# Posts Tagged: Conway

• math, number theory

## Aaron Siegel on transfinite number hacking

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One of the coolest (pure math) facts in Conway’s book ONAG is the explicit construction of the algebraic closure $\overline{\mathbb{F}_2}$ of the field with two elements as the set of all ordinal numbers smaller than $(\omega^{\omega})^{\omega}$ equipped with nimber addition and multiplication. Some time ago we did run a couple of posts on this. In… Read more »

• games

## n-dimensional and transfinite Nimbers

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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 15-th 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 Nimbers-position of last time… Read more »

• games

## How to play Nimbers?

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

• absolute, noncommutative, number theory

## Seating the first few billion Knights

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The odd Knight of the round table problem asks for a consistent placement of the n-th 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 non-zero elements with the group… Read more »

• stories

## So, who did discover the Leech lattice?

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For the better part of the 30ties, Ernst Witt (1) did hang out with the rest of the ‘Noetherknaben’, the group of young mathematicians around Emmy Noether (3) in Gottingen. In 1934 Witt became Helmut Hasse‘s assistent in Gottingen, where he qualified as a university lecturer in 1936. By 1938 he has made enough of… Read more »

• stories

## Who discovered the Leech lattice?

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The Leech lattice was, according to wikipedia, ‘originally discovered by Ernst Witt in 1940, but he did not publish his discovery’ and it ‘was later re-discovered in 1965 by John Leech’. However, there is very little evidence to support this claim. The facts What is certain is that John Leech discovered in 1965 an amazingly… Read more »

• games, number theory

## Seating the first few thousand Knights

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

• featured

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• stories

## Finding Moonshine

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On friday, I did spot in my regular Antwerp-bookshop Finding Moonshine by Marcus du Sautoy and must have uttered a tiny curse because, at once, everyone near me was staring at me… To make matters worse, I took the book from the shelf, quickly glanced through it and began shaking my head more and more,… Read more »