A game played on a rectangle

game-theory

Solution 1:

The $n \times 1$ version is Dawson's Chess. The OP's sequence is A215721 in the OEIS, after adding 1 to each term. I wrote a program too, and found that the proportion of losing initial positions seems to tend to a constant -- there are 1473 losing positions in the first 10000, and 14709 losing positions in the first 100000. I found an explanation at "Sprague-Grundy values for Dawson's Chess" (A002187 in the OEIS):

Has period 34 with the only exceptions at n=0, 14, 16, 17, 31, 34 and 51.

5/34 is 0.14705...

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