Can folding a napkin ever make its perimeter bigger instead of smaller?
Take a square napkin and fold it, any number of times, along any lines you like. Common sense says the folded shape's outer edge, its perimeter, can only ever shrink or stay the same, never grow. Mathematician Vladimir Arnold made this his first-listed problem in 'Arnold's Problems,' nicknaming it the 'rumpled rouble problem.' Can a sequence of folds actually increase a napkin's perimeter?
Reveal the answer
It depends on what counts as a fold: if every fold has to reflect the entire stack of paper layers at once, the perimeter provably never increases. But if a fold is allowed to flip just one connected piece of paper rather than the whole stack, mathematicians still don't know whether some clever sequence of folds could sneak the perimeter larger, making this version of the problem genuinely unsolved.