suits.)

Deal n cards to each of the players.

Players arrange their n cards in any order in a row, face up.

(One row of cards per player.)

After the cards are arranged, each player then tries to find as many distinct palindromes (symmetric patterns of of redness and blackness) within their opponent's row of cards, where each palindrome starts and ends with a red card. By DISTINCT palindrome I mean that each particular arrangement of reds and blacks counts only once. (Also, different distinct palindromes may share some of the same cards. And a palindrome may consist of exactly one red card.)

For example, if we have an n of 12 and we have the following row:

R B B R R B B B R B B R

Then the palindrome (R B B R) would be counted once, even though it occurs twice in the row.

(The palindrome (BBRBB) would not count at all because it starts and ends with black cards.)

A player gets this many points:

(number of red cards in the player's row) - (number of DISTINCT palindromes found by the player's opponent in the player's row).

(This score will always be 0 or higher.)

Highest score wins.

Thanks,

Leroy Quet

PS: See the post (to my blog "Amorphous Trapezoid") about games-related poetry at:

http://prism-of-spirals.blogspot.com/2008/12/blog-post_21.html

## Friday, December 12, 2008

### "Maze" Of Polygonal Sections, Game

This game has elements in common with an earlier game of mine, Slice And Fill.

See:

http://gamesconceived.blogspot.com/2008/09/slice-and-fill.html

This game will work with any number of players.

Start with an n-by-n grid lightly and carefully drawn on paper.

Darken in the 4 grid-lines that form the square boundary of the grid. (All of the vertexes along the grid's edge are thereafter each considered to be drawn-to by a line-segment.)

Players take turns drawing straight line-segments, one segment per move, each segment drawn from any vertex of the grid that has a line segment passing through it or terminating at it, to any vertex that touches no line-segments, such that the line segments don't cross any others or coincide with any others.

(Any number of segments may be drawn FROM any single vertex. Line-segments may be diagonal and of any slope. Each line-segment may pass through any number of vertices. But, I repeat, line segments must each be drawn from a vertex of the grid to another vertex of the grid, not from an intersection of a line-segment and a grid-line if that intersection is not a vertex of the grid.)

The first line-segment (after the perimeter of the grid is filled in) is drawn from a vertex along the edge of the grid, of course.

When all vertexes of the grid are touching line-segments, we have a maze (without an entrance or exit), and then the next phase of the game begins.

One player starts the second phase by filling in any "section" of the subdivided grid. A section is defined by* the lines of the grid and/or by straight line-segments drawn by players (a section may be a square, or it may be a polygon which is a subset of a square).

*[By "where the section is defined by...", I mean "where the section is BORDERED by" the lines of the grid and/or by straight line-segments drawn by players. There are no internal line-segments within any given "section".]

Then the players take turns filling in, if possible, any UNFILLED section that is immediately adjacent to the previously filled in section (filled in by another player) and that is not separated from the previously filled in section by a line-segment drawn in the earlier phase of the game. (So, consecutively filled sections must not only be adjacent, but must be in the same "corridor" of the maze.)

If a section can be filled in under the rules, then a section must be filled in.

If, however, a section cannot be filled in by a player (either because it is surrounded by already filled in sections, or it is at one of the maze's dead-ends), then the previous player to move gets a point. The player who cannot fill in a section under the rules above then fills in any unfilled section (so as to start a new string of filled in sections).

The game continues until all sections are filled in.

The player with the greatest number of points wins.

Thanks,

Leroy Quet

See:

http://gamesconceived.blogspot.com/2008/09/slice-and-fill.html

This game will work with any number of players.

Start with an n-by-n grid lightly and carefully drawn on paper.

Darken in the 4 grid-lines that form the square boundary of the grid. (All of the vertexes along the grid's edge are thereafter each considered to be drawn-to by a line-segment.)

Players take turns drawing straight line-segments, one segment per move, each segment drawn from any vertex of the grid that has a line segment passing through it or terminating at it, to any vertex that touches no line-segments, such that the line segments don't cross any others or coincide with any others.

(Any number of segments may be drawn FROM any single vertex. Line-segments may be diagonal and of any slope. Each line-segment may pass through any number of vertices. But, I repeat, line segments must each be drawn from a vertex of the grid to another vertex of the grid, not from an intersection of a line-segment and a grid-line if that intersection is not a vertex of the grid.)

The first line-segment (after the perimeter of the grid is filled in) is drawn from a vertex along the edge of the grid, of course.

When all vertexes of the grid are touching line-segments, we have a maze (without an entrance or exit), and then the next phase of the game begins.

One player starts the second phase by filling in any "section" of the subdivided grid. A section is defined by* the lines of the grid and/or by straight line-segments drawn by players (a section may be a square, or it may be a polygon which is a subset of a square).

*[By "where the section is defined by...", I mean "where the section is BORDERED by" the lines of the grid and/or by straight line-segments drawn by players. There are no internal line-segments within any given "section".]

Then the players take turns filling in, if possible, any UNFILLED section that is immediately adjacent to the previously filled in section (filled in by another player) and that is not separated from the previously filled in section by a line-segment drawn in the earlier phase of the game. (So, consecutively filled sections must not only be adjacent, but must be in the same "corridor" of the maze.)

If a section can be filled in under the rules, then a section must be filled in.

If, however, a section cannot be filled in by a player (either because it is surrounded by already filled in sections, or it is at one of the maze's dead-ends), then the previous player to move gets a point. The player who cannot fill in a section under the rules above then fills in any unfilled section (so as to start a new string of filled in sections).

The game continues until all sections are filled in.

The player with the greatest number of points wins.

Thanks,

Leroy Quet

## Tuesday, December 2, 2008

### Co-Compositeness (a game)

This is a game for any number of players. Start with an n-by-n grid, where n is larger if there are more players. (I suggest an n of at least 16 if there are 2 players.)

The first player to move places a 1 in any of the grid's squares.

Players take turns placing numbers in the grid's squares as follows:

*Each player places in a grid-square the next higher odd integer than the (odd) integer previously put in a square by the previous player to move. (So, let m be the number of total moves made by all the players so far; then the next player to move places a {2m+1} in the next square.)

*Players place the (odd) integer in a square that is immediately adjacent (in any of the 8 directions of: above, below, left of, right of, or diagonally) to the square the previous (odd) number was last put inside.

*Each integer is either placed in an empty square or in a square that already contains just one number that is NOT coprime to the integer the player is now placing in the square. There may be no more than 2 integers in any one square.

Scoring:

Every time a player places an integer in a square with an integer already in it, then that player gets a point. (Any pair of integers in the same square must be "co-composite", ie non-coprime.)

Players continue filling in the squares until a player cannot move anywhere. (If a player can move, then the player must move.) Then the game is over.

Highest score wins.

(Note: Part of the strategy of this game might be for a player to try to force an early ending to the game if that player has the highest score so far.)

Thanks,

Leroy Quet

The first player to move places a 1 in any of the grid's squares.

Players take turns placing numbers in the grid's squares as follows:

*Each player places in a grid-square the next higher odd integer than the (odd) integer previously put in a square by the previous player to move. (So, let m be the number of total moves made by all the players so far; then the next player to move places a {2m+1} in the next square.)

*Players place the (odd) integer in a square that is immediately adjacent (in any of the 8 directions of: above, below, left of, right of, or diagonally) to the square the previous (odd) number was last put inside.

*Each integer is either placed in an empty square or in a square that already contains just one number that is NOT coprime to the integer the player is now placing in the square. There may be no more than 2 integers in any one square.

Scoring:

Every time a player places an integer in a square with an integer already in it, then that player gets a point. (Any pair of integers in the same square must be "co-composite", ie non-coprime.)

Players continue filling in the squares until a player cannot move anywhere. (If a player can move, then the player must move.) Then the game is over.

Highest score wins.

(Note: Part of the strategy of this game might be for a player to try to force an early ending to the game if that player has the highest score so far.)

Thanks,

Leroy Quet

## Monday, November 24, 2008

## Tuesday, November 18, 2008

### f(g(x)) = g(f(x)) (a game)

This game is for 2 players.

First, the players each secretly come up with two mathematical functions (two functions per player), f(x) and g(x).

The functions, f(x) and g(x), are defined and real for all real x, and they are continuous.

Next, the players take turns each being the "proposer" and the "solver".

First in a round the proposer reveals his/her functions.

The solver tries to determine (within a given time period) whether

f(g(x)) = g(f(x))

either for (1 of 3 choices) no real x's, a finite number of real x's, or an infinite number of real x's.

If the solver can't do this in a specific finite period of time, then the proposer gets a point. Otherwise the solver gets a point.

The players then switch who is the proposer and who is the solver.

(So, with only the two rounds, this is a low-scoring game.)

Example: f(x) = sin(x). g(x) = e^(x+1).

So, f(g(x)) = sin(e^(x+1)). g(f(x)) = e^(sin(x)+1).

The number of x's where sin(e^(x+1)) = e^(sin(x)+1) is precisely the number of pairs of integers (m,n) such that:

e^(1- pi/2 +2*n*pi) = pi/2 + 2*pi*m (I think).

I don't personally know if this equation is solvable at all for any integers m and n, let alone if there are a finite number of pairs (m,n) or an infinite number of pairs of integers m and n. So, if I was the solver and if the proposer proposed f(x) = sin(x) and g(x) = e^(x+1), then the proposer would get a point that round, and I would get no points that round.

First, the players each secretly come up with two mathematical functions (two functions per player), f(x) and g(x).

The functions, f(x) and g(x), are defined and real for all real x, and they are continuous.

Next, the players take turns each being the "proposer" and the "solver".

First in a round the proposer reveals his/her functions.

The solver tries to determine (within a given time period) whether

f(g(x)) = g(f(x))

either for (1 of 3 choices) no real x's, a finite number of real x's, or an infinite number of real x's.

If the solver can't do this in a specific finite period of time, then the proposer gets a point. Otherwise the solver gets a point.

The players then switch who is the proposer and who is the solver.

(So, with only the two rounds, this is a low-scoring game.)

Example: f(x) = sin(x). g(x) = e^(x+1).

So, f(g(x)) = sin(e^(x+1)). g(f(x)) = e^(sin(x)+1).

The number of x's where sin(e^(x+1)) = e^(sin(x)+1) is precisely the number of pairs of integers (m,n) such that:

e^(1- pi/2 +2*n*pi) = pi/2 + 2*pi*m (I think).

I don't personally know if this equation is solvable at all for any integers m and n, let alone if there are a finite number of pairs (m,n) or an infinite number of pairs of integers m and n. So, if I was the solver and if the proposer proposed f(x) = sin(x) and g(x) = e^(x+1), then the proposer would get a point that round, and I would get no points that round.

## Tuesday, November 11, 2008

### Grid-Squiggly Game

This game is played on an n-by-n grid (n by n squares, {n+1} by {n+1} lines) that is lightly drawn on paper. The game is for 2 or more players.

First, players each secretly pick a positive integer m between 1 and n^2. (See below.)

The first player to move draws a line segment, one grid-square in length, along any vertical or horizontal line of the grid. (But don't draw along the border of the grid.)

Players then take turns drawing a line-segment each turn, where the line-segment is one grid-square in length, and goes from any vertex with a line-segment drawn to it (by any player) to any adjacent vertex that does not yet have a line segment drawn to it. (The drawn-to vertex is immediately above, below, right of, left of the drawn-from vertex.)

No line segments go along the border of the grid, but line segments can connect to vertices along the border of the grid.

Players continue to draw segments until there is no place they can draw them. (A total of n^2 + 2n - 4 segments will be drawn.)

Next, with a pencil of a color different that their opponents' pencil colors, each player takes turns (completely) filling in sections of the grid, one section each move. Each "section" is bounded by the lines the players drew and by the border of the grid.

When the whole grid has been colored in, count the number of squares filled in by each player.

Players then reveal the numbers (m) they picked at the game's beginning.

The player whose number of squares filled in is closest to the number they picked at the game's beginning (m) wins.

Thanks,

Leroy Quet

First, players each secretly pick a positive integer m between 1 and n^2. (See below.)

The first player to move draws a line segment, one grid-square in length, along any vertical or horizontal line of the grid. (But don't draw along the border of the grid.)

Players then take turns drawing a line-segment each turn, where the line-segment is one grid-square in length, and goes from any vertex with a line-segment drawn to it (by any player) to any adjacent vertex that does not yet have a line segment drawn to it. (The drawn-to vertex is immediately above, below, right of, left of the drawn-from vertex.)

No line segments go along the border of the grid, but line segments can connect to vertices along the border of the grid.

Players continue to draw segments until there is no place they can draw them. (A total of n^2 + 2n - 4 segments will be drawn.)

Next, with a pencil of a color different that their opponents' pencil colors, each player takes turns (completely) filling in sections of the grid, one section each move. Each "section" is bounded by the lines the players drew and by the border of the grid.

When the whole grid has been colored in, count the number of squares filled in by each player.

Players then reveal the numbers (m) they picked at the game's beginning.

The player whose number of squares filled in is closest to the number they picked at the game's beginning (m) wins.

Thanks,

Leroy Quet

## Tuesday, October 28, 2008

### Within The Curve

Here is a game for any number of players.

Each player has m (n-by-n) grids, where m is the number of players. (So there are m^2 grids all together.)

I suggest an n of at least 12.

On one of their grids each player secretly draws a closed non-self-intersecting curve. (The curve is bounded within the n-by-n grid.) Each player's curve does not go through any intersections of the grid-lines.

Next, on one of each of the other player's blank n-by-n grids each player copies his/her curve over.

The copies of each curve must go through the same respective squares of each grid as the original curve did.

So, there are m copies each of m curves, each player in possession of one copy of each curve.

Next, secretly and simultaneously, each player fills in the squares each curve goes through on any particular grid with 1,2,3,...., the integers placed in order and next to each other along the curve. The numbers can start anywhere along a curve, and can go either clockwise or counterclockwise.

Next, each player secretly fills in the squares within each curve's interior with 1,2,3,..., the numbers placed in order, each number placed in any empty interior square such that all other numbers (including possibly numbers along the curve) above, right of, left of, or below the number are coprime to that number.

(Any pair of adjacent numbers that are both in squares a curve passes through don't have to be coprime. Only interior numbers have to be coprime to adjacent numbers along the curve, or to adjacent numbers that are also on the curve's interior.)

Players continue to fill the interior of each curve with numbers until the players can't fill in any more numbers under the rules.

When each player has filled in each curve as far as they can, the score for each player is the sum of the top numbers in the interior squares of each of the m curves the player filled (partially) in.

Highest score wins.

Players may check their opponents' grids after the game is over to make sure that all applicable pairs of adjacent numbers are actually coprime. If a player made a mistake, that player automatically loses the game.

Thanks,

Leroy Quet

Each player has m (n-by-n) grids, where m is the number of players. (So there are m^2 grids all together.)

I suggest an n of at least 12.

On one of their grids each player secretly draws a closed non-self-intersecting curve. (The curve is bounded within the n-by-n grid.) Each player's curve does not go through any intersections of the grid-lines.

Next, on one of each of the other player's blank n-by-n grids each player copies his/her curve over.

The copies of each curve must go through the same respective squares of each grid as the original curve did.

So, there are m copies each of m curves, each player in possession of one copy of each curve.

Next, secretly and simultaneously, each player fills in the squares each curve goes through on any particular grid with 1,2,3,...., the integers placed in order and next to each other along the curve. The numbers can start anywhere along a curve, and can go either clockwise or counterclockwise.

Next, each player secretly fills in the squares within each curve's interior with 1,2,3,..., the numbers placed in order, each number placed in any empty interior square such that all other numbers (including possibly numbers along the curve) above, right of, left of, or below the number are coprime to that number.

(Any pair of adjacent numbers that are both in squares a curve passes through don't have to be coprime. Only interior numbers have to be coprime to adjacent numbers along the curve, or to adjacent numbers that are also on the curve's interior.)

Players continue to fill the interior of each curve with numbers until the players can't fill in any more numbers under the rules.

When each player has filled in each curve as far as they can, the score for each player is the sum of the top numbers in the interior squares of each of the m curves the player filled (partially) in.

Highest score wins.

Players may check their opponents' grids after the game is over to make sure that all applicable pairs of adjacent numbers are actually coprime. If a player made a mistake, that player automatically loses the game.

Thanks,

Leroy Quet

## Tuesday, October 14, 2008

### Polygons In Permutation Grid

Here is a game for 2 players.

Start with an n-by-n grid (n-lines by n-lines, or n-1 squares by n-1 squares). I suggest that n be at least 10.

First, players take turns placing a total of n dots at intersections of the grid.

Each dot is placed at an intersection of any two lines that do not have any other dots on either of them.

So, after n dots are placed on the grid, the dots represent a permutation of (1,2,3,...n).

Reading the dots from top to bottom, let the dot on the mth horizontal line be p(m).

Reading the dots from left to right, let the dot on the mth vertical line be q(m).

Draw a straight line-segment from p(m) to p(m+1) for all m where 1<= m <= n-1.

Draw a straight line-segment from q(m) to q(m+1) for all m where 1<= m <= n-1.

Player 1 gets a point for every triangle that is formed by the line-segments.

Player 2 gets a point for every non-triangle (4 or more sides) that is formed by the line-segments.

Only polygons completely bounded by parts of line-segments (not counting the grid's lines) score any points.

For a triangle or non-triangle to score a point, the polygon must not be subdivided by any line-segments (but may be subdivided by grid-lines).

I suspect that there is a bias either towards player 1 or player 2. So, play an even number of rounds with the same-sized grids, each player playing player 1 and player 2 an equal number of times, and add up each player's score to get the players' grand total scores.

Highest grand total score wins.

Thanks,

Leroy Quet

Start with an n-by-n grid (n-lines by n-lines, or n-1 squares by n-1 squares). I suggest that n be at least 10.

First, players take turns placing a total of n dots at intersections of the grid.

Each dot is placed at an intersection of any two lines that do not have any other dots on either of them.

So, after n dots are placed on the grid, the dots represent a permutation of (1,2,3,...n).

Reading the dots from top to bottom, let the dot on the mth horizontal line be p(m).

Reading the dots from left to right, let the dot on the mth vertical line be q(m).

Draw a straight line-segment from p(m) to p(m+1) for all m where 1<= m <= n-1.

Draw a straight line-segment from q(m) to q(m+1) for all m where 1<= m <= n-1.

