## 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

### 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

## 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

### 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

### 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

### 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

### 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

### 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

### 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
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.)

### 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

### (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.
(like this post) because of all the spam coming from Google.
The question is, has anyone seen this post? Or is Google blocked by
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
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.
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

### 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

## 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

### 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

### 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

### 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

### 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.

### 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

### 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

### 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

### 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

### 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.
(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
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
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.
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,
Game is for 2 or more players.
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

### Grid Subdividing Solitaire Game

Here is a solitaire game.
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

### 3 Colors Filling Grid

For 2 players. Played with 3 colored pencils, each of a different
color.
(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
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
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.)
(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.
(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

### Dream Math Game

I saw this game in a dream.
I post it here only as a prototype for others to improve upon.
(Half the fun of playing this game is not playing it, but is coming up
with rules for it.)
Game is for 2 players and is played using an n-by-n grid drawn on
paper.
Players take turns writing, in order, 1 through n^2 into the grid's
empty squares,
one integer per square,until the grid is filled.
(So, player 1 writes in the odd integers, player 2 writes in the even
integers.)
Player 1 gets the sum of the greatest common divisors (GCDs) of the
n*(n-1)
_horizontally_ adjacent pairs of integers in the grid.
Player 2 gets the sum of the greatest common divisors (GCDs) of the
n*(n-1)
_vertically_ adjacent pairs of integers in the grid.
Example:
If we have the 3-by-3 grid
1 2 4
3 9 8
6 5 7
Player 1 gets
GCD(1,2)+GCD(2,4)+
GCD(3,9)+GCD(9,8)+
GCD(6,5)+GCD(5,7)
= 9 points.
Player 2 gets
GCD(1,3)+GCD(3,6)+
GCD(2,9)+GCD(9,5)+
GCD(4,8)+GCD(8,7)
= 11 points.
Now, this would be a good place to stop so as to have a simple game,
but my dream took the game further.
point for this pair, would get the minimum element of the pair added to
the score.
And, in addition, player 1 would get 5 bonus points added for every
row-score which added up to a prime, player 2 would get 5 bonus points
for every column-score which added up to a prime.
So, back to the example:
1 2 4
3 9 8
6 5 7
Player 1 gets
1 + 2 + 5 (add 5 because 1+2 is prime)
+ 3 + 8 (8 is minimum of 8 and 9) + 5
+ 5 + 5
= 34 points.
Player 2 gets
1 + 3
+ 2 + 5 + 5
+ 4 + 7 + 5
= 32 points
And any who wish to play this game can add their own rules for other
values of GCDs.
For example, if the GCD of a pair is 2, you can add the maximum of the
pair to the score;
if GCD = 3, you can add the product of the pair;
or you can add the floor of the maximum of the pair divided by the
minimum of the pair;
or you can add the number of divisors of the sum of the terms of the
pair;
etc...
Have fun!
thanks,
Leroy Quet

Here is another one of my games.
(Its rules are slightly more complex than the rules of most of my games
I have posted, but the rules are still relatively simple when compared
with the rules of many famous games.)
Game is for 2 players, each with a differently colored pencil.
Player start by taking turns drawing a predetermined number of dots on
a piece of paper (so that no 3 dots are colinear). Draw about a total
of 25 to 60 dots.
Players alternate drawing straight line-segments with their pencils and
a straight-edge, one line-segment per move, so that a player's
line-segment goes from the last dot drawn to by the other player to any
dot not yet drawn to.
(Player 1 starts by drawing a line-segment from any dot to any other
dot.)
If the game gets this far, the last player draws his/her line-segment
back to the first dot drawn from, so every dot (at game's end) has one
line-segment drawn to it and one-segment drawn from it.
A player gets a point for every preexisting line-segment of his/her
opponent that his/her own line passes through.
A player can draw his/her line through ONE of his/her own preexisting
lines per move, but may not pass through more than one of his/her own
lines per move.
If a player cannot move (because there are 2 or more of his/her own
line-segments between him/her and every dot not yet drawn to), the
player loses a turn (the other player gets to move again).
If neither player can move, the game is over.
(Again -- if the game gets this far, the game is over when the last dot
a first dot are connected by a segment.)
(And there probably should be a rule that a line-segment cannot pass
through any dots on its way to another dot, and that no 2 line-segments
of the same slope can coincide, if 3 dots are colinear.)
thanks,
Leroy Quet

