A long paper strip is folded in half, then in half again, and then in half once more, always the same way round. One straight cut is then made right across the folded bundle, at a place well away from every fold. How many separate pieces of paper are there once the cut is made?

Think about the strip opened out rather than about the bundle. Three folds mark the strip into $2\times 2\times 2=8$ equal sections, and the single cut through the bundle passes through every layer, so on the opened strip it leaves exactly one cut mark in each of those 8 sections. A straight strip cut in 8 places falls into 8 + 1 = 9 pieces. Check the idea on a strip folded just once: two sections, two cut marks, three pieces — a U-shaped piece and the two loose ends. The traps: 8 counts the layers, or equivalently the cut marks, forgetting that cuts always number one less than pieces; 16 assumes each layer falls into two free pieces, forgetting that the folds still hold layers together; 5 is the answer after only two folds.