Player 1 gets a point for every triangle that is formed by the line-segments.

Player 2 gets a point for every non-triangle (4 or more sides) that is formed by the line-segments.

Only polygons completely bounded by parts of line-segments (not counting the grid's lines) score any points.

For a triangle or non-triangle to score a point, the polygon must not be subdivided by any line-segments (but may be subdivided by grid-lines).

I suspect that there is a bias either towards player 1 or player 2. So, play an even number of rounds with the same-sized grids, each player playing player 1 and player 2 an equal number of times, and add up each player's score to get the players' grand total scores.

Highest grand total score wins.

Thanks,

Leroy Quet

## Wednesday, October 8, 2008

### Cross, Don't Cross Circle Game

Here is a game for 2 players.

As in many of my games, the players play an even number of rounds, half of the rounds where one player is offense and the other player is defense, and the other half of the rounds with the players switching who is defense and offense, and then the players adding up their scores for their grand total scores.

A round starts with a carefully drawn circle on paper. (No grids this time. Sorry.)

The defense player starts the game by drawing m (m is fixed number for all rounds) straight line segments from anywhere on the circumference of the circle to anywhere else on the circumference of the circle. (I suggest an m of 4 to 6 for beginning players.) The defense player's line segments may cross each other (but don't have to cross).

(I suggest that the bigger m is, the bigger the circle is drawn.)

After the defense player has drawn the m line segments, then it is the offense player's turn to make his/her moves for the round. Clarification: After the defense player draws her/his m line segments, he/she does not move any more during the round.

(So, in a round, first the defense player draws all his/her line segments, then the offense player draws all his/her line-segment-- see below.)

On a move the offense player draws a straight line-segment from an intersection to another intersection*.

*An intersection is either where any line segment (drawn by the defense player) touches the circle, or is where any pair of previously-drawn line segments (drawn by either player) cross.

On every odd-numbered move (the first move, the third move, the fifth move, etc) the offense player's line segment must not cross any other previously-drawn line segments.

On every even-numbered move the offense player's line segment MUST cross exactly one previously-drawn line segment (crossing no fewer, no more than one segment).

And, oh yeah, neither the defense player's nor the offense player's line segments may coincide (coincide along more than one point) with any other previously-drawn line-segment.

The offense player moves until he/she can't move anywhere, otherwise she/he MUST move.

By the way, the defense player may find possible moves for the offense player if the offense player wrongly claims that he/she can't move any more. (It is advantageous for the defense player if the offense player keeps drawing line segments.)

As the offense player draws line segments, the number of these line segments drawn is tabulated.

After playing all the rounds, the winner of this game is the player who, during all rounds that they were the offense player, drew the FEWEST line-segments all together.

--

Question:

I wonder, is there a pattern of line segments the defense player can draw that will guarantee a larger number of moves by the offense player than with any other pattern of line segments drawn by the defense player?

(By "pattern" I mean, as an example, lines drawn parallel, all lines crossing at a center point, the lines forming the perimeter of an m- gon, etc.)

Thanks,

Leroy Quet

PS: After I post this game to my blog, the list of 66 or so games I posted in September will be hidden. Just click on the triangle next to the September link to get that list of games back.

As in many of my games, the players play an even number of rounds, half of the rounds where one player is offense and the other player is defense, and the other half of the rounds with the players switching who is defense and offense, and then the players adding up their scores for their grand total scores.

A round starts with a carefully drawn circle on paper. (No grids this time. Sorry.)

The defense player starts the game by drawing m (m is fixed number for all rounds) straight line segments from anywhere on the circumference of the circle to anywhere else on the circumference of the circle. (I suggest an m of 4 to 6 for beginning players.) The defense player's line segments may cross each other (but don't have to cross).

(I suggest that the bigger m is, the bigger the circle is drawn.)

After the defense player has drawn the m line segments, then it is the offense player's turn to make his/her moves for the round. Clarification: After the defense player draws her/his m line segments, he/she does not move any more during the round.

(So, in a round, first the defense player draws all his/her line segments, then the offense player draws all his/her line-segment-- see below.)

On a move the offense player draws a straight line-segment from an intersection to another intersection*.

*An intersection is either where any line segment (drawn by the defense player) touches the circle, or is where any pair of previously-drawn line segments (drawn by either player) cross.

On every odd-numbered move (the first move, the third move, the fifth move, etc) the offense player's line segment must not cross any other previously-drawn line segments.

On every even-numbered move the offense player's line segment MUST cross exactly one previously-drawn line segment (crossing no fewer, no more than one segment).

And, oh yeah, neither the defense player's nor the offense player's line segments may coincide (coincide along more than one point) with any other previously-drawn line-segment.

The offense player moves until he/she can't move anywhere, otherwise she/he MUST move.

By the way, the defense player may find possible moves for the offense player if the offense player wrongly claims that he/she can't move any more. (It is advantageous for the defense player if the offense player keeps drawing line segments.)

As the offense player draws line segments, the number of these line segments drawn is tabulated.

After playing all the rounds, the winner of this game is the player who, during all rounds that they were the offense player, drew the FEWEST line-segments all together.

--

Question:

I wonder, is there a pattern of line segments the defense player can draw that will guarantee a larger number of moves by the offense player than with any other pattern of line segments drawn by the defense player?

(By "pattern" I mean, as an example, lines drawn parallel, all lines crossing at a center point, the lines forming the perimeter of an m- gon, etc.)

Thanks,

Leroy Quet

PS: After I post this game to my blog, the list of 66 or so games I posted in September will be hidden. Just click on the triangle next to the September link to get that list of games back.

## Monday, September 29, 2008

### Connectin'-It Game

This is a game for 2 or more players.

Start with an n-by-n grid taken from graph paper. (I suggest an n of about 10 to 20 for beginners.)

On a move a player can do one of two things:

He/she can draw a dot at any intersection of the grid that doesn't already have a dot and isn't on a line-segment.

Or she/he can draw a straight line-segment that connects any two dots, provided that the line segment does not pass through any other line segment and doesn't pass over any intermediate dots.

A player gets a point every time she/he connects two dots with a line-segment.

The first player to get a pre-determined score wins.

(I suggest a winning score between n^2/4 and n^2/2, if there are two players.)

Thanks,

Leroy Quet

Start with an n-by-n grid taken from graph paper. (I suggest an n of about 10 to 20 for beginners.)

On a move a player can do one of two things:

He/she can draw a dot at any intersection of the grid that doesn't already have a dot and isn't on a line-segment.

Or she/he can draw a straight line-segment that connects any two dots, provided that the line segment does not pass through any other line segment and doesn't pass over any intermediate dots.

A player gets a point every time she/he connects two dots with a line-segment.

The first player to get a pre-determined score wins.

(I suggest a winning score between n^2/4 and n^2/2, if there are two players.)

Thanks,

Leroy Quet

### Wind Around Solitaire

Here is a 1-player game played on an n-by-n grid, where n is odd. (I suggest an n of about 13 to 21 for beginners.)

The player places the numbers 1,2,3,...n^2 IN ORDER into the grid.

1 goes in the center square.

Each number (2k+1) must go either below, above, left of, or right of the square with (2k-1) in it.

Each number (2k) can go in any square that is immediately adjacent to a square with an integer already in it.

Numbers can only be placed in empty squares.

The player's score is the number of times the path of odd integers goes completely around the center square clockwise before the player is unable to place any more numbers in the grid.

Thanks,

Leroy Quet

The player places the numbers 1,2,3,...n^2 IN ORDER into the grid.

1 goes in the center square.

Each number (2k+1) must go either below, above, left of, or right of the square with (2k-1) in it.

Each number (2k) can go in any square that is immediately adjacent to a square with an integer already in it.

Numbers can only be placed in empty squares.

The player's score is the number of times the path of odd integers goes completely around the center square clockwise before the player is unable to place any more numbers in the grid.

Thanks,

Leroy Quet

## Monday, September 22, 2008

### Slice Through The Boundaries

This is a game played by any number of people.

It is played on an n-by-n section of grid taken from graph paper. (I

suggest an n of about 12 for beginners, if there are only 2 players.)

Players take turns who is the offense player. (The other players play

defense on a round.) A round is played for each player playing the

game. An empty n-by-n grid is used each round (with the same n as in

the other rounds).

Players take turns each filling in the empty squares of the grid, one

empty square filled in each move by each player.

If there are m players, then each player fills in floor(n^2/(2m))

squares. (That is a total of m*floor(n^2/(2m)) squares filled in all

together.)

Then the offense player draws a straight line (with a straight-edge)

from any side of the n-by-n grid to any other side.

The line must not be perfectly vertical or perfectly horizontal.

The offense player gets a point for every boundary between a filled-in

square and an empty square that the line passes through.

Highest score wins.

Example:

Filled-in square = *. Empty square = o.

n = 6. (View with fixed-width font.)

\ 1 2 3 4 5 6

A o * * * o *

B o * * o o o

C * o * o o *

D * * o o o o

E o o * * o o

F * o * * * *

Let us say that the line goes from just below the upper-left corner of

the grid to just left of the lower right corner. (The line meets the

perimeter of the grid less than one square's length from each of these

corners.)

This is kind of hard to depict here, because the squares of the grid

are literally squares, while they are not in my diagram; but hopefully

it is clear anyway.

The line first crosses the boundary between 1B and 2B. Then it crosses

the boundary between 2B and 2C. Then it crosses the boundary 2C to

3C. Then 3C-3D. Then 4D-4E. Then 4E-5E. And finally, 5E-5F.

Note: Technically we are concerned about the number of times the line

crosses from a filled-in square into an unfilled-in square, or vice

versa. So if a line crosses from a square to a diagonally adjacent

square via the vertex that joins them, then what we are concerned

about is the two squares' status. In other words, the vertex is

considered the "boundary" in that case.

Thanks,

Leroy Quet

It is played on an n-by-n section of grid taken from graph paper. (I

suggest an n of about 12 for beginners, if there are only 2 players.)

Players take turns who is the offense player. (The other players play

defense on a round.) A round is played for each player playing the

game. An empty n-by-n grid is used each round (with the same n as in

the other rounds).

Players take turns each filling in the empty squares of the grid, one

empty square filled in each move by each player.

If there are m players, then each player fills in floor(n^2/(2m))

squares. (That is a total of m*floor(n^2/(2m)) squares filled in all

together.)

Then the offense player draws a straight line (with a straight-edge)

from any side of the n-by-n grid to any other side.

The line must not be perfectly vertical or perfectly horizontal.

The offense player gets a point for every boundary between a filled-in

square and an empty square that the line passes through.

Highest score wins.

Example:

Filled-in square = *. Empty square = o.

n = 6. (View with fixed-width font.)

\ 1 2 3 4 5 6

A o * * * o *

B o * * o o o

C * o * o o *

D * * o o o o

E o o * * o o

F * o * * * *

Let us say that the line goes from just below the upper-left corner of

the grid to just left of the lower right corner. (The line meets the

perimeter of the grid less than one square's length from each of these

corners.)

This is kind of hard to depict here, because the squares of the grid

are literally squares, while they are not in my diagram; but hopefully

it is clear anyway.

The line first crosses the boundary between 1B and 2B. Then it crosses

the boundary between 2B and 2C. Then it crosses the boundary 2C to

3C. Then 3C-3D. Then 4D-4E. Then 4E-5E. And finally, 5E-5F.

Note: Technically we are concerned about the number of times the line

crosses from a filled-in square into an unfilled-in square, or vice

versa. So if a line crosses from a square to a diagonally adjacent

square via the vertex that joins them, then what we are concerned

about is the two squares' status. In other words, the vertex is

considered the "boundary" in that case.

Thanks,

Leroy Quet

### Lines-By-Lines

This is a game for 2 players.

Two rounds are played. In each round one player is offense while the

other is defense.

Players switch who is offense and who is defense for the 2nd round.

Start a round with an n-by-n grid.

Both rounds are played on the same sized grid drawn carefully on

paper.

(It is preferable to use graph paper.)

The offense player moves first in a round.

The first player to move draws a straight line-segment (with a

straight-edge, preferably) from any vertex of the grid to any other,

provided that no intermediate vertexes are crossed. (The only vertexes

to coincide with the line-segment are at the segment's end-points.)

Players take turns each drawing a straight line-segment (from the

vertex where the other player last drew a line-segment to) to another

line-segment. So all the line-segments together form a continuous

path.

Line-segments must not cross or coincide with each other.

Line-segments must not coincide with any vertexes of the grid, with

the exception of at the line-segments' end-points.

Line segments must not travel horizontally or vertically.

Play continues until it is not possible to draw a line-segment to any

other vertex, given the rules.

The offense player gets a point for every square of the grid the path

of line-segments passes through.

(A player gets at most one point for each square, no matter how many

line-segments pass through that square.)

Players switch who is offense, then play another round.

Highest score wins.

Note: It may be more interesting if it is required that the first

player to move draws his/her line-segment from the center of the grid.

Thanks,

Leroy Quet

Two rounds are played. In each round one player is offense while the

other is defense.

Players switch who is offense and who is defense for the 2nd round.

Start a round with an n-by-n grid.

Both rounds are played on the same sized grid drawn carefully on

paper.

(It is preferable to use graph paper.)

The offense player moves first in a round.

The first player to move draws a straight line-segment (with a

straight-edge, preferably) from any vertex of the grid to any other,

provided that no intermediate vertexes are crossed. (The only vertexes

to coincide with the line-segment are at the segment's end-points.)

Players take turns each drawing a straight line-segment (from the

vertex where the other player last drew a line-segment to) to another

line-segment. So all the line-segments together form a continuous

path.

Line-segments must not cross or coincide with each other.

Line-segments must not coincide with any vertexes of the grid, with

the exception of at the line-segments' end-points.

Line segments must not travel horizontally or vertically.

Play continues until it is not possible to draw a line-segment to any

other vertex, given the rules.

The offense player gets a point for every square of the grid the path

of line-segments passes through.

(A player gets at most one point for each square, no matter how many

line-segments pass through that square.)

Players switch who is offense, then play another round.

Highest score wins.

Note: It may be more interesting if it is required that the first

player to move draws his/her line-segment from the center of the grid.

Thanks,

Leroy Quet

### Triangle Grid Of Coprimality

Here is a game for 2 players.

The game is played on a "triangle"-shaped grid. 1 square on the top

row; 2 squares on the 2nd row; 3 squares on the 3rd row; etc; n

squares on the nth and bottom row.

(I suggest an n of about 12 for beginners.)

So, we have this:

(Each @ represents a square.)

@

@ @

@ @ @

@ @ @ @

@ @ @ @ @

etc

The players take turns filling in squares of the triangle, one empty

square filled in per move.

Each player fills in floor(n^2 /8) squares. (So, 2*floor(n^2 /8)

squares are filled in by both players together.)

Player 1 gets a point for every COLUMN of the triangle where the

number of filled in squares (filled in by either player) is coprime to

the number of total squares in the column.

Player 2 gets a point for every ROW of the triangle where the number

of filled in squares is coprime to the number of total squares in the

row.

(And for the purposes of this game, 0 is considered to be coprime only

to 1.)

For example, let's say we have a triangle filled like this at the

game's end:

(n= 6. @ = empty square. X = filled square.)

X

@ X

X X @

@ @ X @

X X @ @ @

@ @ X @ @ @

Player 1: column 1 (from left), 3 is not coprime to 6 (no point);

column 2, 3 is coprime to 5 (1 point); column 3, 2 is not coprime 4

(no point); column 4, 0 is not coprime to 3 (no point); column 5, 0 is

not coprime to 2 (no point); column 6, 0 is coprime to 1 (1 point).

Player 2: row 1 (from top), 1 is coprime to 1 (1 point); row 2, 1 is

coprime to 2 (1 point); row 3, 2 is coprime to 3 (1 point); row 4, 1

is coprime to 4 (1 point); row 5, 2 is coprime to 5 (1 point); row 6,

1 is coprime to 6 (1 point).

So, player 1 gets 2 points, while player 2 gets 6 points. Player 2

wins.

Is there a strategy that guarantees a win for one player?

Thanks,

Leroy Quet

The game is played on a "triangle"-shaped grid. 1 square on the top

row; 2 squares on the 2nd row; 3 squares on the 3rd row; etc; n

squares on the nth and bottom row.

(I suggest an n of about 12 for beginners.)

So, we have this:

(Each @ represents a square.)

@

@ @

@ @ @

@ @ @ @

@ @ @ @ @

etc

The players take turns filling in squares of the triangle, one empty

square filled in per move.

Each player fills in floor(n^2 /8) squares. (So, 2*floor(n^2 /8)

squares are filled in by both players together.)

Player 1 gets a point for every COLUMN of the triangle where the

number of filled in squares (filled in by either player) is coprime to

the number of total squares in the column.

Player 2 gets a point for every ROW of the triangle where the number

of filled in squares is coprime to the number of total squares in the

row.

(And for the purposes of this game, 0 is considered to be coprime only

to 1.)

For example, let's say we have a triangle filled like this at the

game's end:

(n= 6. @ = empty square. X = filled square.)

X

@ X

X X @

@ @ X @

X X @ @ @

@ @ X @ @ @

Player 1: column 1 (from left), 3 is not coprime to 6 (no point);

column 2, 3 is coprime to 5 (1 point); column 3, 2 is not coprime 4

(no point); column 4, 0 is not coprime to 3 (no point); column 5, 0 is

not coprime to 2 (no point); column 6, 0 is coprime to 1 (1 point).

Player 2: row 1 (from top), 1 is coprime to 1 (1 point); row 2, 1 is

coprime to 2 (1 point); row 3, 2 is coprime to 3 (1 point); row 4, 1

is coprime to 4 (1 point); row 5, 2 is coprime to 5 (1 point); row 6,

1 is coprime to 6 (1 point).

So, player 1 gets 2 points, while player 2 gets 6 points. Player 2

wins.

Is there a strategy that guarantees a win for one player?

Thanks,

Leroy Quet

### Line-Segments In Circle

Here is a game of mine that is NOT based on an n-by-n grid...

The game is for 2 players.

Draw a circle on a piece of paper. (Preferably the circle is drawn

with a compass, but this is not necessary. The circle should be drawn

carefully, however.)

Put a dot at the circle's center.

Players take turns drawing straight line-segments (preferably with a

straight-edge) within the circle as follows:

*A segment can go from the dot at the circle's center to the edge of

the circle.

