
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 »

The odd Knight of the round table problem asks for a consistent placement of the nth Knight in the row at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements. The first identifies the multiplicative group of its nonzero elements with the group… Read more »

The Knightseating problems asks for a consistent placing of nth Knight at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements.

Here’s a tiny problem illustrating our limited knowledge of finite fields : “Imagine an infinite queue of Knights ${ K_1,K_2,K_3,\ldots } $, waiting to be seated at the unitcircular table. The master of ceremony (that is, you) must give Knights $K_a $ and $K_b $ a place at an odd root of unity, say $\omega_a… Read more »
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