## Friday, September 19, 2008

### Diagonal/Straight: A Cooperative Game

Here is yet another game of mine played on a grid drawn on paper.
It is a somewhat unusual in that the game is a cooperative game,
the 2 players get the same number of points.
Start with an n-by-n grid drawn on paper, where n is odd
(and I suggest n be at least 7, maybe much higher).
Players alternatingly take turns filling in, on each move,
an *empty* square immediately next to the last square filled
in by the other player.
Player 1 can only fill in an empty square immediately either
to the left of, to the right of, above, or below the last
square filled.
Player 2 can only fill in an empty square diagonally adjacent
to the last square filled.
Fill in the center square.
Player 1 then fills in a square above,below, left of, or
right of the center square.
Player 2 then fills in a square diagonally touching the
previously filled square.
Etc.
Play continues until no more squares can be filled in.
The score (for both players) is the number of squares filled
in before play must stop.
(And there should be very little communication between players.)
If you insist on playing a competitive game, you can do so
with 3 or more players.
Just have each pair of players play a game,
for a total (if m= number of players) of m(m-1)/2 games,
and add every player's particular scores to get a total
score for each player.
You can also play this game solitaire, but I believe
that perhaps playing with 2 people might be more fun.
Thanks,
Leroy Quet

Update: I suggest that, instead of simply filling in the squares of the grid, the players instead fill in the squares with the integers 1 then 2 then 3 then 4, etc. (Player 1 writes in the odd integers. Player 2 writes in the even integers.) Doing this has two advantages. One, it it easy to see where each player last moved. And, two, it is easy to see how many points the players each get, since this is simply the largest integer in the grid when the game is over.

### Dots Intersected By Sloping Lines

Here is another game of mine played on a grid drawn on paper.
It seems like it might be a fun way to teach slopes or how to reduce
fractions.
Start by lightly drawing an n-by-n grid on paper, or better yet, use
graph paper.
(I suggest an n of about 8 for starters.)
Game is for any number of players >= 2.
First, player 1 draws a dot at any intersection of grid-lines (at a
lattice-point).
He then connects one of the grid's corners to this dot with a straight
line-segment.
Players alternate moving. On each move, a player first draws a dot at a
lattice-point of the grid and then connects the last dot drawn by the
previous player to the newly drawn dot by a
*straight* line-segment (where the segment terminates at these 2 dots).
The new dot must be drawn so that no line-segments coincide. (The
segments may cross, but they cannot be on top of each other with the
same slope.)
Also, the dots cannot be drawn on previously drawn segments or dots.
A player drawing a line-segment gets a point for every previously drawn
dot his/her line-segment intersects, considering the *exact* slope of
the segment and *exact* positions of the intersected dots
*in theory*, as if the dots and line-segments and grid had been drawn
perfectly.
So, for instance, if we have a dot at (2,2), and the last drawn dot is
at (0,1), and the player makes a new dot at (6,4), no matter how poorly
the lines and dots and grid are drawn, the player still gets a point
for the "intersected" point (2,2) (which was theoretically intersected
by the line-segment of slope -1/2).
The game continues until either a predetermined score is reached by one
player, or until every lattice-point is intersected by a lines-segment
or has a dot drawn on it.

thanks,
Leroy Quet

### Intersection: Grid Solitaire Game

You start with an 8-by-8 grid, which is 9-by-9 lines, lightly drawn on
paper.
Start on any corner of the grid.
You draw darker lines along the light lines from vertex to vertex
(vertex = intersection of light lines), alternating drawing horizontal
and vertical lines, not drawing any darker lines where darker lines
You can make your line any length in one direction (up, down, left, or
right) as long as the line goes from the end of the last line to a
vertex of the grid.
No darker lines should coincide, except where the darker lines
intersect (and continue past each other). Your path of connected darker
lines should not even meet itself at any of its corners (meeting at a
single vertex).
You get a point every time an intersection is 'perpendicular' to the
last intersection made (AS the darker lines are being drawn).
(And you get a point for your first intersection.)
By 'perpendicular' intersection, I mean that if your previous
intersection was a horizontal newer dark line crossing a vertical older
dark line, then your current intersection is a vertical newer dark line
crossing a horizontal older dark line, and vice versa.
Your path of connected darker lines starts and stops at the same
corner.
I got 13 points. Can you get 13 points or better?
I also got 14 points by drawing a completely different path. Can you
get 14 points or better?
Example of perpendicular crossings:
(View with fixed-width font.)
.----+...+---+
.....!...!...!
.+---+---+---+
.!...!...!....
.+---+...!....
Example of non-perpendicular crossings:
.----+........
.....!........
.+---+---+....
.!...!...!....
.+---+...!....
.........!....
......---+---+
.........!...!
.........+---+