*A segment can go from {an intersection of another segment and the

circle} to {another intersection of a segment and the circle}.

*A segment can pass through at most ONE line-segment that has already

been drawn.

*A segment can not coincide with a previously drawn segment. (ie. Two

segments can coincide at at most one point.)

*A segment cannot be drawn from the center to the circle that is 180

degrees away from the segment last drawn by the other player, if the

last segment was drawn from the center to the circle. (This prevents a

player from simply matching the other player's moves, and so

automatically winning.)

The last player able to move is the winner.

Is there a strategy that makes this game easy to win?

Thanks,

Leroy Quet

The game is for 2 players.

Draw a circle on a piece of paper. (Preferably the circle is drawn

with a compass, but this is not necessary. The circle should be drawn

carefully, however.)

Put a dot at the circle's center.

Players take turns drawing straight line-segments (preferably with a

straight-edge) within the circle as follows:

*A segment can go from the dot at the circle's center to the edge of

the circle.

*A segment can go from {an intersection of another segment and the

circle} to {another intersection of a segment and the circle}.

*A segment can pass through at most ONE line-segment that has already

been drawn.

*A segment can not coincide with a previously drawn segment. (ie. Two

segments can coincide at at most one point.)

*A segment cannot be drawn from the center to the circle that is 180

degrees away from the segment last drawn by the other player, if the

last segment was drawn from the center to the circle. (This prevents a

player from simply matching the other player's moves, and so

automatically winning.)

The last player able to move is the winner.

Is there a strategy that makes this game easy to win?

Thanks,

Leroy Quet

### Turn It Around -- Multiply/Add Grid

Here is another game played on an n-by-n grid drawn on paper,

where n is even.

(Actually, it would be MUCH easier to play this game on a computer.)

2 players.

Player 1 starts the game by placing a 1 in any square of the grid.

The players take turns placing numbers in the empty squares of the

grid, one number in one square per move.

Integers can only be placed in any square that is adjacent (in the

direction of up, right, down, left) to a square that is already filled

in with a number. The integer placed in a square must be exactly 1

more than the number in any filled-in square that is adjacent to the

square being filled in on the move.

Play continues until all squares are filled in.

Scoring:

For player 1, imagine another grid filled in the same way as the game

grid. Rotate the the second grid 180 degrees and place on top of the

first grid. Now multiply each number in the second grid by the number

immediately below it, getting n^2 total products. Player 1 gets the

sum of these products as the score. (So, the score is relatively

large, as far as my games go.)

For player 2, do the same thing, but rotate the top grid by only 90

degrees. (It doesn't matter if you rotate clockwise or

counterclockwise.)

Here is a small example:

n = 3.

Finished grid looks like this:

1 2 3

4 3 6

5 4 5

Rotated 180 degrees, multiply and add:

Player 1's score =

1*5 + 2*4 + 3*5 +

4*6 + 3*3 + 6*4 +

5*3 + 4*2 + 5*1 =

113.

Rotated 90 degrees, multiply and add:

Player 2's score =

1*5 + 2*4 + 3*1 +

4*4 + 3*3 + 6*2 +

5*5 + 4*6 + 5*3 =

73.

Player 1 wins.

Thanks,

Leroy Quet

where n is even.

(Actually, it would be MUCH easier to play this game on a computer.)

2 players.

Player 1 starts the game by placing a 1 in any square of the grid.

The players take turns placing numbers in the empty squares of the

grid, one number in one square per move.

Integers can only be placed in any square that is adjacent (in the

direction of up, right, down, left) to a square that is already filled

in with a number. The integer placed in a square must be exactly 1

more than the number in any filled-in square that is adjacent to the

square being filled in on the move.

Play continues until all squares are filled in.

Scoring:

For player 1, imagine another grid filled in the same way as the game

grid. Rotate the the second grid 180 degrees and place on top of the

first grid. Now multiply each number in the second grid by the number

immediately below it, getting n^2 total products. Player 1 gets the

sum of these products as the score. (So, the score is relatively

large, as far as my games go.)

For player 2, do the same thing, but rotate the top grid by only 90

degrees. (It doesn't matter if you rotate clockwise or

counterclockwise.)

Here is a small example:

n = 3.

Finished grid looks like this:

1 2 3

4 3 6

5 4 5

Rotated 180 degrees, multiply and add:

Player 1's score =

1*5 + 2*4 + 3*5 +

4*6 + 3*3 + 6*4 +

5*3 + 4*2 + 5*1 =

113.

Rotated 90 degrees, multiply and add:

Player 2's score =

1*5 + 2*4 + 3*1 +

4*4 + 3*3 + 6*2 +

5*5 + 4*6 + 5*3 =

73.

Player 1 wins.

Thanks,

Leroy Quet

### Unique Count In Rows/Columns

This game is for 2 players.

Start by drawing an n-by-n grid on paper. (I suggest an n of about 12

for beginners.)

Players take turns filling in the grid's squares, one empty square

being filled in on each move.

Each player fills in a total of floor(n^2 /4) squares.

(So a total of 2*floor(n^2 /4) squares are filled in all together.)

Scoring is as follows:

Player 1 gets a point for every ROW with a unique number of filled-in

squares in it. In other words, every row that counts does not have a

number of filled-in squares that any other row also has.

Player 2 gets a point for every COLUMN with a unique number of filled-

in squares in it.

Variation:

Players on their moves each draw cards from a regular card deck (no

jokers).

The cards are not placed back, unless n is big enough that the card

would run out -- in which case just reshuffle and reuse the deck as

needed.

The player moving then writes down the number of the card in the empty

square, instead of simply filling the square in.

(Ace = 1, jack = 11, queen = 12, king = 13.)

Then, as before, each player fills in floor(n^2/4) squares. Scoring is

as follows:

Player 1 gets a point for every ROW with a unique SUM of the values of

the filled-in squares in it. In other words, every row that counts

does not have a SUM of the values of filled-in squares that any other

row also has.

Player 2 gets a point for every COLUMN with a unique sum of the values

of the filled-in squares in it.

Thanks,

Leroy Quet

Start by drawing an n-by-n grid on paper. (I suggest an n of about 12

for beginners.)

Players take turns filling in the grid's squares, one empty square

being filled in on each move.

Each player fills in a total of floor(n^2 /4) squares.

(So a total of 2*floor(n^2 /4) squares are filled in all together.)

Scoring is as follows:

Player 1 gets a point for every ROW with a unique number of filled-in

squares in it. In other words, every row that counts does not have a

number of filled-in squares that any other row also has.

Player 2 gets a point for every COLUMN with a unique number of filled-

in squares in it.

Variation:

Players on their moves each draw cards from a regular card deck (no

jokers).

The cards are not placed back, unless n is big enough that the card

would run out -- in which case just reshuffle and reuse the deck as

needed.

The player moving then writes down the number of the card in the empty

square, instead of simply filling the square in.

(Ace = 1, jack = 11, queen = 12, king = 13.)

Then, as before, each player fills in floor(n^2/4) squares. Scoring is

as follows:

Player 1 gets a point for every ROW with a unique SUM of the values of

the filled-in squares in it. In other words, every row that counts

does not have a SUM of the values of filled-in squares that any other

row also has.

Player 2 gets a point for every COLUMN with a unique sum of the values

of the filled-in squares in it.

Thanks,

Leroy Quet

### Up And Up, Grid Solitaire

This game is played solitaire. (If more than one player wants to play,

then each player plays this game with the same sized grid, and players

compare final scores.)

A player starts with an n-by-n grid drawn on paper. (I suggest an n of

about 8 for beginners.)

The player starts the game by placing a "1" in any square of the grid.

The player, on move number m (m = positive integer), places the number

m in an EMPTY square. Square number m must be either left of, right

of, above, or below square number (m-1), for all m >= 2.

On move 2, the player places the 2 one square from the 1. On move 3,

the player places the 3 two squares from the 2. On move 4, the player

places the 4 three squares from the 3, etc.

Now, k is a variable that increases by 1 on each move, on occasion

being set back to 1 (see below).

A player MUST place the number m a total of k moves from the (m-1), if

the (m-1) was (k-1) moves from the (m-2), UNLESS the player cannot do

so (either because there are no empty squares k squares from the

(m-1), or k squares would be off the grid in all directions).

If a player cannot put an m, for whatever reason, exactly k squares

from square number (m-1), then the count starts over at k=1, and the

player places a number m in an empty square ONE square above, right

of, left of, or below square (m-1).

Then k then becomes 2. And on the next move, the player fills in the

square two squares from the last square, etc.

Play continues until the player cannot move any more, even if k is set

back to 1.

The player's score is the number of squares filled = the last number

written in a square.

Sample game: 6-by-6 grid.

23 * * * * 6

9 15 14 8 10 7

* 3 12 2 11 *

18 16 * 1 17 19

22 * * 21 * 20

24 4 13 * * 5

Score = 24.

Math question: What is the highest possible score for a given n-by-n

grid? Is there an interesting sequence as n = 1,2,3,4, etc... Or is

there an easy pattern to the highest possible scores?

Thanks,

Leroy Quet

then each player plays this game with the same sized grid, and players

compare final scores.)

A player starts with an n-by-n grid drawn on paper. (I suggest an n of

about 8 for beginners.)

The player starts the game by placing a "1" in any square of the grid.

The player, on move number m (m = positive integer), places the number

m in an EMPTY square. Square number m must be either left of, right

of, above, or below square number (m-1), for all m >= 2.

On move 2, the player places the 2 one square from the 1. On move 3,

the player places the 3 two squares from the 2. On move 4, the player

places the 4 three squares from the 3, etc.

Now, k is a variable that increases by 1 on each move, on occasion

being set back to 1 (see below).

A player MUST place the number m a total of k moves from the (m-1), if

the (m-1) was (k-1) moves from the (m-2), UNLESS the player cannot do

so (either because there are no empty squares k squares from the

(m-1), or k squares would be off the grid in all directions).

If a player cannot put an m, for whatever reason, exactly k squares

from square number (m-1), then the count starts over at k=1, and the

player places a number m in an empty square ONE square above, right

of, left of, or below square (m-1).

Then k then becomes 2. And on the next move, the player fills in the

square two squares from the last square, etc.

Play continues until the player cannot move any more, even if k is set

back to 1.

The player's score is the number of squares filled = the last number

written in a square.

Sample game: 6-by-6 grid.

23 * * * * 6

9 15 14 8 10 7

* 3 12 2 11 *

18 16 * 1 17 19

22 * * 21 * 20

24 4 13 * * 5

Score = 24.

Math question: What is the highest possible score for a given n-by-n

grid? Is there an interesting sequence as n = 1,2,3,4, etc... Or is

there an easy pattern to the highest possible scores?

Thanks,

Leroy Quet

### Crossing The Rings

Unlike most, but not all, of my earlier games, this game is NOT played

on an n-by-n grid.

Instead, carefully draw (preferably with a compass or a computer) n

equally-spaced concentric circles, where n is about 5 or more for

beginners (and when just 2 players are playing), a much larger n for

advanced players or if there are more players. (The concentric circles

should end up looking something like a target.)

(I suggest you photocopy the concentric circles, so as to make playing

multiple rounds much easier.)

Players (any number >=2) take turns being the "divider".

The divider, on his/her round, subdivides each ring between each pair

of consecutive circles into anywhere from m to 2^m sections, where m

is the order of a ring from the center of the "target". (The range of

allowable numbers of sections per ring can be modified, if players

chose.)

The line-segments dividing the rings should be perfectly straight, and

should meet the consecutive circles they connect at a right angle to

the tangent of the circles at the points where the circles and the

line-segments meet.

But it is up to the divider as to where along the rings the dividing

line-segments go exactly.

Next, the players who are not the divider take turns each drawing a

straight line-segment (with a straight-edge) from any intersection

where a dividing line-segment and a circle meet to any other such

intersection.

The player who is moving then fills in any sections (section = the

sections of the rings that are subdivided by the dividing line-

segments) that his/her line passes through using a colored pencil of a

color different from the colors of the other players' pencils.

The players' lines may intersect each other, and may pass through

already-filled-in sections. But a player can only fill in sections

that are not yet filled in.

Lastly, the divider then draws a line segment from any intersection to

any other, and fills in the sections his/her line passes through. (As

before, the players' lines may cross, but only sections not filled in

before may be filled in.)

Each player gets a point for each section of his/her color.

Rounds are played -- each round with the same number of concentric

circles -- so that each player plays the divider the same number of

times. Players also switch the order they play on each round, so that

each player plays in any given order the same number of times as every

other player does.

Scores are added from the rounds, and the player with the greatest

number of points wins, of course.

Thanks,

Leroy Quet

PS:

Clarification: Either, players should be banned from drawing their

lines to coincide with dividing

line segments, or if a game line and a dividing line segment coincide

then the player drawing the game line should not be able to fill in

either segment on the two sides of the dividing line-segment.

(The players can chose which game-rule they chose.)

on an n-by-n grid.

Instead, carefully draw (preferably with a compass or a computer) n

equally-spaced concentric circles, where n is about 5 or more for

beginners (and when just 2 players are playing), a much larger n for

advanced players or if there are more players. (The concentric circles

should end up looking something like a target.)

(I suggest you photocopy the concentric circles, so as to make playing

multiple rounds much easier.)

Players (any number >=2) take turns being the "divider".

The divider, on his/her round, subdivides each ring between each pair

of consecutive circles into anywhere from m to 2^m sections, where m

is the order of a ring from the center of the "target". (The range of

allowable numbers of sections per ring can be modified, if players

chose.)

The line-segments dividing the rings should be perfectly straight, and

should meet the consecutive circles they connect at a right angle to

the tangent of the circles at the points where the circles and the

line-segments meet.

But it is up to the divider as to where along the rings the dividing

line-segments go exactly.

Next, the players who are not the divider take turns each drawing a

straight line-segment (with a straight-edge) from any intersection

where a dividing line-segment and a circle meet to any other such

intersection.

The player who is moving then fills in any sections (section = the

sections of the rings that are subdivided by the dividing line-

segments) that his/her line passes through using a colored pencil of a

color different from the colors of the other players' pencils.

The players' lines may intersect each other, and may pass through

already-filled-in sections. But a player can only fill in sections

that are not yet filled in.

Lastly, the divider then draws a line segment from any intersection to

any other, and fills in the sections his/her line passes through. (As

before, the players' lines may cross, but only sections not filled in

before may be filled in.)

Each player gets a point for each section of his/her color.

Rounds are played -- each round with the same number of concentric

circles -- so that each player plays the divider the same number of

times. Players also switch the order they play on each round, so that

each player plays in any given order the same number of times as every

other player does.

Scores are added from the rounds, and the player with the greatest

number of points wins, of course.

Thanks,

Leroy Quet

PS:

Clarification: Either, players should be banned from drawing their

lines to coincide with dividing

line segments, or if a game line and a dividing line segment coincide

then the player drawing the game line should not be able to fill in

either segment on the two sides of the dividing line-segment.

(The players can chose which game-rule they chose.)

### Slice And Fill

Another game from Leroy Quet Inc.:

Start with an n-by-n grid either on graph paper or drawn carefully. I

suggest a relatively small n as far as my games usually go, about 4 to

11.

This game is for two players, each player with a pen/pencil of a

different color than his/her opponent's.

On each move, a player must first take one action (see below), and may

take a second action as well.

First, on her/his move, a player MUST fill an empty section of the

grid entirely, where the section is defined by* the lines of the grid

and/or by straight line-segments drawn by players (see below -- a

section may be a square, or it may be a polygon which is a subset of a

square).

*[By "where the section is defined by...", I mean "where the section is

BORDERED by" the lines of the grid

and/or by straight line-segments drawn by players.

There are no internal line-segments within any given "section" (at

least until lines are added later in the game, crossing the section

and subdividing the section into plural sections). ]

Unless this is the first move by a player or in case of some

other circumstances (see below), then the player must fill in a

section that is adjacent to the last section filled in by the same

player. (By "adjacent", the new section must be a previously unfilled

section that touches the player's last filled-in section along a line,

and not just touching at a point.)

Second, a player MAY on his/her move, after filling in a section,

connect any two vertices of the grid with a straight line-segment

(drawn carefully) (The vertices at the endpoints of any line-segment

must be contained within the n-by-n grid, possibly being on the grid's

border. Line segments may be diagonal, of course.) The line segment

may not cross another line segment previously drawn by either player

as part of the game. But line segments can cross grid-lines and cross

previously filled-in sections.

If a player fills in a section that is adjacent (touching along a

line, not just at a point) to a section previously filled in by the

player's opponent, a "point" is given to the player. (I put "point" in

quotes, because the goal of the game is to get as FEW points as

possible.)

Also, if a player has just filled in a section that is adjacent to a

section previously filled in by the player's opponent, then the player

may fill in any empty section of the grid on her/his next move,

starting a new path of adjoining sections.

The previous paragraph tells one situation where a player can fill any

section of the grid, not necessarily a section next to the section

previously filled in by that player. The other situation is when the

player cannot move because all adjacent sections to the player's last

filled-in section are already filled in.

Play continues until all sections are filled in.

The winner has the FEWEST number of points.

Note: Officially, the line-segments and grids act as if they were

drawn perfectly -- the line-segments pass through the appropriate grid-

vertices, given the slope of the lines.

Any strategies for this game? Any way to ensure a win for one player

or the other?

(If so, I need to fix the rules.)

Thanks,

Leroy Quet

Start with an n-by-n grid either on graph paper or drawn carefully. I

suggest a relatively small n as far as my games usually go, about 4 to

11.

This game is for two players, each player with a pen/pencil of a

different color than his/her opponent's.

On each move, a player must first take one action (see below), and may

take a second action as well.

First, on her/his move, a player MUST fill an empty section of the

grid entirely, where the section is defined by* the lines of the grid

and/or by straight line-segments drawn by players (see below -- a

section may be a square, or it may be a polygon which is a subset of a

square).

*[By "where the section is defined by...", I mean "where the section is

BORDERED by" the lines of the grid

and/or by straight line-segments drawn by players.

There are no internal line-segments within any given "section" (at

least until lines are added later in the game, crossing the section

and subdividing the section into plural sections). ]

Unless this is the first move by a player or in case of some

other circumstances (see below), then the player must fill in a

section that is adjacent to the last section filled in by the same

player. (By "adjacent", the new section must be a previously unfilled

section that touches the player's last filled-in section along a line,

and not just touching at a point.)

