Monthly Archives: November 2012

Geomagic squares

In 2001, Lee Sallows generalised the concept of a magic square, replacing the integers in each cell with subsets of the plane. He calls such a configuration a ‘geomagic square’ if the subsets in every row, column and diagonal can … Continue reading

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Cipher 5: 99 problems or 23?

Due to popular demand, the frequency of Cipher Tuesdays has increased from every four weeks to its theoretical limit of every week. Also by popular demand, this cipher is given as both an image (for manual solvers) and delineated text … Continue reading

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Miscellany

Tsukamoto and Miyazaki of Kyoto University have discovered that three states are sufficient to enable gliders to exist on a Penrose tiling, rather than four. The following video demonstrates their discovery together with a periodic emitter in an augmented rule: [youtube=http://www.youtube.com/watch?v=6xVLpP25DRQ] On … Continue reading

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Outer automorphism of S6

Let’s consider the group S6 of permutations of {1,2,3,4,5,6}. For fairly boring reasons, we can find S5 living inside there — the subgroup which fixes the element ‘6’. More generally, S(n-1) is contained within Sn. S6 is special, though, since … Continue reading

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Hyperboloid of one sheet

Suppose we took a solid unit cube and balanced it on a vertex. Next, we spin the cube really quickly about the vertical axis, producing a solid of revolution: Clearly, the top and bottom are two identical cones produced by revolving a line segment passing … Continue reading

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Betts revisited

As a brief summary, I posted about the three-dimensional generalisation of a particular two-dimensional problem by Alexander Betts. In case you haven’t read it, or didn’t understand it, I’ll explain the ideas again here. A finite set S of n points in d-dimensional … Continue reading

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Cipher 4: Think rationally

To make a welcome change from the monoalphabetic substitution ciphers embedded in images, I have decided to publish this particular cipher in a text-only format for ease of access. WII MJCVKWV FZN ZQLHWMH HA ZLH GMFKTJQ DKOQXT XM UNN LAADANPNFT EOWZXJRX … Continue reading

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Spin groups

If you take an object and rotate it by 360° about a particular axis, it will return to its original orientation. Certainly, every object you’re likely to have seen obeys this basic principal. If you’re naive, you may actually believe … Continue reading

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Field with one element

So, what’s up with that, right…? In case you haven’t heard of the field with one element, it’s a totally stupid non-existent thing that doesn’t exist. Nevertheless, its blatant non-existence did not deter several people from extensively investigating it, with … Continue reading

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Mouse-astrophe

Suppose we have a particularly ravenous cat (Felis sylvestris) chasing some particularly tasty mice (Mus musculus, if I remember correctly) across the vast expanse of the real Eculidean plane R². The most interesting case is when mice and cats move at … Continue reading

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