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Peter Taylor's user avatar
Peter Taylor's user avatar
Peter Taylor
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Can you consistently add axioms about the Busy Beaver function to ZF?
Also note that if $k \ge 64$ you potentially have a proof of the non-existence of rank-into-rank cardinals (or the inconsistency of rank-into-rank cardinals with ZFK): cheddarmonk.org/papers/laver.pdf
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Can you consistently add axioms about the Busy Beaver function to ZF?
@EricTowers, that's a factor of 4 larger than it needs to be: github.com/sorear/metamath-turing-machines
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Computation of a minimal polynomial
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A thickening map on integer partitions
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What is the smallest uniquely hamiltonian graph with minimum degree at least 3?
If you have access to it, A census of maximum uniquely hamiltonian graphs by Barefoot and Entringer claims in the abstract to provide an algorithm for constructing maximal uniquely Hamiltonian graphs of order $n$.
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Sophisticated treatments of topics in school mathematics
We were taught it in terms of linear ODEs in the maths and physics curricula followed at my school.
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Mathematical software wish list
Ganesalingam and Barnet-Lamb were working on this; I'm not sure whether they still are.
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Help solving a recurrence relation
Cross-post on MSE
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What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?
Yes, I also get 17932. I had the correct order for the notation, but I was reading "unopposed" as "passed" and trying to account for cases where a capture by a pawn didn't pass the pawn.
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What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?
The notation is copied from the answer, except that I've skipped the $+$ for brevity. The key point, which the use of a 6-tuple obscures, is that the definition of "opposed" means that both players must have the same number of opposed pawns. Thanks for your comment, which highlighted this for me. For the benefit of anyone else trying to reproduce this, it's actually 285 possible projections to pawn type triples, because with $0$ opposed pawns there are a further $9^2$ cases.
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What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?
I'm trying to reproduce the first stage of your calculations and failing. With vectors $(-2,2,0,-2,1,0), (-1,1,0,0,0,-1), (0,0,0,-1,0,0), (0,0,0,0,-1,0), (0,0,0,0,0,-1)$ (representing respectively pawn takes pawn, pawn takes piece, piece takes piece, and piece takes pawn in two forms) and the symmetric vectors for the other colour, I already get 105464 different vectors - and I think there are other deltas required in some cases. Would you mind expanding your description of that phase?
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