In part 2 of Game Theory by Thomas Ferguson, example 2 'Turning Corners' on page 33, Thomas Ferguson mentions a so-called 'flipping-coin' game, where the Sprague-Grundy functions g(x, y) equals nim multiplication of x and y.
A move consists of turning over four distinct coins at the corners of a rectangle, i.e.
(a, b), (a, y), (x, b) and (x, y), where 0 ≤ a < x and 0 ≤ b < y, the coin at (x, y) going from
heads to tails.
This is under the (in part 2 of this book) usual assumptions that the game is for 2 players, and the last player that makes a move wins (i.e., you lose when you can't make a valid move according to the rules of the game). This might be the game that Alex Fink referred to, but I didn't find it to be such a contrived example, so it might be worth looking at (to be honest, at this point I don't understand the game completely and I don't have time for it now: this might be the same game that Lenstra mentions in the article linked to by Kevin O'Bryant: both are 'coin-turning' games).
Edit: In fact, the Sprague-Grundy value of a move in any 'product' of two coin-turning games G1 and G2 is given by the nimber product of the SG-value of the corresponding move in G1 and the corresponding move in G2. What this means exactly is explained more clearly in the document I linked to.
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