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Math puzzles

Puzzles you can play on the page, and the mathematics that explains them

These are the classics of recreational mathematics: puzzles with pieces, boards and moves. Most of them can be played right here. Play first; then read how a little mathematics turns a puzzle that seems to need luck or patience into one you fully understand: how many moves it takes, whether it can be solved at all, and why.

Tower of Hanoi Move the tower; the fewest moves doubles with each disc The 15 puzzle Slide the tiles; half of all starts are impossible Magic squares Every line the same total, from China to Dürer Eight queens Eight queens, no attacks: 92 ways The knight's tour Visit every square once with a knight Lights Out Press to toggle; solved by algebra with only 0 and 1 The Monty Hall problem Stay or switch? The answer that fooled experts The bridges of Königsberg The walk that started graph theory

What the mathematics gives you

A count. How many moves does the Tower of Hanoi need? How many ways are there to place eight queens? Counting turns "it takes a while" into an exact number, and often shows why the number grows so fast. See Counting.

A proof of impossibility. Some puzzles can't be solved, and no amount of trying will show that. A single clever quantity that never changes, an invariant, can. The 15 puzzle and the bridges of Königsberg are the two most famous examples. See Invariants and parity.

A method. Backtracking for the queens, a rule of thumb for the knight, a few lines of algebra for Lights Out: each puzzle comes with a way of solving every case, not just the one in front of you. See Search and backtracking.