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
Tuesday, October 28, 2008
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.
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