## Squares of squares, and the group of rational points on the circle

2016-08-02 § 1 Comment

The purpose of this post is to describe a slightly different way of thinking about the existence – or otherwise – of a 3×3 magic square of squares. Of course it may not lead to any real progress, but it does at least suggest some natural weaker questions that may be more amenable. It follows on from part 1 and part 2. « Read the rest of this entry »

## Almost-magic squares of squares

2016-07-18 § 2 Comments

In the last post we saw that every 3×3 almost-magic square is a rearrangement of three three-term arithmetic progressions that have the same common difference. In other words, if we pick any three numbers *x*, *y* and *z*, and any common difference *d*, then the nine numbers

x–d, |
x, |
x+d |

y–d, |
y, |
y+d |

z–d, |
z, |
z+d |

can be rearranged to make an almost-magic square

x | z+d | y-d |

z-d | y | x+d |

y+d | x-d | z |

where all the rows and columns, and the leading diagonal, add up to *x*+*y*+*z*.

We’re interested in finding almost-magic squares where all the entries are square numbers, so that means we’re particularly interested in three-term arithmetic progressions of *squares*. Fortunately there is a well-developed theory of these things, which are closely related to Pythagorean triples.

« Read the rest of this entry »

## Magic squares of squares: Part I

2016-07-13 § 2 Comments

A recent Numberphile video discussed an intriguing unsolved problem in number theory: is there a 3×3 magic square whose entries are all square numbers? (Matt Parker proposed a solution which doesn’t *quite* work: see the video for more. The “Parker square” even has its own Twitter account.)

It turns out that this question leads to some interesting mathematics, which I’m going to have a go at explaining. At the very least I hope to give some insight into why it’s a hard problem.

To reduce the risk that I’ll run out of time and energy before pressing Publish, I’m going to split this post into two or possibly three parts, which I’ll post separately. The first part, which you’re reading **right now**, gives some background on 3×3 magic squares. The second will explain how to find *almost-magic* squares of squares using a bit of number theory and computer algebra. By *almost-magic* I mean that the rows and the columns and *one* of the two diagonals have the same sum.

The third, if I write it, will be more speculative and look at some possible approaches to either finding the mythical square of squares or proving that it doesn’t exist.

« Read the rest of this entry »

## The Government Statistical Service’s terrible spreadsheet advice

2014-12-05 § 6 Comments

The UK Government Statistical Service recently released its good practice guidance for releasing statistics in spreadsheets. While this advice is clearly well-intentioned^{*}, and parts of it are good, the overall effect is to encourage the release of data in formats that are difficult to process by computer. This is a disappointing retrograde step.

## Tackling the Minimal Superpermutation Problem

2014-08-22 § 5 Comments

What’s the shortest string that contains every possible permutation of ABCD somewhere inside it? As it happens, it’s 33 letters long: ABCDABCADBCABDCABACDBACBDACBADCBA. A string like this is called a *minimal superpermutation*.

So what’s the shortest string that contains every possible permutation of ABCDE? It was recently shown that 153 letters is the shortest possible, and that there are eight different superpermutations of this length.

Okay, what about ABCDEF? The answer is that **nobody knows**. Until this week the shortest known superpermutation of ABCDEF was 873 letters long:

