The people of Königsberg wondered whether they could cross each of their city's seven bridges exactly once in a single walk. In 1736, Leonhard Euler proved they couldn't: more than two districts were reached by an odd number of bridges. To show it, he ignored distances and shapes and kept only the connections. Graph theory, which now guides our GPS routes, was born.
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.