Second, a player MAY on his/her move, after filling in a section,

connect any two vertices of the grid with a straight line-segment

(drawn carefully) (The vertices at the endpoints of any line-segment

must be contained within the n-by-n grid, possibly being on the grid's

border. Line segments may be diagonal, of course.) The line segment

may not cross another line segment previously drawn by either player

as part of the game. But line segments can cross grid-lines and cross

previously filled-in sections.

If a player fills in a section that is adjacent (touching along a

line, not just at a point) to a section previously filled in by the

player's opponent, a "point" is given to the player. (I put "point" in

quotes, because the goal of the game is to get as FEW points as

possible.)

Also, if a player has just filled in a section that is adjacent to a

section previously filled in by the player's opponent, then the player

may fill in any empty section of the grid on her/his next move,

starting a new path of adjoining sections.

The previous paragraph tells one situation where a player can fill any

section of the grid, not necessarily a section next to the section

previously filled in by that player. The other situation is when the

player cannot move because all adjacent sections to the player's last

filled-in section are already filled in.

Play continues until all sections are filled in.

The winner has the FEWEST number of points.

Note: Officially, the line-segments and grids act as if they were

drawn perfectly -- the line-segments pass through the appropriate grid-

vertices, given the slope of the lines.

Any strategies for this game? Any way to ensure a win for one player

or the other?

(If so, I need to fix the rules.)

Thanks,

Leroy Quet

### (Convex) Hull Lot Of Fun

Start by carefully drawing a square grid that has an odd number of

lines -- and an even number of squares -- on each side.

I suggest about 9 or more lines -- 8 or more squares -- on each side,

for beginners.

I suggest bigger grids with more lines for advanced players.

The game begins with the players (two in number) taking turns drawing

dots, one dot per move, each dot at an empty (no dot there yet)

intersection of grid-lines.

Each player draws some fixed number of dots. I suggest that, if there

are m^2 intersections in the grid (where m is the number of lines on a

side of the grid), then each player draws m^2/6 (approximately) dots,

for m^2/3 total number of dots.

Player 1 moves first, followed by player 2, then player 1, then player

2,....

On the first move by player 1, the dot CANNOT go in the center

intersection of the grid.

After the dots are drawn, either player draws line-segments to connect

the dots in the boundary of the convex hull of the dots. (See below

for convex hull definition.) -- The convex hull boundary connects the

same dots that it would connect if the grid happened to have been

perfectly drawn. (So, this game might be a good teaching tool for

learning slopes.)

(By the way: If the player not drawing the line-segments thinks that

the "convex hull's boundary" being drawn is not exactly the true

convex hull boundary, because it is connecting the wrong dots, then

the "convex hull's boundary" can be challenged for accuracy.)

After the first convex hull boundary is drawn, the boundary of the

convex hull of all the dots inside (and not on the edge of) the first

convex hull boundary is drawn.

Then the next convex hull boundary within the previous boundary is

drawn...

This continues until in the center of the innermost convex hull there

are zero dots, 1 dot, or several dots in a line.

If there are an odd number of dots in the center, then player 1 wins.

If there are an even number of dots in the center, then player 2 wins.

I suggest that players play an even number of rounds, and switch who

is player 1 and who is player 2. Then who wins the most number of

rounds is the winner. (This suggestion is in case there is a bias in

this game towards either player 1 or player 2.)

What is a good strategy for this game?

Convex hull: A convex hull of a set of points in a plane is the

smallest CONVEX polygon that contains all the points. (Convex has

pretty much the intuitive meaning here: There are no concave sections

along the polygon's boundary. Also, ANY point within the polygon can

be connected with ANY other point within the polygon by a straight

line-segment without the line-segment ever leaving the polygon.)

Intuitively: (Following plagiarized from Wikipedia.) Imagine the dots

of the game being pins sticking out of a board. Imagine an elastic

band that stretches around all the pins. Releasing the band, it

contracts around the outermost pins to form the boundary of the convex

hull.

By the way, I know that many newsreaders block posts made from Google

(like this post) because of all the spam coming from Google.

The question is, has anyone seen this post? Or is Google blocked by

about everyone now?

Thanks,

Leroy Quet

PS:

I feel I should mention in words why I have the rule that the player 1

cannot put a dot at the center intersection of an odd-by-odd grid on

the first move.

Say player 1 places his first dot at the center intersection. Then

player 2 puts her dot anywhere else. Then player 1 needs only to

continue to match player 2's moves so that the finished grid has

rotational symmetry.

Then, either there will be one dot inside the center convex hull at

game's end, or there will be a line of an odd number of dots. Player 1

automatically wins.

It would be advised for player 2 to try to continually keep the grid

from getting symmetric about any point (not necessarily the very

center dot) so that player 1 cannot just reflect player 2's moves

about the point.

If we have an even number of lines on each side of the grid, then

player 2 simply matches player 1's moves so that the grid has

rotational symmetry. Player 2 wins automatically.

I wonder if there are any necessary rule changes that are needed to

prevent easy-wins for one side or the other. (By easy-win, I mean some

strategic move that would ensure a win for a given player.)

Thanks,

Leroy Quet

lines -- and an even number of squares -- on each side.

I suggest about 9 or more lines -- 8 or more squares -- on each side,

for beginners.

I suggest bigger grids with more lines for advanced players.

The game begins with the players (two in number) taking turns drawing

dots, one dot per move, each dot at an empty (no dot there yet)

intersection of grid-lines.

Each player draws some fixed number of dots. I suggest that, if there

are m^2 intersections in the grid (where m is the number of lines on a

side of the grid), then each player draws m^2/6 (approximately) dots,

for m^2/3 total number of dots.

Player 1 moves first, followed by player 2, then player 1, then player

2,....

On the first move by player 1, the dot CANNOT go in the center

intersection of the grid.

After the dots are drawn, either player draws line-segments to connect

the dots in the boundary of the convex hull of the dots. (See below

for convex hull definition.) -- The convex hull boundary connects the

same dots that it would connect if the grid happened to have been

perfectly drawn. (So, this game might be a good teaching tool for

learning slopes.)

(By the way: If the player not drawing the line-segments thinks that

the "convex hull's boundary" being drawn is not exactly the true

convex hull boundary, because it is connecting the wrong dots, then

the "convex hull's boundary" can be challenged for accuracy.)

After the first convex hull boundary is drawn, the boundary of the

convex hull of all the dots inside (and not on the edge of) the first

convex hull boundary is drawn.

Then the next convex hull boundary within the previous boundary is

drawn...

This continues until in the center of the innermost convex hull there

are zero dots, 1 dot, or several dots in a line.

If there are an odd number of dots in the center, then player 1 wins.

If there are an even number of dots in the center, then player 2 wins.

I suggest that players play an even number of rounds, and switch who

is player 1 and who is player 2. Then who wins the most number of

rounds is the winner. (This suggestion is in case there is a bias in

this game towards either player 1 or player 2.)

What is a good strategy for this game?

Convex hull: A convex hull of a set of points in a plane is the

smallest CONVEX polygon that contains all the points. (Convex has

pretty much the intuitive meaning here: There are no concave sections

along the polygon's boundary. Also, ANY point within the polygon can

be connected with ANY other point within the polygon by a straight

line-segment without the line-segment ever leaving the polygon.)

Intuitively: (Following plagiarized from Wikipedia.) Imagine the dots

of the game being pins sticking out of a board. Imagine an elastic

band that stretches around all the pins. Releasing the band, it

contracts around the outermost pins to form the boundary of the convex

hull.

By the way, I know that many newsreaders block posts made from Google

(like this post) because of all the spam coming from Google.

The question is, has anyone seen this post? Or is Google blocked by

about everyone now?

Thanks,

Leroy Quet

PS:

I feel I should mention in words why I have the rule that the player 1

cannot put a dot at the center intersection of an odd-by-odd grid on

the first move.

Say player 1 places his first dot at the center intersection. Then

player 2 puts her dot anywhere else. Then player 1 needs only to

continue to match player 2's moves so that the finished grid has

rotational symmetry.

Then, either there will be one dot inside the center convex hull at

game's end, or there will be a line of an odd number of dots. Player 1

automatically wins.

It would be advised for player 2 to try to continually keep the grid

from getting symmetric about any point (not necessarily the very

center dot) so that player 1 cannot just reflect player 2's moves

about the point.

If we have an even number of lines on each side of the grid, then

player 2 simply matches player 1's moves so that the grid has

rotational symmetry. Player 2 wins automatically.

I wonder if there are any necessary rule changes that are needed to

prevent easy-wins for one side or the other. (By easy-win, I mean some

strategic move that would ensure a win for a given player.)

Thanks,

Leroy Quet

### Another Criss-Cross Grid Game

Here is another one of my games. It isn't as fun, probably, as some of

the other games I have posted. But maybe someone will find it

enjoyable anyway.

Start with an n-by-n grid (n-by-n lines, (n-1)-by-(n-1) row/columns)

on graph-paper. (n should be at least 5, maybe in the range of 10 or

more.)

There are two players who switch roles after each round, an offensive

player and a defensive player.

Say the grid is n lines wide((n-1) columns) and n lines high ((n-1)

rows). The defensive player starts the round by writing the integers 1

through (2n) in any order to the left of the left-most vertical line

(each integer lined up with a different horizontal line of the grid)

and above the top-most horizontal line (each integer lined up with a

different vertical line of the grid).

Whether a particular integer is written either along the left side of

the grid or along the top of the grid is up to the defensive player.

Example: 6 lines -by- 6 lines:

. 2 6 11 7 1 8

9 -------------

4 | + + + + +

10| + + + + +

12| + + + + +

3 | + + + + +

5 | + + + + +

(Note: In case ascii art doesn't look correct, the pluses are the

intersections of the grid, and are supposed to each be directly below

an integer of the top row of numbers and directly to the right of the

left column of numbers.)

The two players each take turns drawing a straight line-segment (with

a straightedge) from the intersection of the grid last drawn-to by the

opposing player to an intersection determined by the order of the move

within the round.

If the move is move m -- the defensive player moves on even-numbered

moves, and the offensive player moves on odd-numbered moves -- then

the player can move to any intersection in the same row/column lined

up with the m along the grid's edges. In other words, if the value m

is written along the top of the grid, then the player on the mth move

can move to any one of the n intersections in the same COLUMN as the

value m. And if the value m is written along the left side of the

grid, then the player on the mth move can move to any one of the n

intersections in the same ROW as the value m.

Players cannot draw line-segments along already drawn line-segments.

(ie Line-segments can only intersect at most at one point.)

Line-segments cannot cross intersections that are already the

endpoints of other line-segments.

After 2n total moves (n moves for each player), the round is over.

The offensive player gets a point for every time a line-segment (drawn

by either player) crosses another segment.

If k line-segments intersect at a common point, the offensive player

gets k(k-1)/2 points for that intersection.

This is the same as counting the number of line-segments intersected

by a line-segment AS the line-segment is being drawn. (If your line

segment is crossing an intersection with (k-1) line-segments already

intersecting there, then add (k-1) to the offensive player's score. Or

wait until after the game is over to enumerate the crossings, and give

the offensive player k(k-1)/2 points for those k line-segments

intersecting at a common point.)

So, the defensive player moves so as to try to keep the lines from

crossing. The offensive player moves to try to get as many crossings

as possible.

After a round is complete the players switch who is the offensive

player and who is the defensive player. The highest score wins, after

playing an even number of rounds, of course.

What is a good strategy for this game?

Thanks,

Leroy Quet

the other games I have posted. But maybe someone will find it

enjoyable anyway.

Start with an n-by-n grid (n-by-n lines, (n-1)-by-(n-1) row/columns)

on graph-paper. (n should be at least 5, maybe in the range of 10 or

more.)

There are two players who switch roles after each round, an offensive

player and a defensive player.

Say the grid is n lines wide((n-1) columns) and n lines high ((n-1)

rows). The defensive player starts the round by writing the integers 1

through (2n) in any order to the left of the left-most vertical line

(each integer lined up with a different horizontal line of the grid)

and above the top-most horizontal line (each integer lined up with a

different vertical line of the grid).

Whether a particular integer is written either along the left side of

the grid or along the top of the grid is up to the defensive player.

Example: 6 lines -by- 6 lines:

. 2 6 11 7 1 8

9 -------------

4 | + + + + +

10| + + + + +

12| + + + + +

3 | + + + + +

5 | + + + + +

(Note: In case ascii art doesn't look correct, the pluses are the

intersections of the grid, and are supposed to each be directly below

an integer of the top row of numbers and directly to the right of the

left column of numbers.)

The two players each take turns drawing a straight line-segment (with

a straightedge) from the intersection of the grid last drawn-to by the

opposing player to an intersection determined by the order of the move

within the round.

If the move is move m -- the defensive player moves on even-numbered

moves, and the offensive player moves on odd-numbered moves -- then

the player can move to any intersection in the same row/column lined

up with the m along the grid's edges. In other words, if the value m

is written along the top of the grid, then the player on the mth move

can move to any one of the n intersections in the same COLUMN as the

value m. And if the value m is written along the left side of the

grid, then the player on the mth move can move to any one of the n

intersections in the same ROW as the value m.

Players cannot draw line-segments along already drawn line-segments.

(ie Line-segments can only intersect at most at one point.)

Line-segments cannot cross intersections that are already the

endpoints of other line-segments.

After 2n total moves (n moves for each player), the round is over.

The offensive player gets a point for every time a line-segment (drawn

by either player) crosses another segment.

If k line-segments intersect at a common point, the offensive player

gets k(k-1)/2 points for that intersection.

This is the same as counting the number of line-segments intersected

by a line-segment AS the line-segment is being drawn. (If your line

segment is crossing an intersection with (k-1) line-segments already

intersecting there, then add (k-1) to the offensive player's score. Or

wait until after the game is over to enumerate the crossings, and give

the offensive player k(k-1)/2 points for those k line-segments

intersecting at a common point.)

So, the defensive player moves so as to try to keep the lines from

crossing. The offensive player moves to try to get as many crossings

as possible.

After a round is complete the players switch who is the offensive

player and who is the defensive player. The highest score wins, after

playing an even number of rounds, of course.

What is a good strategy for this game?

Thanks,

Leroy Quet

### Quite A Stretch -- Up/Down Card Game

Here is a card game for two players. (Although it could be easily

modified for more players.)

This game seems slightly familiar to me. Are major aspects of it taken

from pre-existing card games?

All that matters in this game as far as the cards are concerned is

each card's numerical value (with Ace = 1, Jack = 11, Queen = 12, King

= 13).

Start by shuffling a deck of playing cards. (No jokers.)

Divide up the cards evenly between players. A player is not allowed to

see his/her opponent's cards until the cards are played. (The cards

belonging to each player that have not yet been played I will call a

player's "hidden hand".)

All cards once played in this game are placed face-up.

Each player starts their pile of cards by placing any one card they

choose (face-up, of course) down between the players. (So we have two

piles, one for each player.)

On each round (of two moves each) players take turns being the

offensive player and the defensive player.

The offensive player puts a card down (face-up) next to his pile. Say

that the top card (ie, the last card played by that player in the

previous round) in the offensive player's pile has a numerical value n

and the card he/she just put down next to the pile has a value m. And

say the card on top of the defensive player's pile has a value k. Then

the defensive player must, if he/she can, place a card (any card he/

she chooses from his/her hidden hand) down on top of his/her pile that

differs from k by less than or equal to the absolute value of (n-m)

AND is in the same numerical direction from k as m is from n. So, in

other words, if the card the defensive player plays has a value j,

then (m-n) has the same sign (+, -, or is zero) as (j-k).

And |m-n| is >= |j-k|. (|j-k|, the absolute value of the difference between the

card the defensive player last played and the card the defensive

player is now playing, must be any value from 0 to |m-n| {ie, from 0

to the absolute difference between the card that the offensive player

played in the previous round and the card the offensive player has

currently played}.)

(See example.)

If the defensive player cannot move, then he/she skips his defensive

move, and therefore does not remove a card from his/her hidden hand in

that round.

The players then move the cards that are next to their piles onto the

top of each pile (face-up).

Then, for the next round, the players switch who is offense and who is

defense, and the formerly defensive player then plays offense.

(So players play cards like this {if both players can move}: {player

1, player 2}, {player 2, player 1}, {player 1, player 2}, {player 2,

player 1}, etc.)

(Even if the defensive player did not play a card in the previous

round, the defensive player then immediately becomes the offensive

player of the next round and plays a card then anyway, whatever the

outcome of the previous round.)

The first player to run out of cards in his/her hidden hand wins.

Here is the beginning of a sample game:

Start: Player 1 puts down a 5, and player 2 puts down a 6.

Round 1:

Player 1 puts down a 10. (So player 2 must put down a 6,7,8,9,10, or

11.)

Player 2 puts down a 7.

Round 2:

Player 2 is now offense, and he puts down a 3. (So player 1 must put

down a 6,7,8,9, or 10. -- Since the card put down in round 1 by player

2 was a 7, and last card put down by player 1 was a 10.)

Player 1 places down an 8.

Round 3:

Player 1 then places down another 8. (So player 2 must put down a 3.)

Player 2 does not have a 3, so player 2 skips defensive move.

Round 4:

Player 2 places down a 7. (So player 1 must put down a 8,9,10,11, or

12. -- Note: Last card played by player 2 was a 3, from round 2.)

Player 1 places down a 12.

ETC.

What would be a good strategy for this game?

Thanks,

Leroy Quet

modified for more players.)

This game seems slightly familiar to me. Are major aspects of it taken

from pre-existing card games?

All that matters in this game as far as the cards are concerned is

each card's numerical value (with Ace = 1, Jack = 11, Queen = 12, King

= 13).

Start by shuffling a deck of playing cards. (No jokers.)

Divide up the cards evenly between players. A player is not allowed to

see his/her opponent's cards until the cards are played. (The cards

belonging to each player that have not yet been played I will call a

player's "hidden hand".)

All cards once played in this game are placed face-up.

Each player starts their pile of cards by placing any one card they

choose (face-up, of course) down between the players. (So we have two

piles, one for each player.)

On each round (of two moves each) players take turns being the

offensive player and the defensive player.

