what is the fairest ratio of k to n for this game

Question

Here is a description of a two player game with players plr1 and plr2 that is played on a n by n grid.

Players take turns filling cells in the grid. plr1 goes first and wins by completing a straight (not diagonal) line of n filled cells on the grid. plr2 wins if plr1 has not won before k cells have been filled.

Just to be clear, plr1 must complete the line n filled cells, that is to say, if plr2 were to complete such a line this would not result in a win for plr1. Additionally this line must be exactly of length n. It cannot exceed n for it to be a win.

Obviously, n < k < n²

The question is to find what the "fairest" ratio of k to n is for this type of game. By fair I mean a game that would been won approximately equally often by plr1 and plr2 assuming they were of relatively similar skill. Of course, this ratio may vary as n increases, so the question might also be stated as finding the fairest k for a given n.

Thanks

Answer 1

Nice question. Wanted to make this a comment, but don't have the reputation for it. A few questions, observations:

• on a $2 \times 2$ grid, it would seem that plr1 wins always. yes?
• on a $3 \times 3$ grid, it would seem that plr2 can always prevent plr1 from winning
• on a $4 \times 4$ grid, I can't find a way for plr1 to win either as it seems that plr2 can usually get a "rook covering" of the board [putting rooks on a chessboard so that no rook is attacking another rook, but the entire board is under rook surveillance] or a weaker form of it [the rooks can attack each other].

can plr1 ever win for any $k$ for $n > 2$? or maybe I understood the question incorrectly? Are the players alternating turns or do they take turns like plr1, plr2, plr2, plr1, plr1, plr2, plr2, plr1, plr1, etc.?