My 13 point solution:
(Move from * down first.)
*--------+..+-----+......
!........!..!.....!......
!.....+--+--+--+..!......
!.....!..!..!..!..!......
!..+--+--+..!..!..!......
!..!..!.....!..!..!......
!..!..!..+--+--+--+--+...
!..!..!..!..!..!..!..!...
+--+--+--+--+--+--+--+--+
...!..!..!..!..!..!..!..!
...!..!..+--+--+..!..!..!
...!..!.....!.....!..!..!
...!..+-----+-----+..!..!
...!........!........!..!
...+--------+--------+..!
............!...........!
............+-----------+

I will give my 14-point solution now.
(View with fixed-width font.)
Start at * and move upwards first.
+--+..............+--+...
!..!..............!..!...
+--+--+........+--+--+...
...!..!........!..!......
...+--+--+..+--+--+......
......!..!..!..!.........
......+--+--+--+..+--+...
.........!..!.....!..!...
......+--+--+..+--+--+...
......!..!.....!..!......
...+--+--+..+--+--+--+...
...!..!.....!..!..!..!...
+--+--+..+--+--+..+--+--+
!..!.....!..!........!..!
!..+-----+--+........+--+
!........!...............
*--------+...............
Note how different this solution is to the 13-point solution I give
above.

Thanks,
Leroy Quet

### Criss-cross Grid Game

Here is yet another of my games played on an n-by-n grid drawn on
paper.
For 2 players,each with a different colored pencil.
Players alternate placing an X or an O of their color in the grid's
squares as follows:
*Player 1 places the first O in the upper-left square.
*Each player places their symbol immediately next to the symbol last
drawn
by the other player in the directions of vertically, horizontally, or
diagonally.
*If the square a player draws his symbol in is empty, he/she draws an
O.
*If the square contains an O, he/she writes an X in the square with the
O.
*A player may not draw in (move to) a square already occupied by an X
(and an O).
Play continues until moving is impossible or when one player first gets
a predetermined number of points.
Scoring:
*If a player places an X in the same square as an O, the player who did
*not* draw
the O in that square gets a point.
*If a player places an O in an empty square diagonally adjacent to the
last square
drawn in by the other player, and the other 2 squares in the 2-by-2
bigger square
-- formed by the empty square being drawn in and the last drawn-in
square at opposite
corners of the bigger square -- are both filled, then the player
'completing the (bigger 2-by-2)
square' (the player whose move it is) gets a point.
Example to make above clearer:
+---+---+
!.O.!...!
+---+---+
!OX.!.O.!
+---+---+
Last move put an X in lower left square (getting a point for the player
who
did *not* place the O there earlier).
Player now wants to place an O in upper right square, which would get
him/her
a point (since the upper-left and lower-right squares of the 2-by-2
bigger square are both not empty).
What is a good strategy for this game?
thanks,
Leroy Quet

### Even/Odd: Another Game Of Contiguousness

Here is yet another grid-based game.
For 2 players, each player with a different colored pencil (or pen or
crayon).
Players take turns filling in any of the empty squares with their
colors, one square per move.
Play continues until every square is filled in.
I give two possible variations on scoring:
1) Player 1 gets a point for every 'contiguous region', of either
color, which is formed from an odd number of squares.
Player 2 gets a point for every 'contiguous region', of either color,
which is formed from an even number of squares.
2) Player 1 gets a point for every square in a 'contiguous region', of
either color, which is formed from an odd number of squares.
Player 2 gets a point for every square in a 'contiguous region', of
either color, which is formed from an even number of squares.
By "contiguous region", I mean:
A connected region of one color completely surrounded by squares of
the other color and/or the grid's border (surrounded in the directions
of up, down, left, or right,but not diagonally).
Example completed game:
* * # # # *
* * # # * *
# * * * # #
# # * * # *
* * # * # *
# # # * # #
So, we have contiguous regions of:
11 *'s, 5 #'s, 3 #'s, 3 *'s, 2*'s, 6 #'s, 2*'s, and 4 #'s.
So, under scoring method 1,
player 1 gets 4 points,
player 2 gets 4 points also.
Under scoring method 2,
player 1 gets 22 points,
player 2 gets 14 points.
Is there a bias under either scoring method for one player or the
other?
How does the scoring method affect strategy?
thanks,
Leroy Quet