The offensive player puts a card down (face-up) next to his pile. Say

that the top card (ie, the last card played by that player in the

previous round) in the offensive player's pile has a numerical value n

and the card he/she just put down next to the pile has a value m. And

say the card on top of the defensive player's pile has a value k. Then

the defensive player must, if he/she can, place a card (any card he/

she chooses from his/her hidden hand) down on top of his/her pile that

differs from k by less than or equal to the absolute value of (n-m)

AND is in the same numerical direction from k as m is from n. So, in

other words, if the card the defensive player plays has a value j,

then (m-n) has the same sign (+, -, or is zero) as (j-k).

And |m-n| is >= |j-k|. (|j-k|, the absolute value of the difference between the

card the defensive player last played and the card the defensive

player is now playing, must be any value from 0 to |m-n| {ie, from 0

to the absolute difference between the card that the offensive player

played in the previous round and the card the offensive player has

currently played}.)

(See example.)

If the defensive player cannot move, then he/she skips his defensive

move, and therefore does not remove a card from his/her hidden hand in

that round.

The players then move the cards that are next to their piles onto the

top of each pile (face-up).

Then, for the next round, the players switch who is offense and who is

defense, and the formerly defensive player then plays offense.

(So players play cards like this {if both players can move}: {player

1, player 2}, {player 2, player 1}, {player 1, player 2}, {player 2,

player 1}, etc.)

(Even if the defensive player did not play a card in the previous

round, the defensive player then immediately becomes the offensive

player of the next round and plays a card then anyway, whatever the

outcome of the previous round.)

The first player to run out of cards in his/her hidden hand wins.

Here is the beginning of a sample game:

Start: Player 1 puts down a 5, and player 2 puts down a 6.

Round 1:

Player 1 puts down a 10. (So player 2 must put down a 6,7,8,9,10, or

11.)

Player 2 puts down a 7.

Round 2:

Player 2 is now offense, and he puts down a 3. (So player 1 must put

down a 6,7,8,9, or 10. -- Since the card put down in round 1 by player

2 was a 7, and last card put down by player 1 was a 10.)

Player 1 places down an 8.

Round 3:

Player 1 then places down another 8. (So player 2 must put down a 3.)

Player 2 does not have a 3, so player 2 skips defensive move.

Round 4:

Player 2 places down a 7. (So player 1 must put down a 8,9,10,11, or

12. -- Note: Last card played by player 2 was a 3, from round 2.)

Player 1 places down a 12.

ETC.

What would be a good strategy for this game?

Thanks,

Leroy Quet

### Horizontal/Vertical (Sometimes Diagonal) Grid Game

Start with an n-by-n grid drawn on paper. (I suggest an n of at least

8, if not much larger. But not too large if you don't have a long time

to play.)

The players (2 in number) take turns filling in the squares of the

grid, one square per move.

Player 1 fills in the first square anywhere in the grid.

Play continues like this:

Player 1 fills in any EMPTY square that is either left of or right of

the last square filled in by player 2.

Player 2 fills in any EMPTY square that is either above or below the

last square filled in by player 1.

Either player may fill in an empty square that is diagonally touching

the last square filled in by the other player IF BOTH players agree

that such a move is acceptable at that time.

The players alternatingly fill in a string of squares this way until

one player cannot move. (If a player can move, the player must move.)

Then the player that could not move fills in any empty square with his/

her initials or unique symbol (chosen before play). (The other player

then moves from that position as before in the next move.)

(Note: by "empty" square, I mean a square that neither has been filled

in nor has any initials or symbols in it.)

Play continues until all squares (including isolated single squares)

are filled in or have symbols/initials in them.

The player with the MOST symbols/initials is the winner.

So it is good to NOT be able to move as much as possible during play.

Note: In the first move of the game, player 1 does NOT write his/her

symbol/initials into the first square. Only after a player cannot move

does that player write their symbol/initials in a square.

Clarification: If a player is in a situation where he cannot move

either (up/down)(left/right), but he/she can move diagonally, then

that player must move diagonally IF the other player agrees to this

move. Most often the other player will agree, since this denies the

first player a point. But sometimes for strategic reasons the other

player may deny the first player the right to move diagonally.

What would be a good strategy for this game?

Thanks,

Leroy Quet

8, if not much larger. But not too large if you don't have a long time

to play.)

The players (2 in number) take turns filling in the squares of the

grid, one square per move.

Player 1 fills in the first square anywhere in the grid.

Play continues like this:

Player 1 fills in any EMPTY square that is either left of or right of

the last square filled in by player 2.

Player 2 fills in any EMPTY square that is either above or below the

last square filled in by player 1.

Either player may fill in an empty square that is diagonally touching

the last square filled in by the other player IF BOTH players agree

that such a move is acceptable at that time.

The players alternatingly fill in a string of squares this way until

one player cannot move. (If a player can move, the player must move.)

Then the player that could not move fills in any empty square with his/

her initials or unique symbol (chosen before play). (The other player

then moves from that position as before in the next move.)

(Note: by "empty" square, I mean a square that neither has been filled

in nor has any initials or symbols in it.)

Play continues until all squares (including isolated single squares)

are filled in or have symbols/initials in them.

The player with the MOST symbols/initials is the winner.

So it is good to NOT be able to move as much as possible during play.

Note: In the first move of the game, player 1 does NOT write his/her

symbol/initials into the first square. Only after a player cannot move

does that player write their symbol/initials in a square.

Clarification: If a player is in a situation where he cannot move

either (up/down)(left/right), but he/she can move diagonally, then

that player must move diagonally IF the other player agrees to this

move. Most often the other player will agree, since this denies the

first player a point. But sometimes for strategic reasons the other

player may deny the first player the right to move diagonally.

What would be a good strategy for this game?

Thanks,

Leroy Quet

## Sunday, September 21, 2008

### Move By The Numbers

This game is for two players, and it involves (as do many of my games)

an n-by-n grid drawn on paper.

First, each player makes a list of n positive integers (where n is

the number of squares along a side of the grid), each integer in the

list being <= n-1. (The order of the integers is significant.) Each

player makes his/her list without knowlege of the other player's list.

Players then write their numbers, in order, along the edges of the

grid.

Player 1 writes her/his integers above the top row of the grid,

exactly

one integer above each square. Player 2 writes his/her integers to the

left of the left-most column of the grid, exactly one integer to the

left of each square.

One of the players plays offense, the other defense. After the round

is

complete, the players switch who is offense and who is defense, using

the same numbers in the same order, but using a new unmarked grid.

To start, player 1 places a "1" in any of the grid's squares.

Players take turns placing integers in the grid. Player 1 places

1,3,5,7,..the odd positive integers, in order, in the grid. Player 2

places 2,4,6,8,... the even positive integers, in order, in the grid.

A player, on move k of the game, places the integer k in any EMPTY

grid

square. He/she places the k directly to the right, to the left, above,

or below the square with (k-1) in it. ((k-1) is in the last number put

in the grid by the other player.)

Let the integer written above the column the integer (k-1) is written

in

be c. Let the integer written to the left of the row (k-1) is written

in

be r.

The player placing the integer k in a square must place that integer

EITHER c or r squares (in any of the 4 main directions) from the

square

the (k-1) is written in.

(The side of the grid that an integer is written next to does not have

anything to do with what direction the k-square is from the (k-1)-

square.)

The players act as if the top and bottom of the grid are connected,

and

act as if the left and right sides of the grid are connected.

(Toroidal

topology.)

So if a move, say, is off the grid to the right, the players acts as

if

the row forms a circle, and continues counting from the left side,

counting to the right.

A move to the left off the grid continues from the right side on the

same

row, continuing to the left. A move upward off the grid continues from

the

bottom of the same column, continuing upward. And a move downward off

the

grid continues from the top of the same column, continuing downward.

And remember. All moves must end up on empty squares.

If a player can move, she/he must.

The round is over when the players can't move anymore.

The offensive player gets a point for every square that has a number

in it.

(Ie the offensive player gets a score equal to the largest number in

any

square of the grid.)

The players switch who is defense and who is offense, as I said above,

with

the same numbers along the edges of the grid, but with a new unmarked

grid.

Highest score wins.

Thanks,

Leroy Quet

an n-by-n grid drawn on paper.

First, each player makes a list of n positive integers (where n is

the number of squares along a side of the grid), each integer in the

list being <= n-1. (The order of the integers is significant.) Each

player makes his/her list without knowlege of the other player's list.

Players then write their numbers, in order, along the edges of the

grid.

Player 1 writes her/his integers above the top row of the grid,

exactly

one integer above each square. Player 2 writes his/her integers to the

left of the left-most column of the grid, exactly one integer to the

left of each square.

One of the players plays offense, the other defense. After the round

is

complete, the players switch who is offense and who is defense, using

the same numbers in the same order, but using a new unmarked grid.

To start, player 1 places a "1" in any of the grid's squares.

Players take turns placing integers in the grid. Player 1 places

1,3,5,7,..the odd positive integers, in order, in the grid. Player 2

places 2,4,6,8,... the even positive integers, in order, in the grid.

A player, on move k of the game, places the integer k in any EMPTY

grid

square. He/she places the k directly to the right, to the left, above,

or below the square with (k-1) in it. ((k-1) is in the last number put

in the grid by the other player.)

Let the integer written above the column the integer (k-1) is written

in

be c. Let the integer written to the left of the row (k-1) is written

in

be r.

The player placing the integer k in a square must place that integer

EITHER c or r squares (in any of the 4 main directions) from the

square

the (k-1) is written in.

(The side of the grid that an integer is written next to does not have

anything to do with what direction the k-square is from the (k-1)-

square.)

The players act as if the top and bottom of the grid are connected,

and

act as if the left and right sides of the grid are connected.

(Toroidal

topology.)

So if a move, say, is off the grid to the right, the players acts as

if

the row forms a circle, and continues counting from the left side,

counting to the right.

A move to the left off the grid continues from the right side on the

same

row, continuing to the left. A move upward off the grid continues from

the

bottom of the same column, continuing upward. And a move downward off

the

grid continues from the top of the same column, continuing downward.

And remember. All moves must end up on empty squares.

If a player can move, she/he must.

The round is over when the players can't move anymore.

The offensive player gets a point for every square that has a number

in it.

(Ie the offensive player gets a score equal to the largest number in

any

square of the grid.)

The players switch who is defense and who is offense, as I said above,

with

the same numbers along the edges of the grid, but with a new unmarked

grid.

Highest score wins.

Thanks,

Leroy Quet

### Claim The Grid's Squares

Game for 2 players.

Draw the grid large enough so that each square can contain

the game-piece (such as a coin) of each player.

Play starts with each player marking any one of the squares,

a different square for each player, with the player's symbol

or filling in a square, using colored pencils or pens, with

the player's color.

Players each then place their game-piece on their starting

square.

The game consists of "turns" where one player moves and then

the other. Who moves first in each turn alternates. So the

moves are done like this:

(player 1, player 2) (player 2, player 1) (player 1, player 2) (player

2, player 1), etc.

Before either player moves in a turn, the player who moves

second in the turn calls out how many spaces both players

will move in the turn. The number of spaces called out

is an integer from 1 to (n-1).

Players, in the appropriate order, then move their game-

pieces either up, down, left, or right the same number of

squares on the grid. (Both players must move the same number

of positions in a turn, but each can move in their own

direction.)

If a player cannot move because the number called is too big,

and the move would take the player off the grid, then that

player simply does not move on that turn.

When a player is the first to land his/her game-piece on a

square, the player then marks that square with his/her symbol,

or fills the square in with her/his color.

Players can always land on squares that are marked (by either

player), but they can only claim empty squares for their own.

Players can't move onto squares where their opponent's game-

piece is located.

Play continues until every square is filled in, or until a

predetermined number of turns have passed.

The winner of the game has the most number of squares marked

with their symbol or with their color.

Clarification: a player must move if he/she is able to move, whether she/he wants to or not.

Thanks,

Leroy Quet

Draw the grid large enough so that each square can contain

the game-piece (such as a coin) of each player.

Play starts with each player marking any one of the squares,

a different square for each player, with the player's symbol

or filling in a square, using colored pencils or pens, with

the player's color.

Players each then place their game-piece on their starting

square.

The game consists of "turns" where one player moves and then

the other. Who moves first in each turn alternates. So the

moves are done like this:

(player 1, player 2) (player 2, player 1) (player 1, player 2) (player

2, player 1), etc.

Before either player moves in a turn, the player who moves

second in the turn calls out how many spaces both players

will move in the turn. The number of spaces called out

is an integer from 1 to (n-1).

Players, in the appropriate order, then move their game-

pieces either up, down, left, or right the same number of

squares on the grid. (Both players must move the same number

of positions in a turn, but each can move in their own

direction.)

If a player cannot move because the number called is too big,

and the move would take the player off the grid, then that

player simply does not move on that turn.

When a player is the first to land his/her game-piece on a

square, the player then marks that square with his/her symbol,

or fills the square in with her/his color.

Players can always land on squares that are marked (by either

player), but they can only claim empty squares for their own.

Players can't move onto squares where their opponent's game-

piece is located.

Play continues until every square is filled in, or until a

predetermined number of turns have passed.

The winner of the game has the most number of squares marked

with their symbol or with their color.

Clarification: a player must move if he/she is able to move, whether she/he wants to or not.

Thanks,

Leroy Quet

### Another Dots-Lines Game

Here is another dots and line-segments game (like the

game, Subdivide, I posted earlier).

(I am not sure which game, this one or Subdivide, is

more fun -- probably Subdivide. This game seems to me

to be less original too.)

This game is for two players, and is played using a

pencil/pen and blank pieces of paper.

The game consists of an even number of rounds, each

player playing offense or defense the same number of

rounds.

On a round the players take turn placing a

predetermined number of dots (say, 24, 12 per player)

anywhere on one side of a piece of paper. The same

number of dots are drawn on each round.

After the dots are drawn, players take turns (defensive

player first) connecting pairs of dots with straight

line-segments, connecting one pair of dots with one

line-segment per move.

The line-segments must not cross any other lines or

coincide with any other line-segments or pass over any

other dots other than the two dots at each line-

segment's ends.

The offensive player gets n points whenever a line-

segment drawn by either player connects to a dot with

n line-segments PREVIOUSLY drawn to it. (So, if either

player draws a line-segment from a dot with n line-

segments drawn to it previously by the players, to a

dot with m line-segments drawn to it previously by the

players, then the offensive player has {m+n} added to

his/her score on that move.) There isn't an upper limit

on how many line-segments can be drawn to any dot. (And

there aren't any points awarded on a particular move

for the line-segment drawn on that move.)

It should be noted that, therefore and obviously, the

offensive player probably wants to draw line-segments

to dots with many lines already connecting to them.

While the defensive player may try to draw segments

in such a way so as to block the offensive player from

connecting to the many-line-segment dots so as to

minimize the number of points the offensive player

gets on her/his rounds. The defensive player may also

try to connect to dots with a fewer number of line-

segments already drawn to them, of course.

A round is complete as soon as every dot has at least

two line-segments connected to them.

Highest total score (after all rounds are played) wins.

Thanks,

Leroy Quet

game, Subdivide, I posted earlier).

(I am not sure which game, this one or Subdivide, is

more fun -- probably Subdivide. This game seems to me

to be less original too.)

This game is for two players, and is played using a

pencil/pen and blank pieces of paper.

The game consists of an even number of rounds, each

player playing offense or defense the same number of

rounds.

On a round the players take turn placing a

predetermined number of dots (say, 24, 12 per player)

anywhere on one side of a piece of paper. The same

number of dots are drawn on each round.

After the dots are drawn, players take turns (defensive

player first) connecting pairs of dots with straight

line-segments, connecting one pair of dots with one

line-segment per move.

The line-segments must not cross any other lines or

coincide with any other line-segments or pass over any

other dots other than the two dots at each line-

segment's ends.

The offensive player gets n points whenever a line-

segment drawn by either player connects to a dot with

n line-segments PREVIOUSLY drawn to it. (So, if either

player draws a line-segment from a dot with n line-

segments drawn to it previously by the players, to a

dot with m line-segments drawn to it previously by the

players, then the offensive player has {m+n} added to

his/her score on that move.) There isn't an upper limit

on how many line-segments can be drawn to any dot. (And

there aren't any points awarded on a particular move

for the line-segment drawn on that move.)

It should be noted that, therefore and obviously, the

offensive player probably wants to draw line-segments

to dots with many lines already connecting to them.

While the defensive player may try to draw segments

in such a way so as to block the offensive player from

connecting to the many-line-segment dots so as to

minimize the number of points the offensive player

gets on her/his rounds. The defensive player may also

try to connect to dots with a fewer number of line-

segments already drawn to them, of course.

A round is complete as soon as every dot has at least

two line-segments connected to them.

Highest total score (after all rounds are played) wins.

Thanks,

Leroy Quet

### Subdivide, The Game

The game consists of any number of rounds, but the

number is a multiple of the number of players playing

the game.

Players take turns being the offensive player (snicker

snicker) for each round.

On each round players take turns drawing a predetermined

number of dots on a blank piece of paper.

After the dots are drawn, players take turns (offensive

player first) drawing straight line segments (preferably

with a straight-edge) that connect pairs of dots, one

line segment connecting two dots per move.

The segments can connect any two dots provided that:

*The segments don't cross over any intermediate dots

and don't cross other previously drawn line-segments.

*There is a maximum of 3 line-segments connected to

every internal dot. But as for the dots making up the

perimeter of the convex-hull of the set of dots, any

number of line-segments can connect to these dots.

The round is over when the perimeter of the convex-hull

of the set of dots is completed.

The number of points the offensive player gets is equal

to the maximum number of dots making up the perimeter

of any one polygon drawn during the game.

Here by 'polygon', it is meant that the scoring polygon

has no internal line-segments subdividing it into

smaller polygons. But it may have internally drawn

line-segments, as long as these segments don't

subdivide the polygon.

(As to what to do if the winning polygon has a smaller

polygon within it, touching the permimeter of the

winning polygon at at most 1 dot, I leave that up to

the players to decide.)

Here is a sample round:

There are 2 players, player 1 (offense) drawing the

odd line-segments, player 2 drawing the even segments.

The dots happen to have been drawn as a 4-by-4 array

of dots.

The dots each are, for purpose of illustrating the game,

labeled a,b,c, or d for the row they occur in, and

1,2,3 or 4 for the column they occur in.

Segment 1: a1 - b2. Segment 2: a1 - a2. Segment 3: b1 - c2.

