Multiply 142,857 by 2, 3, 4, 5 or 6 and you get 285,714, 428,571, 571,428, 714,285 and 857,142: always the same six digits rotated in a circle. Multiply it by 7, and everything flips: 999,999. It's no accident: these six digits are the ones that repeat forever when you divide 1 by 7, hence the name “cyclic number”.
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.