Buttons & Scissors is NP-Complete

Buttons & Scissors is a popular single-player puzzle. A level is played on an n-by-n grid, where each position is empty or has a single coloured button sewn onto it. The player’s goal is to remove all of the buttons using a sequence of horizontal, vertical, and diagonal scissor cuts. Each cut removes all buttons between two distinct buttons of the same colour, and is not valid if there is an intermediate button of a different colour. We prove that deciding whether a given level can be completed is NP-complete. In fact, NP-completeness holds when only horizontal and vertical cuts are allowed, and each colour is used by at most 7 buttons. Our framework was also used in an NP-completeness proof when each colour is used by at most 4 buttons, which is best possible.