### Polygons-&-Dots Game

Here is another game.
For 2 players.
First, players alternatingly take turns placing the same
number of dots on a piece of paper.
Each dot can be anywhere on the paper as long as no 3 or
more dots are lined up (along any imaginary straight lines),
and each dot is placed where no dot existed previously.
Second, the players take turns connecting the dots by drawing
straight line-segments (with a straight-edge, if necessary).
Each segment is drawn from the last connected-to dot
(connected to by the opposing player) to any unconnected-to dot.
The first player can connect any 2 dots.
The last dot to be connected-to and the first dot are connected
when the game is done, so the line-segments form a closed path
which visits every dot on the paper exactly once.
Scoring:
Player 1 gets a point for every polygon with an odd number of sides,
player 2 gets a point for every polygon with an even number of sides.
By "polygon", I mean those polygons formed by intersecting
line-segments and by dots as vertexes, but with no internal
line-segments.
For example, let us say that we had 5 dots (in reality, we would
only have an even number of dots) arranged at the vertexes of a
regular pentagon. The dots are connected to make a 5-pointed star,
ie: each vertex is connected to the 2 vertexes not adjacent to it.
We would then have 6 polygons, 5 triangles and one smaller pentagon.
The 5 larger triangles, for example, would not count.
So, player 1 would get 6 points, player 2 would get 0 points.
I suggest, especially if there are a large number of polygons to
consider, that the polygons be shaded in after their sides are
counted.
Shading them in with different colors might be aesthetically pleasing.
:)

The no-colinear-points rule is somewhat difficult to enforce in
practice if this game is not played on a computer.
If neither player notices that a player is drawing a point so that 3
points are colinear, this is no big deal, really.

thanks,
Leroy Quet

### Contiguous: Another Grid Game

Here is another simple game of mine (probably unoriginal) which is
played using an n-by-n grid drawn on paper.
(I would suggest an n of at least 5 or 6.)
Game is for any number of players >=2.
Each player uses a different colored pencil/pen, or each player uses a
different symbol (o, x, #, etc).
Players alternatingly take turns filling one square of the grid each
turn with either their color or symbol.
The square filled must, before being filled, be empty.
The square filled must be immediately adjacent to (in the direction of
either up, down, left, right, or diagonally) the most recently filled square
(which was filled by the previous player), if possible.
If no empty squares are adjacent to the last filled square, any empty
square in the grid can be filled.
(And any square can be the first filled square of the game.)
The game continues until each square of the grid is filled in.

which result...) is the scoring.
Each player's score is the product of the numbers of squares in the
"islands" formed of their colors/symbols.
An island is any contiguous group of squares formed of one
color/symbol and surrounded completely by squares of other
colors/symbols (or by the edge of the grid).
So, the number of terms in the product making up a player's score is
the number of islands of a player's color/symbol.
Example (n=5) completed game:
x o # # x
# o x x o
x # o x #
o x o # o
# x o # x
Scoring:
x: 1*1*3*1*2*1 = 6.
o: 2*1*3*1*1 = 6.
#: 2*1*1*1*1*2 = 4.
So x and o have tied for first place.
Note:
If you have, say, 2 contiguous symbols of yours separated by one empty
square from, say, 4 contiguous symbols of yours, it would *not* be at
your advantage to place a symbol of yours in the empty square between.
For with someone else's symbol in the separating square, you would
have 2*4 =points for the 2 islands. But with the separating square
filled with your symbol, you get 2+1+4 = 7 points for the one larger
island.

Whether two diagonally-adjacent squares of the same color/symbol are
to be necessarily considered as part of the same island or not, I
leave for the players of this game to agree amongst themselves.

thanks,
Leroy Quet