Well before Alan Turing, Pole Marian Rejewski reconstructed the inner wiring of the Enigma machine in 1932 without ever seeing one, using permutation theory. With Jerzy Różycki and Henryk Zygalski, he read German messages for years. In July 1939, weeks before the invasion, Poland handed everything over to the French and British. Bletchley Park built on those foundations.
How many different structures can you build with six classic 2-by-4 Lego bricks of the same colour? The company used to quote just over 102 million. In 2005, Danish mathematician Søren Eilers recounted everything by computer: 915,103,765 combinations. The official estimate had been too low by a factor of about nine.
Fold an A4 sheet in half and you get an A5 with the same proportions. That only works for one ratio between length and width: about 1.414, the square root of 2. The A0 format measures exactly one square metre, and A4 is a sixteenth of it. With standard 80-gram-per-square-metre paper, an A4 sheet therefore weighs exactly 5 grams.