In 1924, Stefan Banach and Alfred Tarski proved that a solid ball can be cut into finitely many pieces, five suffice, and reassembled by simple moves into two balls identical to the original. No magic: the result relies on a debated axiom, the axiom of choice, and the “pieces” are clouds of points so strange that no volume can be assigned to them. Sadly, it won't work on an orange.
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.