Grid-Squiggly Game

This game is played on an n-by-n grid (n by n squares, {n+1} by {n+1} lines) that is lightly drawn on paper. The game is for 2 or more players.

First, players each secretly pick a positive integer m between 1 and n^2. (See below.)

The first player to move draws a line segment, one grid-square in length, along any vertical or horizontal line of the grid. (But don't draw along the border of the grid.)

Players then take turns drawing a line-segment each turn, where the line-segment is one grid-square in length, and goes from any vertex with a line-segment drawn to it (by any player) to any adjacent vertex that does not yet have a line segment drawn to it. (The drawn-to vertex is immediately above, below, right of, left of the drawn-from vertex.)
No line segments go along the border of the grid, but line segments can connect to vertices along the border of the grid.

Players continue to draw segments until there is no place they can draw them. (A total of n^2 + 2n - 4 segments will be drawn.)


Next, with a pencil of a color different that their opponents' pencil colors, each player takes turns (completely) filling in sections of the grid, one section each move. Each "section" is bounded by the lines the players drew and by the border of the grid.

When the whole grid has been colored in, count the number of squares filled in by each player.

Players then reveal the numbers (m) they picked at the game's beginning.
The player whose number of squares filled in is closest to the number they picked at the game's beginning (m) wins.


Thanks,
Leroy Quet