List of impossible puzzles

This is a list of puzzles that cannot be solved. An impossible puzzle is a puzzle that cannot be resolved, either due to lack of sufficient information, or any number of logical impossibilities.

• 15 Puzzle – Slide fifteen numbered tiles into numerical order. It is impossible to solve in half of the starting positions.^[1]
• Five room puzzle – Cross each wall of a diagram exactly once with a continuous line.^[2]
• MU puzzle – Transform the string MI to MU according to a set of rules.^[3]
• Mutilated chessboard problem – Place 31 dominoes of size 2×1 on a chessboard with two opposite corners removed.^[4]
• Coloring the edges of the Petersen graph with three colors.^[5]
• Seven Bridges of Königsberg – Walk through a city while crossing each of seven bridges exactly once.^[6]
• Squaring the circle, the impossible problem of constructing a square with the same area as a given circle, using only a compass and straightedge.^[7]
• Three cups problem – Turn three cups right-side up after starting with one wrong and turning two at a time.^[8]
• Three utilities problem – Connect three cottages to gas, water, and electricity without crossing lines.^[9]
• Thirty-six officers problem – Arrange six regiments consisting of six officers each of different ranks in a 6 × 6 square so that no rank or regiment is repeated in any row or column.^[10]