## Wednesday, January 28, 2009

### Polygonix

Here is a game for two players. It can be played on a grid, but this isn't necessary. (Although playing on a bounded grid eliminates a lot of the ambiguity that may occur about the positions of dots.)

The players take turns being the defense player and the offense player.

The defense player starts a round by drawing m (m is a positive integer determined ahead of time) dots on a blank piece of paper. If you are using a grid, the defense player draws the dots at some of the intersections of the grid-lines.

Then the offense player connects pairs of dots with straight line-segments. The offense player continues to do this until all of the dots are each on the perimeter of at least one CONVEX polygon, and none of the dots are on the perimeter of any concave polygons or on the perimeter of any polygon that is not simply-connected.

(By "polygon", here I mean a region bounded completely by line-segments with no line-segments through its interior {except possibly grid-lines}.)

The offense player gets a point for each polygon.

Players play an even number of rounds, switching who is offense and who is defense, then add their scores.

The LOWEST score wins. (So, players try to minimize the number of convex polygons they draw when they play offense.)

Thanks,
Leroy Quet

## Tuesday, January 20, 2009

### Intersections Of Rectangular Loops

Here is a game for 2 player, each player with a colored pen/pencil of a color different than her/his opponent's pen/pencil. This game is played on an n-by-n grid drawn on paper. (I will call the two players, for convenience, player-yellow and player-purple.)

Players take turns filling in empty squares of the grid, one square filled on each move, such that each player fills in exactly n squares during play (2n squares filled in by both players together) and such that there are exactly 2 squares (no fewer, no more) in each row of the grid and exactly 2 squares in each column.

After the squares are filled in, someone draws a straight horizontal line segment between the centers of each pair of squares in the same row, and draws a vertical line-segment between each pair of squares in the same column. You then should have 1 or more closed curves consisting of straight line-segments and 90-degree turns.

Scoring is as follows:

What matters in this game are the intersections of the line-segments. (Which closed-curves the line-segments of an intersection belong to is unimportant in this game. A closed curve may even intersect itself, of course.)

Call the 4 squares that are in the same row and column as an intersection of 2 perpendicular line-segments the 4 "extremities" of the intersection.
Call the pair of extremities that are both along the vertical line-segment of the intersection, or are both along the horizontal line-segment of the intersection, a pair of "opposing extremities".

Look at each intersection. (I suggest putting a circle in the square with each intersection, just to make the intersections easier to see.)
For every intersection where: {{the horizontal opposing extremities are of the same color} and {the vertical opposing extremities are of the same color}} or {{the horizontal opposing extremities are of differing colors} and {the vertical opposing colors are of differing colors}}, player-yellow gets a point.

On the other hand, for every intersection where one pair of opposing extremities consists of 2 squares of the same color and the other pair of opposing extremities consists of 2 squares of differing color, then player-purple gets a point.

Here is a much simpler way, probably, to figure out who gets a point at any intersection.

Count the number of the intersection's extremities colored by player-yellow or count the number of extremities colored by player-purple. If the number of an intersection's extremities filled by either one player is even, then player-yellow gets a point for that intersection. If the number of an intersection's extremities filled by either one player is odd, then player-purple gets a point for the intersection.

Highest score wins.

Thanks,
Leroy Quet

## Friday, January 16, 2009

### Doppel-game

(Title is taken from "doppelganger".)

This game seems to be familiar. And the rules are simple. So, maybe, I might have already posted a game with similar rules. Or a similar game might have been invented by someone else. (Actually, I could include this disclaimer with almost any of my games.)

Here is a game for 2 players played on an n-by-n grid.

First, fill in any one randomly chosen square of the grid.

Players then take turns filling in empty squares of the grid, one square per move, such that any square being filled in is immediately next to -- and in the direction of above, below, right of, or left of -- any square anywhere on the grid that has already been filled in (by either player).

Say, a player (player A) fills in a square that is immediately next to -- and in the direction of above, below, right of, or left of -- the square that same player (player A) filled in in their last move. Then let the direction from {the previously filled-in square (from the previous move of the same player, player A)} to {the newly filled-in square} be the direction d.
If the direction from {ANY filled-in square immediately next to {the square the other player (player B) last filled in}} to {the square the other player (player B) last filled in} is d, then player A gets a point.

No point is obtained if player A doesn't fill in a square immediately next to the square previously filled in by the same player (player A) or if the direction from {player A's previously filled in square} to {the current filled in square on player A's move} does not equal {a direction from any filled in square (immediately adjacent to the last square filled in by player B)} to {the last square filled in by player B}.

Got that?...
(I said these rules were simple!...Ha! -- Well, the rules ARE simple, once you figure out what they are!)

Play continues until all the squares of the grid are filled in.

The player with the most points wins.

Clarification:
First of all, player A refers to either player, but is the player currently moving.

Call 3 consecutive moves "move m", "move (m+1)", and "move (m+2)".
Player A made moves m and m+2, and player B made move m+1.

The direction from the square filled in on move m to the square filled in on move m+2, if those two square are immediately adjacent (in the direction of up, down, left, or right), is direction d.

Let "square (m+1)" be the square filled in by player B on move (m+1). If the direction from {ANY filled-in square immediately next to square (m+1)} to {square (m+1) itself} is direction d, then player A gets a point on move (m+2).

Thanks,
Leroy Quet

## Wednesday, January 7, 2009

### Diagon -- The Game

Here is a game for any plural number of players.

Start with an n-by-n grid drawn on paper, where n is an odd positive integer.

Fill in the 4 corner squares of the grid.

Players take turns filling in grid-squares, one square per move. On a move a player fills in any empty square that is adjacent to and immediately above, below, right-of, or left-of any filled-in square.

A player gets a point when that player fills in a square such that that square is in a 2-by-2 group of squares where two diagonally adjacent squares are filled in (including the square just filled in) and the other two diagonally-adjacent squares are empty (empty just after the move when the point is scored).

For instance, a point is scored if we have a 2-by-2 group of squares that looks like this:
* o
o *

or this:
o *
* o

(o = empty square, * = filled in square, where one of the filled in squares is the square just filled in by the scoring player.)

The game continues until there is no possibility that any more points can be scored.

Highest score wins.

--

If during the game we have a situation like so, say:
o *
* o
o *
(newly filled in square is the middle filled-in square), then how many points has the player scored? One or two?I leave how many points that can be scored on any single move, one at most or more, be up to the players to agree to among themselves.

--

Question: Is there a simple way for one player, say the first or second to move in a 2-person game, to always win? (If there is, I probably should edit this game to eliminate the possibility of using the simple strategy.)

Thanks,
Leroy Quet