This page tells us that “4 is the smallest number of colors sufficient to color all planar maps”. It’s of course true, but I’d rather focus on another fine property of 4, namely the fact that 4 is the smallest non-trivial square. Squares are pretty ubiquitous things, and here are just two stories about them that I can remember now.

### Squaring the square

When everybody got tired of squaring the circle, some mathematicians started to investigate a similarly sounding, but actually quite different (and more solvable) problem of squaring the square. The problem asks you to divide a square into non-overlapping squares, all of which have different sizes. It was solved by Brooks, Smith, Stone and Tutte* after two years of unsuccessful attempts. Here is the smallest possible squared square:

This particular one was found by a computer search; the first discovered example was far less optimal. Further info about squared squares is availible here.

*Lexicographical order.

### Magic squares of squares

This problem asks to literally make a magic square with all entries being distinct perfect squares. A 4×4 square possessing this property was found by (no prizes for guessing) Euler and no 3×3 examples have been found. Further information here.

### The cipher

You have probably been anticipating this part of the page. Here it is:

`...7?4055157?6?023??543994999345608083801?9074?53006005605574?818709692785?9977?918??0?75416?28527708162011350?46806?5816327617167676526?93752???684421448619396049983447280672?906?70417240?942344661978?24266907875359446?66?8508064636137?6638404902??19?4188190958?659?24477861846140912878298438431703248?734288?65727376631465191??9880?9447960814673760503957196893714671801375?19?5?4629968?476??639039530073191081?98029385?98?006?16?5?9?8086381?000557423?23?3089?1??0041?6619977?92256259918212890625`

Good luck.

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