Segment 4: b1 - a1. Segment 5: c2 - b4. Segment 6: b3 - b4.

Segment 7: b2 - a4. Segment 8: a3 - a4. Segment 9: c2 - d3.

Segment 10: a4 - b4. Segment 11: c3 - b4. Segment 12: b4 - c4.

Segment 13: c1 - d2. Segment 14: c1 - d1. Segment 15: d3 - c4.

Segment 16: c4 - d4. Segment 17: b1 - d2. Segment 18: b1 - c1.

Segment 19: d2 - d3. Segment 20: a2 - a3. Segment 21:d3 - d4.

Segment 22: d1 - d2.

The winning polygon has 6 dots making up its perimeter:

a1, b2, a4, b4, c2, b1.

So player 1 gets 6 points this round.

The line-segment within the winning polygon, segment b3 - b4,

does not affect the score.

Thanks,

Leroy Quet

PS

I was thinking that, even though I like the convex-hull rules,

this game would be a bit more accessible if there is a slight

rule modification.

First, allow a maximum of 3 line-segments to be drawn to any

dot, no matter if the dot is part of the perimeter of the dots'

convex-hull or not.

Second, the game is over whenever no more line-segments can

be drawn, either because all dots have 3 segments drawn to them,

or because the dots with fewer segments aren't accessible to

any other dots with fewer than 3 segments.

Maybe with this rule-change the game will be more accessible to

children and non-math people.

Yet, as I said before, I like the convex-hull rules, even if

they may be too mathy.

Thanks,

Leroy Quet

number is a multiple of the number of players playing

the game.

Players take turns being the offensive player (snicker

snicker) for each round.

On each round players take turns drawing a predetermined

number of dots on a blank piece of paper.

After the dots are drawn, players take turns (offensive

player first) drawing straight line segments (preferably

with a straight-edge) that connect pairs of dots, one

line segment connecting two dots per move.

The segments can connect any two dots provided that:

*The segments don't cross over any intermediate dots

and don't cross other previously drawn line-segments.

*There is a maximum of 3 line-segments connected to

every internal dot. But as for the dots making up the

perimeter of the convex-hull of the set of dots, any

number of line-segments can connect to these dots.

The round is over when the perimeter of the convex-hull

of the set of dots is completed.

The number of points the offensive player gets is equal

to the maximum number of dots making up the perimeter

of any one polygon drawn during the game.

Here by 'polygon', it is meant that the scoring polygon

has no internal line-segments subdividing it into

smaller polygons. But it may have internally drawn

line-segments, as long as these segments don't

subdivide the polygon.

(As to what to do if the winning polygon has a smaller

polygon within it, touching the permimeter of the

winning polygon at at most 1 dot, I leave that up to

the players to decide.)

Here is a sample round:

There are 2 players, player 1 (offense) drawing the

odd line-segments, player 2 drawing the even segments.

The dots happen to have been drawn as a 4-by-4 array

of dots.

The dots each are, for purpose of illustrating the game,

labeled a,b,c, or d for the row they occur in, and

1,2,3 or 4 for the column they occur in.

Segment 1: a1 - b2. Segment 2: a1 - a2. Segment 3: b1 - c2.

Segment 4: b1 - a1. Segment 5: c2 - b4. Segment 6: b3 - b4.

Segment 7: b2 - a4. Segment 8: a3 - a4. Segment 9: c2 - d3.

Segment 10: a4 - b4. Segment 11: c3 - b4. Segment 12: b4 - c4.

Segment 13: c1 - d2. Segment 14: c1 - d1. Segment 15: d3 - c4.

Segment 16: c4 - d4. Segment 17: b1 - d2. Segment 18: b1 - c1.

Segment 19: d2 - d3. Segment 20: a2 - a3. Segment 21:d3 - d4.

Segment 22: d1 - d2.

The winning polygon has 6 dots making up its perimeter:

a1, b2, a4, b4, c2, b1.

So player 1 gets 6 points this round.

The line-segment within the winning polygon, segment b3 - b4,

does not affect the score.

Thanks,

Leroy Quet

PS

I was thinking that, even though I like the convex-hull rules,

this game would be a bit more accessible if there is a slight

rule modification.

First, allow a maximum of 3 line-segments to be drawn to any

dot, no matter if the dot is part of the perimeter of the dots'

convex-hull or not.

Second, the game is over whenever no more line-segments can

be drawn, either because all dots have 3 segments drawn to them,

or because the dots with fewer segments aren't accessible to

any other dots with fewer than 3 segments.

Maybe with this rule-change the game will be more accessible to

children and non-math people.

Yet, as I said before, I like the convex-hull rules, even if

they may be too mathy.

Thanks,

Leroy Quet

### Row/Column Solitaire Game

Start by drawing an r-by-r grid on paper.

(I suggest an r of 5 or so for beginners.)

Put a "1" in each square of the left-most column

and in each square of the top-most row of the grid.

On each move the player chooses any one of the grid's

empty squares. This square is (m,n), which is the

square in the mth column from the left side of the

grid and in the nth row from the top.

The player then sums up all the integers already

written in the mth column and nth row.

So, if the grid looks like this, where the * is

the square the player has chosen to fill in next

with an integer,

1 1 1 1

1 3

1 * 6

1 6

then the sum of the column 2's terms and row 3's

terms is 1+3+6 +1+6 = 17. Let this sum be s.

Next the player counts the number of integers in the

squares to the left of (m,n) and above (m,n)

(ie, the squares with coordinates (j,k), 1 <=j <=m,

1 <=k <=n) which are coprime to s.

If the count is c integers in the given rectangle

which are coprime to s, then the player writes c

in square (m,n).

Play continues until there is an integer in every

square of the grid.

The player's score is the number in the last square

he/she fills in.

(I suggest an r of 5 or so for beginners.)

Put a "1" in each square of the left-most column

and in each square of the top-most row of the grid.

On each move the player chooses any one of the grid's

empty squares. This square is (m,n), which is the

square in the mth column from the left side of the

grid and in the nth row from the top.

The player then sums up all the integers already

written in the mth column and nth row.

So, if the grid looks like this, where the * is

the square the player has chosen to fill in next

with an integer,

1 1 1 1

1 3

1 * 6

1 6

then the sum of the column 2's terms and row 3's

terms is 1+3+6 +1+6 = 17. Let this sum be s.

Next the player counts the number of integers in the

squares to the left of (m,n) and above (m,n)

(ie, the squares with coordinates (j,k), 1 <=j <=m,

1 <=k <=n) which are coprime to s.

If the count is c integers in the given rectangle

which are coprime to s, then the player writes c

in square (m,n).

Play continues until there is an integer in every

square of the grid.

The player's score is the number in the last square

he/she fills in.

### Two Games: Filling Grids With Numbers

Here are two games using the same idea.

Both games are for two people and use an n-by-n grid

drawn on paper. (I suggest an n of 8 to 10 for game 1,

and an n of 12 or so for game 2.)

For both games either player puts a 1 in the upper left

square of the grid.

Players then, on their move, place a number into any

empty square that is next to (in the direction of either

left, right, above, or below) at least one square

with a number already in it. The number the player

puts in the square must be 1 greater than a number

in any of the squares adjacent to the square the player

is writing the number in.

The grid is completely filled in in this way.

Game 1) Players take turns filling in the numbers.

Every time a player writes a prime into a square,

he/she gets a point. Highest score wins.

Game 2) Players make two grids of the same size,

one grid made by each player. When both grids are

complete each player gives their grid to their opponent.

Players then race to see which player can first find

a path (moving up, down, left, right) from the upper

left square of the grid each player is trying to solve

to the grid's lower right square,

moving from 1 to 2 to 3 to...(so that each square

the paths move onto is one number higher than each

path's previous square's number). (Any path that

goes from upper left square to lower right square

while following the rules is a valid path, whether

or not the found path is the path intended by the

maze maker.)

A variation: For game 2, when players

are constructing their mazes for their opponents,

they each get a point for each prime in the grid

they construct. So, in essence, these are two

games, a solitaire version of game 1, and game 2.

thanks,

Leroy Quet

Both games are for two people and use an n-by-n grid

drawn on paper. (I suggest an n of 8 to 10 for game 1,

and an n of 12 or so for game 2.)

For both games either player puts a 1 in the upper left

square of the grid.

Players then, on their move, place a number into any

empty square that is next to (in the direction of either

left, right, above, or below) at least one square

with a number already in it. The number the player

puts in the square must be 1 greater than a number

in any of the squares adjacent to the square the player

is writing the number in.

The grid is completely filled in in this way.

Game 1) Players take turns filling in the numbers.

Every time a player writes a prime into a square,

he/she gets a point. Highest score wins.

Game 2) Players make two grids of the same size,

one grid made by each player. When both grids are

complete each player gives their grid to their opponent.

Players then race to see which player can first find

a path (moving up, down, left, right) from the upper

left square of the grid each player is trying to solve

to the grid's lower right square,

moving from 1 to 2 to 3 to...(so that each square

the paths move onto is one number higher than each

path's previous square's number). (Any path that

goes from upper left square to lower right square

while following the rules is a valid path, whether

or not the found path is the path intended by the

maze maker.)

A variation: For game 2, when players

are constructing their mazes for their opponents,

they each get a point for each prime in the grid

they construct. So, in essence, these are two

games, a solitaire version of game 1, and game 2.

thanks,

Leroy Quet

### Vertex/Intersection Polygon Game

For 2 players, each with a different color of pencil/pen.

First, using a straight-edge, draw a large n-gon

(say, n = 6) on a piece of paper.

(The n-gon does not need to be regular.)

Each player draws straight line segments (with a

straight-edge and their color pencil/pen) within

the n-gon as follows:

The first player draws a line segment from any vertex

of the n-gon to any other.

Players thereafter on each move draw a line segment

between a vertex (of the n-gon) or an intersection point

(made by the crossing of two previously drawn

line-segments) and another vertex/intersection.

Players start each line-segment where the other player

ended his/her latest segment (point A).

Players end each segment at any vertex/intersection

(point B) such that:

*No previously drawn line-segment connects the points

A and B.

*Aside from the 2 previously-drawn crossing segments

(in the case of an intersection) or the two edges of

the n-gon (in the case of a vertex), point B does

not have any other line segments already drawn to it.

*The 2 previously-drawn crossing line segments (in the

case of an intersection) are of two different colors

(ie the lines are made by different players).

The last player able to move is the winner.

(If a player wrongly believes he/she cannot move,

then this player still loses.)

Notes:

Either player may draw a line-segment to/from the

vertex where the first player started their first

line-segment.

Alternative rule: Perhaps a player should be required

that any intersection they draw their segment to be of

the SAME color previously-drawn segments, instead of a

different color. Or maybe one player should draw to

same-color intersections, the other player to

different-colored intersections.

I wonder which rule is the most fun.

I suggest that after a line-segment is drawn to an

intersection/vertex, then the player doing so draws a

dot at the vertex so as to signify that the

vertex/intersection not be drawn to again.

If someone programs this game on a computer, I suggest

allowing the players to zoom in on any part of the game,

and I suggest storing the coordinates of each inersection

with a relatively higher level of precision. This is

because many intersections of some games tend to bunch up

in small areas within the polygons.

Also, you can play this game without any rule at all

requiring intersections drawn to to be of certain colors.

I just had this rule to help prevent absurdly long (and

possibly infinite) games.

Thanks,

Leroy Quet

First, using a straight-edge, draw a large n-gon

(say, n = 6) on a piece of paper.

(The n-gon does not need to be regular.)

Each player draws straight line segments (with a

straight-edge and their color pencil/pen) within

the n-gon as follows:

The first player draws a line segment from any vertex

of the n-gon to any other.

Players thereafter on each move draw a line segment

between a vertex (of the n-gon) or an intersection point

(made by the crossing of two previously drawn

line-segments) and another vertex/intersection.

Players start each line-segment where the other player

ended his/her latest segment (point A).

Players end each segment at any vertex/intersection

(point B) such that:

*No previously drawn line-segment connects the points

A and B.

*Aside from the 2 previously-drawn crossing segments

(in the case of an intersection) or the two edges of

the n-gon (in the case of a vertex), point B does

not have any other line segments already drawn to it.

*The 2 previously-drawn crossing line segments (in the

case of an intersection) are of two different colors

(ie the lines are made by different players).

The last player able to move is the winner.

(If a player wrongly believes he/she cannot move,

then this player still loses.)

Notes:

Either player may draw a line-segment to/from the

vertex where the first player started their first

line-segment.

Alternative rule: Perhaps a player should be required

that any intersection they draw their segment to be of

the SAME color previously-drawn segments, instead of a

different color. Or maybe one player should draw to

same-color intersections, the other player to

different-colored intersections.

I wonder which rule is the most fun.

I suggest that after a line-segment is drawn to an

intersection/vertex, then the player doing so draws a

dot at the vertex so as to signify that the

vertex/intersection not be drawn to again.

If someone programs this game on a computer, I suggest

allowing the players to zoom in on any part of the game,

and I suggest storing the coordinates of each inersection

with a relatively higher level of precision. This is

because many intersections of some games tend to bunch up

in small areas within the polygons.

Also, you can play this game without any rule at all

requiring intersections drawn to to be of certain colors.

I just had this rule to help prevent absurdly long (and

possibly infinite) games.

Thanks,

Leroy Quet

### Up's & Down's Game

Players (2) take turns, each playing m rounds where they are offense

and each playing m rounds where they are defense. A player's grand

score is the sum of the player's scores for rounds where they played

offense.

Both players on a round *take turns* adding one number at a time to a

list.

The number should be from 1 to n (where n is, say, 20) and must not

have

been already put in the list by either player.

After n turns the list is complete and represents a permutation of the

integers 1 through n.

Scoring (by the offensive player) is as follows.

Write below the list of integers (if the integer list was made from

left to right) a list of (n-1) U's and D's, placing each letter between

and below each pair of neighboring integers, a U for up if the

right-most element of the pair above the letter is greater than the

left-most element, or put a D for down if the right-most element of the

pair above the letter is less than the left-most element of the pair.

(See example below.)

The offensive player gets a point for each element in the *longest*

sequence of U's and D's that occurs somewhere else in the U/D list.

And it is possible that a particular U/D sub-sequence shares specific

elements with its duplicate.

For example:

If we have the 10-integer game:

1, 7, 3, 5, 2, 9, 10, 4, 6, 8,

(Player 1 added the 1, 3, 2, 10, and 6 to the list;

Player 2 added the 7, 5, 9, 4, and 8 to the list)

we would have the U/D sequence:

U, D, U, D, U, U, D, U, U.

Now the sub-sequence U, D, U, U occurs twice in the sequence

(with the pair sharing one of their U's).

So the offensive player gets 4 points.

If the finished game looked like:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ,

then we would have the U/D sequence of only U's (9 in number),

U, U, U, U, U, U, U, U, U.

And what would the score be here?

8 points, because the subsequence which equals another subsequence

is made up of 8 U's, and the two subsequences each share the middle 7

U's.

So, as I have implied, the two equal subsequences can share any

number of common elements but not share every element --

they must terminate at different locations in the U/D sequence.

thanks,

Leroy Quet

and each playing m rounds where they are defense. A player's grand

score is the sum of the player's scores for rounds where they played

offense.

Both players on a round *take turns* adding one number at a time to a

list.

The number should be from 1 to n (where n is, say, 20) and must not

have

been already put in the list by either player.

After n turns the list is complete and represents a permutation of the

integers 1 through n.

Scoring (by the offensive player) is as follows.

Write below the list of integers (if the integer list was made from

left to right) a list of (n-1) U's and D's, placing each letter between

and below each pair of neighboring integers, a U for up if the

right-most element of the pair above the letter is greater than the

left-most element, or put a D for down if the right-most element of the

pair above the letter is less than the left-most element of the pair.

(See example below.)

The offensive player gets a point for each element in the *longest*

sequence of U's and D's that occurs somewhere else in the U/D list.

And it is possible that a particular U/D sub-sequence shares specific

elements with its duplicate.

For example:

If we have the 10-integer game:

1, 7, 3, 5, 2, 9, 10, 4, 6, 8,

(Player 1 added the 1, 3, 2, 10, and 6 to the list;

Player 2 added the 7, 5, 9, 4, and 8 to the list)

we would have the U/D sequence:

U, D, U, D, U, U, D, U, U.

Now the sub-sequence U, D, U, U occurs twice in the sequence

(with the pair sharing one of their U's).

So the offensive player gets 4 points.

If the finished game looked like:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ,

then we would have the U/D sequence of only U's (9 in number),

U, U, U, U, U, U, U, U, U.

And what would the score be here?

8 points, because the subsequence which equals another subsequence

is made up of 8 U's, and the two subsequences each share the middle 7

U's.

So, as I have implied, the two equal subsequences can share any

number of common elements but not share every element --

they must terminate at different locations in the U/D sequence.

thanks,

Leroy Quet

### Jumping Farther & Farther

Here is another game that is (guess what) played on an

n-by-n grid drawn on paper. (n is finite.)

(I suggest an n of about 5 to 10 for beginners.)

I do not know if I have stolen this idea from anywhere,

as is the case with all of my games I post here.

But this game does not seem that familiar.

This game can be played either as a two-person game or solitaire.

Two-person variation:

Players take turns placing 1 through n^2 into the empty squares

of the grid, the integers placed in order and a single integer

is placed into an empty grid-square every move.

(Player 1 places the odd numbers into the grid, player 2 places

the even integers into the grid.)

Players could place any marks they choose into the grid's squares,

actually; but I suggest placing numbers so as to keep track of

who moved where when.

An integer can be placed by a player any number of squares

(as long as the integer is placed within the grid) --

and in the direction of either up, down, left, right, or diagonally

-- from the last integer the player's opponent placed into the grid.

Player 1 can place the 1 in any square. Players can "jump over" any

integers already in the grid when going from the last integer by their

opponent to their current integer.

A player gets a point every time the number of squares from the

square he just filled-in to the last square filled in by his

opponent is greater than (not equal to or less than) the number

of squares from the last square filled-in by his opponent to the

square filled-in before by the player in his/her previous move.

In other words, if s(m,m-1) is the number of squares from the

square with the (m-1) in it to the square with the m in it,

then a player (placing the integer m on his current move) gets

a point if s(m,m-1) > s(m-1,m-2).

(And so players cannot score until the second move by player 1

at the earliest.)

