Which convex pentagons can tile a floor with no gaps or overlaps? In 1918, five families were known. In the 1970s, Marjorie Rice, an American with no mathematics degree, read a Martin Gardner column and found four new ones at her kitchen table. The fifteenth was found in 2015, and in 2017 Michaël Rao proved the list was complete.
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.