ABCDEFABCDEAFBCDEABFCDEABCFDEABCDFEABCDAEFBCDAEBFCDAEBCFDAEBCDFAEB CDAFEBCDABEFCDABECFDABECDFABECDAFBECDABFECDABCEFDABCEDFABCEDAFBCED ABFCEDABCFEDABCADEFBCADEBFCADEBCFADEBCAFDEBCADFEBCADBEFCADBECFADBE CAFDBECADFBECADBFECADBCEFADBCEAFDBCEADFBCEADBFCEADBCFEADBCAEFDBCAE DFBCAEDBFCAEDBCFAEDBCAFEDBCABDEFCABDECFABDECAFBDECABFDECABDFECABDC EFABDCEAFBDCEABFDCEABDFCEABDCFEABDCAEFBDCAEBFDCAEBDFCAEBDCFAEBDCAF EBDCABEFDCABEDFCABEDCFABEDCAFBEDCABFEDCABACDEFBACDEBFACDEBAFCDEBAC FDEBACDFEBACDBEFACDBEAFCDBEACFDBEACDFBEACDBFEACDBAEFCDBAECFDBAECDF BAECDBFAECDBAFECDBACEFDBACEDFBACEDBFACEDBAFCEDBACFEDBACBDEFACBDEAF CBDEACFBDEACBFDEACBDFEACBDAEFCBDAECFBDAECBFDAECBDFAECBDAFECBDACEFB DACEBFDACEBDFACEBDAFCEBDACFEBDACBEFDACBEDFACBEDAFCBEDACFBEDACBFEDA CBADEFCBADECFBADECBFADECBAFDECBADFECBADCEFBADCEBFADCEBAFDCEBADFCEB ADCFEBADCBEFADCBEAFDCBEADFCBEADCFBEADCBFEADCBAEFDCBAEDFCBAEDCFBAED CBFAEDCBAFEDCBA

and it was thought that might be the shortest possible.

But we now know it isn’t, because I found a shorter one:

ABCDEFABCDEAFBCDEABFCDEABCFDEACBFDEACFBDEACFDBEACFDEBACFDEABCDFEAB CDAEFBCDAEBFCDAEBCFDAEBCDFAEBCDAFEBCDABEFCDABECFDABECDFABECDAFBECD ABFECDABCEFDABCEDFABCEDAFBCEDABFCEDABCFEDACBFEDCABFDECAFBDCEAFBDCA EFBDCAFEBDCAFBEDCAFBDECAFDBECADFBECADBFECADBEFCADBECFADBECAFDEBCAD FEBCADEFBCADEBFCADEBCFADEBCAFDECBAFDECABFDCEABFDCAEBFDCABEFDCBAEFD BCAEDFBCAEDBFCAEDBCFAEDBCAFEDBCAEFDBACEFDBAECFBDAECFBADECFBAEDCFBA ECDFBACEDFBACDEFBACDFEBACDFBEACDFBAECFDBAEFCDBAFECDBAFCEDBAFCDEBAF CDBEAFCDBAEFDCBEAFDCBEFADCBEFDACBEFDCABFEDCBAFEDCBFAECDBFACEDBFACD EBFACDBEFACDBFEACDBFAECBDFEACBDFECABDFCEABDFCAEBDFCABEDFCBAEDFCBEA DFCBEDAFCBEDFACBEDFCABDEFCBADEFCBDAEFCBDEAFCBDEFACBDEFCABDFECBADFE CBDAFECBDFAECBFDAECBFADECBFAEDCBFEADCFBEADCFEBADCEFBADCEBFADCEBAFD CEBADFCEBADCFEABDCFAEBDCFABEDCFABDECFABDCEFABDCFEADBCEFADBCEAFDBCE ADFBCEADBFCEADBCFEADCBFEDACFBEDACFEBDACEFBDACEBFDACEBDFACEBDAFCEBD ACFEDBACFEDABC

I’ve uploaded a short note about it to the arxiv.

## I hate the Pumping Lemma

2013-08-18 § 39 Comments

I hate the Pumping Lemma for regular languages. It’s a complicated way to express an idea that is fundamentally very simple, and it isn’t even a very good way to prove that a language is not regular.

Here it is, in all its awful majesty: for every regular language L, there exists a positive whole number *p* such that every string *w*∈L that has *p* characters or more can be broken down into three substrings *xyz*, where *y* is not the empty string and the total length of *xy* is at most *p*, and for every natural number *i* the string *xy ^{i}z* is also in L.

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## “Venn diagram” partitioning

2013-07-10 § Leave a comment

Paddy3118 wrote about partitioning elements in the same way a Venn diagram does. So, if we have sets *A*, *B* and *C*, the partitions are