Author Archives: apgoucher

Matrix multiplication update

At the end of the recent post on a combinatorial proof of Houston’s identity, I ended with the following paragraph: This may seem paradoxical, but there’s an analogous situation in fast matrix multiplication: the best known upper bound for the … Continue reading

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Updates and errata

In the Treefoil article, I erroneously described John Rickard’s length-24 cycle in as being the ‘uniquely minimal’ example of a cycle whose three axis-parallel projections are all trees (see here for a more detailed history on this problem). Dan Simms … Continue reading

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A combinatorial proof of Houston’s identity

Robin Houston recently discovered a rather interesting formula for the determinant of an n-by-n matrix. In particular, the formula improves upon the best known upper bound for the tensor rank of the determinant (viewed as a multilinear map which takes … Continue reading

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Tetrational machines

A pair of people called Pavel have independently developed remarkable automata that last record-breakingly long before halting. In both cases, the number of timesteps that it takes for each automaton to halt is so large that it cannot be written … Continue reading

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Infinitely many rational dodecahedra

Thomas Blok and David Madore have recently made significant progress on the problem of finding rational dodecahedra inscribed in the unit sphere, culminating in an infinite parametric family of solutions. In particular, Thomas began with the constrained version of the … Continue reading

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Threelds

A field F consists of two compatible Abelian groups — an additive group on F and a multiplicative group on F \ {0} — such that multiplication distributes over addition. In certain cases, though, this multiplicative group can be the … Continue reading

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29-year-old Conway conjecture settled

Ilkka Törmä and Ville Salo, a pair of researchers at the University of Turku in Finland, have found a finite configuration in Conway’s Game of Life such that, if it occurs within a universe at time T, it must have … Continue reading

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Training a random Gaussian generator

I’ve spent the last couple of months tackling the problem of designing an algorithm to rapidly generate high-quality normally-distributed pseudorandom numbers on a GPU. Whilst this may seem quite pedestrian, it turned out to be much more interesting than I’d … Continue reading

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Involutions on a finite set

Suppose that R is the (assumed to be finite) set of solutions to a certain problem, and you’re interested in determining the parity of |R|. The following proof strategy works surprisingly often, namely in at least two different scenarios, and … Continue reading

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Hamming backups: a 2-of-3 variant of SeedXOR

EDIT (2021-10-14): I’ve written a reference implementation of the Hamming backup idea introduced in this article. SeedXOR is an approach for splitting a Bitcoin wallet seed, adhering to the BIP39 standard, into N ‘parts’ (each the same size as the … Continue reading

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