Remove two opposite corners from a chessboard: can you cover the remaining 62 squares with 31 dominoes, each covering two adjacent squares? You could search for hours, but one observation is enough: the two removed corners share the same colour, leaving 32 squares of one colour and 30 of the other. Yet every domino always covers one white and one black square: the task is impossible.
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.