A game for any plural number of players: (Number of players = m.)

Draw m*k+1 incrementally larger concentric circles on a piece of paper, where k is some positive integer >= 2.

Subdivide the circles by drawing n equally spaced rays from their center, where n is at least 6, say.

(So now you should have a target.)

The players start the game by taking turns, and each player on a turn places an integer -- 1 to n and which has not been written down earlier in the game -- in an empty pie-shaped wedge in the central circle. After n moves, there should be a permutation of (1,2,3,...,n) in the central circle, one integer per wedge.

In the second part of the game, the players take turns, each player "completing a ring" on a move. By completing a ring, the player fills in the n sections of the innermost *empty* ring. The player fills in each section of the ring either with the sum of (the integer immediately adjacent to the section, but in the next ring inward) and (the integer one position clockwise to the section, but in the next ring inward), or with the absolute value of the difference between these particular two integers (in the next ring inward).

Example:

\...8..|....../

-\-----|-----/-

..\.2..|..6./

---\---|---/---

8 = 2+6.

(The 8 could have been a 4.)

After the ring is completed, the player gets the number of primes in his latest ring added to his score, OR, if there is exactly one prime in his ring (no more, no fewer), he gets the value of that prime added to his score.

After all rings are completed, the game is over. Largest score wins.

Thanks,

Leroy Quet

## Saturday, October 30, 2010

## Monday, October 11, 2010

### Grid Game Of Differences

This is a game for two player.

Draw an n-by-n grid on paper, where I suggest that n is at least 8.

The players take turns placing x's in the empty squares of the grid, one x per turn.

No two or more x's may be placed in the same row or in the same column of the grid.

After n total moves (when there is exactly one x in each row and column), the game is over.

Now to determine the score:

Reading left to right, write down the (n-1) absolute values of the differences between the consecutive x's' vertical coordinates, in terms of number of squares.

In another list, reading bottom to top, write down the (n-1) absolute values of the differences between the consecutive x's' horizontal coordinates, in terms of number of squares.

Player 1 gets a point for every distinct numerical value occurring in the first list of differences.

Player 2 gets a point for every distinct numerical value occurring in the second list.

If a particular difference occurs at least once in a single list, then the player gets one point for that particular difference.

Largest score wins.

We may need an example here:

n=9:

. x . . . . . . .

. . . . . . . . x

. . x . . . . . .

. . . . . x . . .

x . . . . . . . .

. . . x . . . . .

. . . . . . . x .

. . . . . . x . .

. . . . x . . . .

Player 1's (vertical) differences (reading left to right) are:

4,2,3,3,5,4,1,5

The unique values that occur are:

1,2,3,4,5

Player 1 gets 5 points.

Player 2's (horizontal) differences (reading bottom to top) are:

2,1,4,3,5,3,6,7

The unique values that occur are:

1,2,3,4,5,6,7

Player 2 gets 7 points.

In another variation of this game, count ONLY those differences that occur exactly once (and no more than once).

In this variation, player 1 would have gotten 2 points, for the differences 1 and 2.

Player 2 would have gotten 6 points, for the differences 1,2,4,5,6,7.

(Since 3 is the only difference in this list that occurs more than once.)

Which variation is more fun?

Thanks,

Leroy Quet

Draw an n-by-n grid on paper, where I suggest that n is at least 8.

The players take turns placing x's in the empty squares of the grid, one x per turn.

No two or more x's may be placed in the same row or in the same column of the grid.

After n total moves (when there is exactly one x in each row and column), the game is over.

Now to determine the score:

Reading left to right, write down the (n-1) absolute values of the differences between the consecutive x's' vertical coordinates, in terms of number of squares.

In another list, reading bottom to top, write down the (n-1) absolute values of the differences between the consecutive x's' horizontal coordinates, in terms of number of squares.

Player 1 gets a point for every distinct numerical value occurring in the first list of differences.

Player 2 gets a point for every distinct numerical value occurring in the second list.

If a particular difference occurs at least once in a single list, then the player gets one point for that particular difference.

Largest score wins.

We may need an example here:

n=9:

. x . . . . . . .

. . . . . . . . x

. . x . . . . . .

. . . . . x . . .

x . . . . . . . .

. . . x . . . . .

