
Last time we introduced the game of transfinite Nimbers and asked a winning move for the transfinite game with stones a at position $~(2,2) $, b at $~(4,\omega) $, c at $~(\omega+2,\omega+3) $ and d at position $~(\omega+4,\omega+1) $. Above is the unique winning move : we remove stone d and by the rectanglerule add… Read more »

Today, we will expand the game of Nimbers to higher dimensions and do some transfinite Nimber hacking. In our identification between $\mathbb{F}_{16}^* $ and 15th roots of unity, the number 8 corresponds to $\mu^6 $, whence $\sqrt{8}=\mu^3=14 $. So, if we add a stone at the diagonal position (14,14) to the Nimbersposition of last time… Read more »

Nimbers is a 2person game, winnable only if you understand the arithmetic of the finite fields $\mathbb{F}_{2^{2^n}} $ associated to Fermat 2powers. It is played on a rectangular array (say a portion of a Goboard, for practical purposes) having a finite number of stones at distinct intersections. Here’s a typical position The players alternate making… Read more »
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