In 1852, Francis Guthrie noticed he needed only four colours to colour a map of England's counties so that no two neighbours shared a shade. Is it true for every map? It took until 1976: Kenneth Appel and Wolfgang Haken proved it by having a computer check nearly 2,000 cases. A first that divided mathematicians: can you trust a proof no human can fully check?
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.