. . . . . . . x .

. . . . . . x . .

. . . . x . . . .

Player 1's (vertical) differences (reading left to right) are:

4,2,3,3,5,4,1,5

The unique values that occur are:

1,2,3,4,5

Player 1 gets 5 points.

Player 2's (horizontal) differences (reading bottom to top) are:

2,1,4,3,5,3,6,7

The unique values that occur are:

1,2,3,4,5,6,7

Player 2 gets 7 points.

In another variation of this game, count ONLY those differences that occur exactly once (and no more than once).

In this variation, player 1 would have gotten 2 points, for the differences 1 and 2.

Player 2 would have gotten 6 points, for the differences 1,2,4,5,6,7.

(Since 3 is the only difference in this list that occurs more than once.)

Which variation is more fun?

Thanks,

Leroy Quet

## Wednesday, October 6, 2010

### Bouncing Pathways Within A Circle: Game

A game for two players:

First, draw a circle on a piece of paper.

Players start by each drawing a different straight line-segment at any angle they choose from the center of the circle to the circumference.

Players thereafter move like so: (Player 2, Player 1), (Player 1, Player 2), (Player 2, Player 1), (Pl 1, Pl 2), (Pl 2, Pl 1), etc.

So, we have "whole moves", consisting of two moves, with a move by each player. And who moves first in the whole moves alternates.

The first player to move in a whole-move decides if the next line-segment will bounce left or bounce right. This player then draws his straight line-segment in the proper direction (relative to the direction his own last line-segment was traveling) from where his own last line segment ended to where the new line-segment comes up against a pre-existing line-segment (drawn by either player) or up against the circumference of the circle. A player's line-segment may pass through a pre-existing line-segment. But each time a player crosses a line-segment with another line-segment, his score is halved. No line-segments may pass outside of the circle.

The second player to move in a full-move then must bounce the same direction, left or right, as the other player did, but relative to the direction this player's own last segment was traveling. And he draws his segment from where his own last line-segment ended to where his new line-segment comes up against another pre-existing segment or up against the circle's circumference. Again, his segment may pass through a pre-existing line-segment (but not pass through the circle's circumference), but doing so halves his score each time he does it.

After a predetermined number of full-moves (such as 10), each player's score = the length of that player's final line-segment divided by 2^(the number of lines crossed by that player).

Largest score wins.

Note: To be clear, there will be two "pathways" within the circle: One pathway belonging to each player, and each pathway made up of the series of connected line-segments drawn by that player.

Also, line-segments may not coincide, except at the points where they intersect.

Thanks,

Leroy Quet

First, draw a circle on a piece of paper.

Players start by each drawing a different straight line-segment at any angle they choose from the center of the circle to the circumference.

Players thereafter move like so: (Player 2, Player 1), (Player 1, Player 2), (Player 2, Player 1), (Pl 1, Pl 2), (Pl 2, Pl 1), etc.

So, we have "whole moves", consisting of two moves, with a move by each player. And who moves first in the whole moves alternates.

The first player to move in a whole-move decides if the next line-segment will bounce left or bounce right. This player then draws his straight line-segment in the proper direction (relative to the direction his own last line-segment was traveling) from where his own last line segment ended to where the new line-segment comes up against a pre-existing line-segment (drawn by either player) or up against the circumference of the circle. A player's line-segment may pass through a pre-existing line-segment. But each time a player crosses a line-segment with another line-segment, his score is halved. No line-segments may pass outside of the circle.

The second player to move in a full-move then must bounce the same direction, left or right, as the other player did, but relative to the direction this player's own last segment was traveling. And he draws his segment from where his own last line-segment ended to where his new line-segment comes up against another pre-existing segment or up against the circle's circumference. Again, his segment may pass through a pre-existing line-segment (but not pass through the circle's circumference), but doing so halves his score each time he does it.

After a predetermined number of full-moves (such as 10), each player's score = the length of that player's final line-segment divided by 2^(the number of lines crossed by that player).

Largest score wins.

Note: To be clear, there will be two "pathways" within the circle: One pathway belonging to each player, and each pathway made up of the series of connected line-segments drawn by that player.

Also, line-segments may not coincide, except at the points where they intersect.

Thanks,

Leroy Quet

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