## Monday, February 8, 2010

### Labyrinthine Loop

Here is an (unoriginal) game for any plural number of players.

The game consists of rounds, where every player is the "offense-player" the same predetermined number of rounds.

At the beginning of each round, draw an array of dots (vertices of a grid) on a piece of paper, n rows of dots by n columns. (n is a predetermined integer, which is the same for all rounds. I suggest n be >= 6.)
There is a new array of dots for each round.

At the beginning of the round, the offense-player moves first, connecting any pair of adjacent dots with a straight line-segment. (By "adjacent", one dot must be one of the closest dots to the other dot, and in the direction of E, SE, S, SW, W, NW, N, or NE.)

Players thereafter continue to take turns. On a turn, a player connects (with a straight line-segment) any dot that has AT LEAST ONE line-segment connected to it already, to any ADJACENT dot that has NO line-segments connected to it.
Again: By "adjacent dots", it is meant that one dot is one of the closest to a second dot, where the two dots are in the direction of either E, SE, S, SW, W, NW, N, or NE to each other.

Diagonal line-segments MAY cross each other.

Players continue taking turns until all dots have line-segments connecting to them.
(ie. Players continue taking turns until a total of n^2 -1 line segments are drawn in a round.)

Then, lastly in the round, the offense-player connects any pair of unconnected adjacent dots with a line-segment.

The offense-player's score for that round is the number of dots in the (single) closed loop of line-segments (including the line-segment the offense-player just drew).
(The "loop" is the simple path from dot to dot that connects back to its starting point. The loop does not including dead-ends, of course.)

After each player has played offense the same predetermined number of rounds, then each player adds up his/her scores from those rounds she/he was offense to get her/his grand total. The player with the highest grand total wins.

PS: It should be noted that when determining the shape of the loop created in the final step of a round, each "line-segment" in the final loop goes strictly from a dot to another dot. When two diagonal line-segments cross, they are considered, for our purposes, to not be touching -- one segment goes "over" another. As a result, there is always one and only one loop made each round.

Thanks,
Leroy Quet