Play continues until every square (in the directions of

up/down/left/right/diagonally from the last square either player

has put an integer in) each already have integers in them.

If there is a natural bias towards one player getting more points

than the other, then do what I suggest for all my games with a bias:

Just play two rounds, switching who is player 1 and who is player 2,

and then each player adds up their scores from both rounds to get

their grand score.

Solitaire version:

Simply play both the even integers and the odd integers as if you

are both players, but getting only one score, trying to get the

highest possible score you can for an n-by-n grid.

The strategies for the solitaire version vary, however, from the

strategies for the competitive version of this game.

(I bet there are some interesting strategies either way.)

Thanks,

Leroy Quet

n-by-n grid drawn on paper. (n is finite.)

(I suggest an n of about 5 to 10 for beginners.)

I do not know if I have stolen this idea from anywhere,

as is the case with all of my games I post here.

But this game does not seem that familiar.

This game can be played either as a two-person game or solitaire.

Two-person variation:

Players take turns placing 1 through n^2 into the empty squares

of the grid, the integers placed in order and a single integer

is placed into an empty grid-square every move.

(Player 1 places the odd numbers into the grid, player 2 places

the even integers into the grid.)

Players could place any marks they choose into the grid's squares,

actually; but I suggest placing numbers so as to keep track of

who moved where when.

An integer can be placed by a player any number of squares

(as long as the integer is placed within the grid) --

and in the direction of either up, down, left, right, or diagonally

-- from the last integer the player's opponent placed into the grid.

Player 1 can place the 1 in any square. Players can "jump over" any

integers already in the grid when going from the last integer by their

opponent to their current integer.

A player gets a point every time the number of squares from the

square he just filled-in to the last square filled in by his

opponent is greater than (not equal to or less than) the number

of squares from the last square filled-in by his opponent to the

square filled-in before by the player in his/her previous move.

In other words, if s(m,m-1) is the number of squares from the

square with the (m-1) in it to the square with the m in it,

then a player (placing the integer m on his current move) gets

a point if s(m,m-1) > s(m-1,m-2).

(And so players cannot score until the second move by player 1

at the earliest.)

Play continues until every square (in the directions of

up/down/left/right/diagonally from the last square either player

has put an integer in) each already have integers in them.

If there is a natural bias towards one player getting more points

than the other, then do what I suggest for all my games with a bias:

Just play two rounds, switching who is player 1 and who is player 2,

and then each player adds up their scores from both rounds to get

their grand score.

Solitaire version:

Simply play both the even integers and the odd integers as if you

are both players, but getting only one score, trying to get the

highest possible score you can for an n-by-n grid.

The strategies for the solitaire version vary, however, from the

strategies for the competitive version of this game.

(I bet there are some interesting strategies either way.)

Thanks,

Leroy Quet

### Amal-Game

Here is a game which is a mix of several other games I

have posted before.

Each of the 2 players has a colored pencil/pen of a color

different than their opponent's.

And I suggest that each player also have some item (such

as a coin) to mark where they had last moved, although

such an item is not a necessity.

Start with an n-by-n grid drawn on paper.

(I suggest an n of about 6 to 10.)

Players place their game-pieces, if they have them,

in opposite corners of the grid.

And each player makes some kind of symbol with their

colored pencil in the square which is at their corner.

Player 1 moves like so:

He can either move up, left, right, or down on any move

(as long as not moving off the grid).

On his first move he moves 1 position, on his second move

he moves 2 positions, then 1, then 2. Generally, he moves

1 position (to an adjacent square) on odd-numbered moves,

and moves 2 positions (possibly skipping over a square

already marked) on even-numbered moves.

Player 2 moves like so:

She can move only one square (to an adjacent square) on any

move (and cannot move off the grid).

On her first move she can move either left, right, up, or down.

On her second move she can move in any of the 4 main diagonal

directions. Generally, she moves up, left, right, or down on

odd-numbered moves, and moves in any of the 4 main diagonal

directions on even-numbered moves.

After a player moves, he/she marks the position he/she was

just at with their colored pencil, if that square was not

already marked with the player's own color.

A player can move onto any square, even those already moved

onto before. But if she/he moves onto a square marked with

his/her own color, even if the other player's color also

marks the square, then the player loses a point.

If, however, a player moves onto a square marked ONLY with

the player's opponent's color (before the player marks the

square with his/her own color, of course), then the player

gets a point.

A round is over as soon as any player gets a predetermined

number of points.

Now, there might be a bias in which player gets more points,

based on who moves first and on the rules for moving.

So there are an even number of rounds, half where each player

is player 1 and the other player is player 2, and half the

rounds the other way.

The scores from each round are added up to get each player's

grand total, which determined who won the entire game.

(Also, if there is only one color of pencil/pen, players can

instead mark each square with their own symbol, such as a

dot or an X or their initials.)

Sample partial-game:

(View with fixed-width font.) Moves numbered:

*..*..x..xo.o..o....*. *. 7. 84 3. 1

*..o..o..*..*..o....*. 6. 5. *. *. 2

o..*..x..*..*..*....7. *. 6. *. *. *

x..x..x..*..*..*....4. 3. 5. *. *. *

*..*..*..*..*..*....*. *. *. *. *. *

x..x..*..*..*..*....1. 2. *. *. *. *

* empty square

x player 1 was here.

o player 2 was here.

x (player 1) has just moved onto a square that o (player 2) had moved

to earlier. (This square is marked with 'xo'.) So player 1 gets a

point.

thanks,

Leroy Quet

have posted before.

Each of the 2 players has a colored pencil/pen of a color

different than their opponent's.

And I suggest that each player also have some item (such

as a coin) to mark where they had last moved, although

such an item is not a necessity.

Start with an n-by-n grid drawn on paper.

(I suggest an n of about 6 to 10.)

Players place their game-pieces, if they have them,

in opposite corners of the grid.

And each player makes some kind of symbol with their

colored pencil in the square which is at their corner.

Player 1 moves like so:

He can either move up, left, right, or down on any move

(as long as not moving off the grid).

On his first move he moves 1 position, on his second move

he moves 2 positions, then 1, then 2. Generally, he moves

1 position (to an adjacent square) on odd-numbered moves,

and moves 2 positions (possibly skipping over a square

already marked) on even-numbered moves.

Player 2 moves like so:

She can move only one square (to an adjacent square) on any

move (and cannot move off the grid).

On her first move she can move either left, right, up, or down.

On her second move she can move in any of the 4 main diagonal

directions. Generally, she moves up, left, right, or down on

odd-numbered moves, and moves in any of the 4 main diagonal

directions on even-numbered moves.

After a player moves, he/she marks the position he/she was

just at with their colored pencil, if that square was not

already marked with the player's own color.

A player can move onto any square, even those already moved

onto before. But if she/he moves onto a square marked with

his/her own color, even if the other player's color also

marks the square, then the player loses a point.

If, however, a player moves onto a square marked ONLY with

the player's opponent's color (before the player marks the

square with his/her own color, of course), then the player

gets a point.

A round is over as soon as any player gets a predetermined

number of points.

Now, there might be a bias in which player gets more points,

based on who moves first and on the rules for moving.

So there are an even number of rounds, half where each player

is player 1 and the other player is player 2, and half the

rounds the other way.

The scores from each round are added up to get each player's

grand total, which determined who won the entire game.

(Also, if there is only one color of pencil/pen, players can

instead mark each square with their own symbol, such as a

dot or an X or their initials.)

Sample partial-game:

(View with fixed-width font.) Moves numbered:

*..*..x..xo.o..o....*. *. 7. 84 3. 1

*..o..o..*..*..o....*. 6. 5. *. *. 2

o..*..x..*..*..*....7. *. 6. *. *. *

x..x..x..*..*..*....4. 3. 5. *. *. *

*..*..*..*..*..*....*. *. *. *. *. *

x..x..*..*..*..*....1. 2. *. *. *. *

* empty square

x player 1 was here.

o player 2 was here.

x (player 1) has just moved onto a square that o (player 2) had moved

to earlier. (This square is marked with 'xo'.) So player 1 gets a

point.

thanks,

Leroy Quet

## Saturday, September 20, 2008

### Enumeration

Here is yet another game of mine played using an n-by-n

grid drawn on paper and involving coprimality.

(I suggest that n be at least 6, maybe much higher.)

In the top row of the grid and in the grid's left-most

column write the integers 1,2,3,... through n, one integer

per grid-square. (1 is in the upper-left corner square.)

Now during play, players (2 in number) take turns writing

integers into the n^2 squares of the grid.

Only one integer is in each square, and a player can only

put an integer in an empty square which is both

immediately below and immediately right of squares which

already have integers in them.

The number that a player writes in an empty square

(square {j,k}) is either:

The number of integers

above and in the same COLUMN (j) as the square being

filled in

that are coprime to the number of the ROW (k) of the

square being filled in;

or

The number of integers

left of and in the same ROW (k) as the square being

filled in

that are coprime to the number of the COLUMN (j) of

the square being filled in.

(The original integers written in the top row and

left-most column indicate the j and k coordinates of

the columns and rows.)

(And the GCD(m,0) will be considered to be m,

for our purposes, so no number but 1 is coprime to

0.)

I have made up two variations of how to score in the game.

(I invite readers of this post to reply with their

own rules for scoring, if they desire.)

Variation 1:

The goal is for one player to get the highest value

in the lower right square (which is the last square

filled in) they can, while the other player tries

to minimize this value.

After playing one round, the players switch order

of play and switch who wants a high score in the

lower right square and who wants the low score.

(The players play the same sized grids for each round.)

The winner is the player who gets the highest score

for her/his round.

Also a solitaire version can be played where a player

simply tries to get the highest score possible.

Variation 2:

Player 1 gets a point for every odd integer in the

grid at game's end. Player 2 gets a point for every

even integer in the grid at game's end.

With the solitaire version of variation 2, a player

simply tries to maximize the number of odd (or even)

integers in the grid at game's end.

Sample partial game:

1 2 3 4 5 6

2 0 1 1 3 2

3 1 1 2 1 *

4 1 3 2

5 3

6

The next player to place in integer in a square,

if he/she will put the integer at position *,

can write a 1 (because 3 {row number} is coprime to

the 2 above the *, but not to 6)

or can write a 3 (because 6 {column number} is coprime

to the three 1's to the left of *, but not to the 2 or 3).

Thanks,

Leroy Quet

grid drawn on paper and involving coprimality.

(I suggest that n be at least 6, maybe much higher.)

In the top row of the grid and in the grid's left-most

column write the integers 1,2,3,... through n, one integer

per grid-square. (1 is in the upper-left corner square.)

Now during play, players (2 in number) take turns writing

integers into the n^2 squares of the grid.

Only one integer is in each square, and a player can only

put an integer in an empty square which is both

immediately below and immediately right of squares which

already have integers in them.

The number that a player writes in an empty square

(square {j,k}) is either:

The number of integers

above and in the same COLUMN (j) as the square being

filled in

that are coprime to the number of the ROW (k) of the

square being filled in;

or

The number of integers

left of and in the same ROW (k) as the square being

filled in

that are coprime to the number of the COLUMN (j) of

the square being filled in.

(The original integers written in the top row and

left-most column indicate the j and k coordinates of

the columns and rows.)

(And the GCD(m,0) will be considered to be m,

for our purposes, so no number but 1 is coprime to

0.)

I have made up two variations of how to score in the game.

(I invite readers of this post to reply with their

own rules for scoring, if they desire.)

Variation 1:

The goal is for one player to get the highest value

in the lower right square (which is the last square

filled in) they can, while the other player tries

to minimize this value.

After playing one round, the players switch order

of play and switch who wants a high score in the

lower right square and who wants the low score.

(The players play the same sized grids for each round.)

The winner is the player who gets the highest score

for her/his round.

Also a solitaire version can be played where a player

simply tries to get the highest score possible.

Variation 2:

Player 1 gets a point for every odd integer in the

grid at game's end. Player 2 gets a point for every

even integer in the grid at game's end.

With the solitaire version of variation 2, a player

simply tries to maximize the number of odd (or even)

integers in the grid at game's end.

Sample partial game:

1 2 3 4 5 6

2 0 1 1 3 2

3 1 1 2 1 *

4 1 3 2

5 3

6

The next player to place in integer in a square,

if he/she will put the integer at position *,

can write a 1 (because 3 {row number} is coprime to

the 2 above the *, but not to 6)

or can write a 3 (because 6 {column number} is coprime

to the three 1's to the left of *, but not to the 2 or 3).

Thanks,

Leroy Quet

### Guessing # Of Uncoprime Walls

Another one of my less-than-famous math games played using a square

grid.

(Maybe this game can, like several other of my games,

help teach students about coprimality.)

Game is for 2 or more players.

Start with an n-by-n grid drawn lightly on paper.

Players take turns placing the integers 1 through n^2 in the

grid's squares, one integer per square. (So, if there are 2 players,

player 1 places the odd integers in the grid, player 2 places

the even integers in the grid.)

Each integer is written in any blank square that is immediately

adjacent (in the directions of either up, down, left of, or

right of) to a square with an integer already in it.

(Player 1 may put the 1 in any of the grid's squares.)

Before the players start to fill in the grid, however, they

each secretly guess how many "walls" will, at game's end,

separate adjacent integers which are not coprime, and then

they write this guess down (not showing their guess to any

other player until the game is over).

(A wall is a vertical or horizontal line-segment of the grid,

of one grid-square in length, separating two adjacent grid-squares.)

After the grid is filled, every wall within the grid separating

an adjacent pair of non-coprime integers is darkened in.

The player whose guess is closest to the actual number of walls

separating non-coprime adjacent integers is the winner.

(Ties are possible.)

(Players may use, as part of their strategy, especially with larger

sized grids, bluffing:

For instance, making it seem they picked a bigger number for the

number of uncoprime pairs than they actually picked, for example,

by at the game's beginning placing many integers next to other

integers they are not coprime to.)

Play m (where m is the number of players) rounds,

then add up the differences between any player's guesses and the actual

number of walls in each round.

The winner is the player whose sum of differences is the *lowest*.

thanks,

Leroy Quet

grid.

(Maybe this game can, like several other of my games,

help teach students about coprimality.)

Game is for 2 or more players.

Start with an n-by-n grid drawn lightly on paper.

Players take turns placing the integers 1 through n^2 in the

grid's squares, one integer per square. (So, if there are 2 players,

player 1 places the odd integers in the grid, player 2 places

the even integers in the grid.)

Each integer is written in any blank square that is immediately

adjacent (in the directions of either up, down, left of, or

right of) to a square with an integer already in it.

(Player 1 may put the 1 in any of the grid's squares.)

Before the players start to fill in the grid, however, they

each secretly guess how many "walls" will, at game's end,

separate adjacent integers which are not coprime, and then

they write this guess down (not showing their guess to any

other player until the game is over).

(A wall is a vertical or horizontal line-segment of the grid,

of one grid-square in length, separating two adjacent grid-squares.)

After the grid is filled, every wall within the grid separating

an adjacent pair of non-coprime integers is darkened in.

The player whose guess is closest to the actual number of walls

separating non-coprime adjacent integers is the winner.

(Ties are possible.)

(Players may use, as part of their strategy, especially with larger

sized grids, bluffing:

For instance, making it seem they picked a bigger number for the

number of uncoprime pairs than they actually picked, for example,

by at the game's beginning placing many integers next to other

integers they are not coprime to.)

Play m (where m is the number of players) rounds,

then add up the differences between any player's guesses and the actual

number of walls in each round.

The winner is the player whose sum of differences is the *lowest*.

thanks,

Leroy Quet

### + - * / Game

Here is a game for 2 or more players.

Start with a deck of cards with n cards labeled 1 to n.

On each move the player whose move it is draws one card.

Do not replace card.

Write down first number drawn.

On each move (after the first) a player draws a card to get

the value k.

He/she may either add k to the written down number,

subtract k from the written down number (as long as the

difference is not negative), multiply k with the written down

number, or divide the written down number by k

(as long as k divides evenly into the written down number).

A new written down number, in this way, is obtained.

A player gets a point every time the written down number

they generate is a prime number.

Play continues for a total of n moves (when the deck

of cards runs out).

Here is an example game (for 2 players). (n=10)

First move: Player 1 draws 7. (Gets a point)

2nd move: Player 2 draws a 4. 7 + 4 = 11. (Gets a point)

3rd move:Player 1 draws a 1. 11/1 = 11. (Gets a point)

4th move: Player 2 draws a 9. 11-9 = 2. (Gets a point)

5th move: Player 1 draws a 2. 2/2=1. (No point)

6th move: Player 2 draws a 3. 1*3 = 3. (Gets a point)

7th move: Player 1 draws a 10. 3+10=13. (Gets a point)

8th move: Player 2 draws a 5. 13*5= 65. (No point)

9th move: Player 1 draws a 8. 65+8=73. (Gets a point)

10th move: Player 2 draws a 6. 73-6=67. (Gets a point)

So player 1 gets 4 points, player 2 gets 4 points too, a tie.

(What do you expect when I play myself?)

In practice, it would probably be better to have a higher n.

Also, if you prefer a pure-strategy game without luck, you can,

instead of using cards, have the integers "drawn" be from a

predetermined sequence (such as 1,2,3,4,....,n or such as

the primes taken in order.)

thanks,

Leroy Quet

Start with a deck of cards with n cards labeled 1 to n.

On each move the player whose move it is draws one card.

Do not replace card.

Write down first number drawn.

On each move (after the first) a player draws a card to get

the value k.

He/she may either add k to the written down number,

subtract k from the written down number (as long as the

difference is not negative), multiply k with the written down

number, or divide the written down number by k

(as long as k divides evenly into the written down number).

A new written down number, in this way, is obtained.

A player gets a point every time the written down number

they generate is a prime number.

Play continues for a total of n moves (when the deck

of cards runs out).

Here is an example game (for 2 players). (n=10)

First move: Player 1 draws 7. (Gets a point)

2nd move: Player 2 draws a 4. 7 + 4 = 11. (Gets a point)

3rd move:Player 1 draws a 1. 11/1 = 11. (Gets a point)

4th move: Player 2 draws a 9. 11-9 = 2. (Gets a point)

5th move: Player 1 draws a 2. 2/2=1. (No point)

6th move: Player 2 draws a 3. 1*3 = 3. (Gets a point)

7th move: Player 1 draws a 10. 3+10=13. (Gets a point)

8th move: Player 2 draws a 5. 13*5= 65. (No point)

9th move: Player 1 draws a 8. 65+8=73. (Gets a point)

10th move: Player 2 draws a 6. 73-6=67. (Gets a point)

So player 1 gets 4 points, player 2 gets 4 points too, a tie.

(What do you expect when I play myself?)

In practice, it would probably be better to have a higher n.

Also, if you prefer a pure-strategy game without luck, you can,

instead of using cards, have the integers "drawn" be from a

predetermined sequence (such as 1,2,3,4,....,n or such as

the primes taken in order.)

thanks,

Leroy Quet

### Grid Subdividing Solitaire Game

Here is a solitaire game.

Start with an (n-1)-by-(n-1) grid (of n-by-n lines) lightly drawn

on paper.

You also need 2 sets of cards, each deck of n cards labelled 1 through

n.

The player draws on each move one card from each deck.

Don't replace the card. Let the value of the cards be A and B.

>From the last position (at a vertex) drawn to on the grid, the player

either draws (along the grid's lines) a vertical line then a horizontal

line, or draws a horizontal then vertical line, drawing to either

vertex (A,B) or vertex (B,A). The line changes directions at most once

(taking a right angle turn). The line may coincide with lines which

were drawn earlier.

(Vertex (1,1) is the upper left corner of the grid. Vertex (n,n) is

the lower right corner of the grid.)

After n moves (when all the cards have been drawn from both decks)

the player gets a score equal to the *product* of the numbers of

squares

in each section the grid has been subdivided into by the player's

lines.

(Imagine the lines drawn during the game as a knife cutting the

grid-cake

into pieces. Multiply the sizes of each piece to get score.)

An example will hopefully make this clear:

(Ignore the ~'s.)

Cards on each move (for n = 9): (A,B) =

(4,2) (1,3) (6,5) (2,4) (8,6) (5,1) (9,8) (3,7) (7,9)

1.~~.~~*= = =+~~.~~.~~.~~.

~~~~~~~~~~!~~!~~~~~~~~~~~

2.~~.~~.~~S~~!~~.~~.~~.~~.

~~~~~~~~~~~~~!~~~~~~~~~~~

3.~~.~~.~~.~~!~~.~~.~~.~~.

~~~~~~~~~~~~~!~~~~~~~~~~~

4.~~* == == =+==+~~.~~.~~.

~~~~!~~~~~~~~!~~!~~~~~~~~~~~

5*= +== == ==+==+~~.~~.~~.

~!~~!~~~~~~~~!~~!~~~~~~~~~

6!~~+== == ==*~~!~~.~~.~~.

~!~~~~~~~~~~~~~~!~~~~~~~~~

7!~~.~~*== == ==+ == =+~~.

~!~~~~~!~~~~~~~~!~~~~~!~~~

8!~~.~~!~~.~~.~~*~~.~~!~~.

~!~~~~~!~~~~~~~~~~~~~~!~~~

9+== ==+== == == ==F =*~~.

.1 .2 .3 .4 .5 .6 .7 .8 .9

Start at S. Finish at F.

*'s are other vertexes that are either (A,B) or (B,A).

Score: 13 * 23 * 3 * 1 * 3 * 11 * 10 = 296010.

thanks,

Leroy Quet

Start with an (n-1)-by-(n-1) grid (of n-by-n lines) lightly drawn

on paper.

You also need 2 sets of cards, each deck of n cards labelled 1 through

n.

The player draws on each move one card from each deck.

Don't replace the card. Let the value of the cards be A and B.

>From the last position (at a vertex) drawn to on the grid, the player

either draws (along the grid's lines) a vertical line then a horizontal

line, or draws a horizontal then vertical line, drawing to either

vertex (A,B) or vertex (B,A). The line changes directions at most once

(taking a right angle turn). The line may coincide with lines which

were drawn earlier.

(Vertex (1,1) is the upper left corner of the grid. Vertex (n,n) is

the lower right corner of the grid.)

After n moves (when all the cards have been drawn from both decks)

the player gets a score equal to the *product* of the numbers of

squares

in each section the grid has been subdivided into by the player's

lines.

(Imagine the lines drawn during the game as a knife cutting the

grid-cake

into pieces. Multiply the sizes of each piece to get score.)

An example will hopefully make this clear:

(Ignore the ~'s.)

Cards on each move (for n = 9): (A,B) =

(4,2) (1,3) (6,5) (2,4) (8,6) (5,1) (9,8) (3,7) (7,9)

1.~~.~~*= = =+~~.~~.~~.~~.

~~~~~~~~~~!~~!~~~~~~~~~~~

2.~~.~~.~~S~~!~~.~~.~~.~~.

~~~~~~~~~~~~~!~~~~~~~~~~~

3.~~.~~.~~.~~!~~.~~.~~.~~.

~~~~~~~~~~~~~!~~~~~~~~~~~

4.~~* == == =+==+~~.~~.~~.

~~~~!~~~~~~~~!~~!~~~~~~~~~~~

5*= +== == ==+==+~~.~~.~~.

~!~~!~~~~~~~~!~~!~~~~~~~~~

6!~~+== == ==*~~!~~.~~.~~.

~!~~~~~~~~~~~~~~!~~~~~~~~~

7!~~.~~*== == ==+ == =+~~.

~!~~~~~!~~~~~~~~!~~~~~!~~~

8!~~.~~!~~.~~.~~*~~.~~!~~.

~!~~~~~!~~~~~~~~~~~~~~!~~~

9+== ==+== == == ==F =*~~.

.1 .2 .3 .4 .5 .6 .7 .8 .9

Start at S. Finish at F.

*'s are other vertexes that are either (A,B) or (B,A).

Score: 13 * 23 * 3 * 1 * 3 * 11 * 10 = 296010.

thanks,

Leroy Quet

### Clockwise-Counterclockwise Game

Here is another game I came up with (which, as is usually

the case, may or may not be original).

For 2 players.

A circle is drawn on paper and the outside of the circle

is marked into 26 equally sized spaces (like a clock

with 26 hours).

Each space is labeled with a different letter, A to Z.

The game starts at position A.

On each move, a player starts at the position last visited

by his opponent, player 1 moves clockwise around the circle,

player 2 moves counter-clockwise.

When a player finishes moving, he/she places a mark at the

space she/he lands upon.

Players move 1 space on their own first move, 2 spaces on

their own second move, 3 spaces on their own third move, etc,

until on their last and tenth moves they move 10 spaces each.

Players on their kth move can do one of two things:

*Move k spaces (in their direction), counting already landed

on spaces. The player should not land on an already moved onto space.

*Move k EMPTY spaces (spaces with no marks) in their direction.

Example:

If part of the circle looks like this (after being flatted out):

-X-|---!-X-!-X-!---!---!---!

.A . B . C . D . E . F . G

and the last player moved to A, and it is the 4th move,

the player whose turn it is (if moving to the right)

can either move to E (4 spaces, whether marked or not)

or to G (4 unmarked spaces).

During an entire kth move, a player either always counts the moved

on spaces or always does not count the moved on spaces as part of

the kth move. In other words, there is no mixing during a move of

counting methods. But player can use different counting methods on

different moves.

Now, at the game's beginning, each player secretly picks a space

they believe the final move of the game (the 2nd player's 10th move)

will land upon or near, and they secretly write the letter of this

space down.

The player whose guess is closest to the actual last position of the

game wins the game.

What would be a good strategy for this game?

(It might seem at first that the last player to move has a lot of

power over where the game ends up. But the last player will be

forced probably to count only empty spaces, since they otherwise

would likely land on an already landed on space, which is forbidden.)

thanks,

Leroy Quet

the case, may or may not be original).

For 2 players.

A circle is drawn on paper and the outside of the circle

is marked into 26 equally sized spaces (like a clock

with 26 hours).

Each space is labeled with a different letter, A to Z.

The game starts at position A.

On each move, a player starts at the position last visited

by his opponent, player 1 moves clockwise around the circle,

player 2 moves counter-clockwise.

When a player finishes moving, he/she places a mark at the

space she/he lands upon.

Players move 1 space on their own first move, 2 spaces on

their own second move, 3 spaces on their own third move, etc,

until on their last and tenth moves they move 10 spaces each.

Players on their kth move can do one of two things:

*Move k spaces (in their direction), counting already landed

on spaces. The player should not land on an already moved onto space.

*Move k EMPTY spaces (spaces with no marks) in their direction.

Example:

If part of the circle looks like this (after being flatted out):

-X-|---!-X-!-X-!---!---!---!

.A . B . C . D . E . F . G

and the last player moved to A, and it is the 4th move,

the player whose turn it is (if moving to the right)

can either move to E (4 spaces, whether marked or not)

or to G (4 unmarked spaces).

During an entire kth move, a player either always counts the moved

on spaces or always does not count the moved on spaces as part of

the kth move. In other words, there is no mixing during a move of

counting methods. But player can use different counting methods on

different moves.

Now, at the game's beginning, each player secretly picks a space

they believe the final move of the game (the 2nd player's 10th move)

will land upon or near, and they secretly write the letter of this

space down.

The player whose guess is closest to the actual last position of the

game wins the game.

What would be a good strategy for this game?

(It might seem at first that the last player to move has a lot of

power over where the game ends up. But the last player will be

forced probably to count only empty spaces, since they otherwise

would likely land on an already landed on space, which is forbidden.)

thanks,

Leroy Quet

### 3 Colors Filling Grid

For 2 players. Played with 3 colored pencils, each of a different

color.

Start with either a 5-by-5 or 6-by-6 grid drawn on paper.

(Actually, this is just a suggestion. You can play with any sized

n-by-n grid.)

Players take turns (player 1, player 2, then player 1, player 2,...)

filling on each move an empty square of the grid with the color which

is up.

The colors follow the same order over and over (say - blue, red,

yellow,

then blue, red, yellow,...),

For example:

Player 1: blue

Player 2: red

Player 1: yellow

Player 2 blue

Player 1: red

Player 2: yellow

The first time a square is filled with a particular color,

any empty square can be filled in.

But after each color has been used to fill a square

(ie. on the 4th or later move), a player can, with a

particular color, only fill a square immediately adjacent

to (in the directions of above, below, right, or left) a

square with the same particular color.

(So, blue, say, can only be used to fill a square

{not yet filled in} which is next to a square with blue in it.)

The winner of this game is the last player who can move.

What is a good strategy for playing this game?

And this game seems familiar. From where have I stolen the idea for it?

thanks,

Leroy Quet

color.

Start with either a 5-by-5 or 6-by-6 grid drawn on paper.

(Actually, this is just a suggestion. You can play with any sized

n-by-n grid.)

Players take turns (player 1, player 2, then player 1, player 2,...)

filling on each move an empty square of the grid with the color which

is up.

The colors follow the same order over and over (say - blue, red,

yellow,

then blue, red, yellow,...),

For example:

Player 1: blue

Player 2: red

Player 1: yellow

Player 2 blue

Player 1: red

Player 2: yellow

The first time a square is filled with a particular color,

any empty square can be filled in.

But after each color has been used to fill a square

(ie. on the 4th or later move), a player can, with a

particular color, only fill a square immediately adjacent

to (in the directions of above, below, right, or left) a

square with the same particular color.

(So, blue, say, can only be used to fill a square

{not yet filled in} which is next to a square with blue in it.)

The winner of this game is the last player who can move.

What is a good strategy for playing this game?

And this game seems familiar. From where have I stolen the idea for it?

thanks,

Leroy Quet

### Unique Sum Game

Here is another game. It it inspired by my recent game, "Prime Sums

Game",

but with a little less math (primes not an issue here).

For 2 players.

(Actually, the game can be adapted for any number of players.)

Start with an n-by-n grid drawn on paper. (like in so many of my games)

Players alternate writing the integers IN ORDER 1, 2, 3,..., n^2

into the empty squares of the grid.

So, for a 2-player game, one player writes odd integers,

the other writes even integers.

Players try to avoid having the sum {of the integer they are

writing} and {of any of the adjacent squares' integers

(in the direction of left, right, up, or down)} from equaling

the sum of any pair of already-written adacent integers.

So, for example,

if we have the grid:

. 7 6 4

1 . 3 .

2 . . 5

we would not want to put the 8 next to the 1 because there is

already the adjacent pair 3 and 6 in the grid, and 1+8 = 3+6.

So, if a player, player A, writes down an integer which, when

summed with an adjacent integer, gets a sum which exists somewhere

else in the grid, it is up to player B to notice this.

If player B notices that player A is making a sum which already

exists, then player B wins the game.

If player B does not notice, then play continues as if the

duplicate sum was not made.

And if the grid is filled in completely without anyone noticing a

duplicate sum, the game is a tie.

What would be a good strategy for this game?

thanks,

Leroy Quet

Game",

but with a little less math (primes not an issue here).

For 2 players.

(Actually, the game can be adapted for any number of players.)

Start with an n-by-n grid drawn on paper. (like in so many of my games)

Players alternate writing the integers IN ORDER 1, 2, 3,..., n^2

into the empty squares of the grid.

So, for a 2-player game, one player writes odd integers,

the other writes even integers.

Players try to avoid having the sum {of the integer they are

writing} and {of any of the adjacent squares' integers

(in the direction of left, right, up, or down)} from equaling

the sum of any pair of already-written adacent integers.

So, for example,

if we have the grid:

. 7 6 4

1 . 3 .

2 . . 5

we would not want to put the 8 next to the 1 because there is

already the adjacent pair 3 and 6 in the grid, and 1+8 = 3+6.

So, if a player, player A, writes down an integer which, when

summed with an adjacent integer, gets a sum which exists somewhere

else in the grid, it is up to player B to notice this.

If player B notices that player A is making a sum which already

exists, then player B wins the game.

If player B does not notice, then play continues as if the

duplicate sum was not made.

And if the grid is filled in completely without anyone noticing a

duplicate sum, the game is a tie.

What would be a good strategy for this game?

thanks,

Leroy Quet

### Prime Sum Game

Here is a simple mathematical game.

(For any number of players.)

You start with an n-by-n grid drawn on paper.

(I suggest an n of about 5 for a 2-player game.)

Players take turns writing, in order, the integers 1, 2, 3,...,n^2

into any of the empty squares of the grid.

(So, for a 2-player game, one player writes the odd integers, the

other player writes the even integers.)

Play continues until every square of the grid has an integer in it.

A player gets a point every time {the integer he/she is writing in a

square} plus {a lower integer in an immediately adjacent

(in the directions of left, right, above, or below) square} is a prime.

For example, if we have the grid, in-part, below:

3 5 2

1 . 6

4

and a player places an 12 into the grid like this:

3 5 2

1 12 6

4

the player gets 2 points for this move, since 12+1 and 12+5 are primes.

And, obviously, the player with the highest score at the end of the

game

is the winner.

What is a good strategy for this game, especially for a 2-person game?

thanks,

Leroy Quet

(For any number of players.)

You start with an n-by-n grid drawn on paper.

(I suggest an n of about 5 for a 2-player game.)

Players take turns writing, in order, the integers 1, 2, 3,...,n^2

into any of the empty squares of the grid.

(So, for a 2-player game, one player writes the odd integers, the

other player writes the even integers.)

Play continues until every square of the grid has an integer in it.

A player gets a point every time {the integer he/she is writing in a

square} plus {a lower integer in an immediately adjacent

(in the directions of left, right, above, or below) square} is a prime.

For example, if we have the grid, in-part, below:

3 5 2

1 . 6

4

and a player places an 12 into the grid like this:

3 5 2

1 12 6

4

the player gets 2 points for this move, since 12+1 and 12+5 are primes.

And, obviously, the player with the highest score at the end of the

game

is the winner.

What is a good strategy for this game, especially for a 2-person game?

thanks,

Leroy Quet

### Prime-Power Divisors Game

Here is a number game

(which might help some students of basic math learn some of the

prime-power divisors of integers).

Players (2) take turns writing down integers in a list

(called the "play list") as follows:

Player 1 starts by writing down any integer >= 2.

A player on his/her move first makes any list (called

the "divisors list") of prime powers (p^k, k = positive integer),

where the primes being raised to the exponents are not

necessarily distinct,

and where the product of the prime-powers equals the previous

integer in the play list (equals the integer most recently

written down in the list by the player's opponent).

For example, if the last integer in the play list was 24, then

the player currently moving can make the divisors list

2,2,2,3 or 2,4,3 or 8,3.

Next, the player currently moving either adds one or subtracts

one from every prime-power in their divisors list

(the player may add 1 to some prime-powers while subtracting

1 from other prime-powers).

Finally the player multiplies these integers together.

For example, if the player had made the divisors list

(from the example above) 2,4,3, he/she may now have the product

1*5*2 = 10.

This product must not be 1 and it must not have occurred earlier

in the play list.

So, in the example, 10 must not have occurred earlier in the

play list.

Players continue taking turns until one player cannot move

(because all possible products already exist in the play list).

The last player to move is the game's winner.

Short sample game:

Play list:

4, 5, 6, 2, 3

Player 1 wins.

What is a good strategy for this game?

thanks,

Leroy Quet

(which might help some students of basic math learn some of the

prime-power divisors of integers).

Players (2) take turns writing down integers in a list

(called the "play list") as follows:

Player 1 starts by writing down any integer >= 2.

A player on his/her move first makes any list (called

the "divisors list") of prime powers (p^k, k = positive integer),

where the primes being raised to the exponents are not

necessarily distinct,

and where the product of the prime-powers equals the previous

integer in the play list (equals the integer most recently

written down in the list by the player's opponent).

For example, if the last integer in the play list was 24, then

the player currently moving can make the divisors list

2,2,2,3 or 2,4,3 or 8,3.

Next, the player currently moving either adds one or subtracts

one from every prime-power in their divisors list

(the player may add 1 to some prime-powers while subtracting

1 from other prime-powers).

Finally the player multiplies these integers together.

For example, if the player had made the divisors list

(from the example above) 2,4,3, he/she may now have the product

1*5*2 = 10.

This product must not be 1 and it must not have occurred earlier

in the play list.

So, in the example, 10 must not have occurred earlier in the

play list.

Players continue taking turns until one player cannot move

(because all possible products already exist in the play list).

The last player to move is the game's winner.

Short sample game:

Play list:

4, 5, 6, 2, 3

Player 1 wins.

What is a good strategy for this game?

thanks,

Leroy Quet

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