Twenty-one sticks. Two players. Take one, two or three at a time, and whoever takes the final stick wins. Both games below can be solved, and the Axiom tutor on the other side plays perfectly.
Every game begins with exactly twenty-one sticks, and nothing else on the table.
On your turn you remove one, two or three sticks. No more and no fewer, and you may not pass.
Whoever takes the final stick, wherever it happens to be sitting, wins the game.
Your opponent never blunders. An Axiom tutor plays both games optimally, so a win is proof that you have found the strategy rather than been lucky.
The sticks lie in a single row. On your turn, take one, two or three of them from the line. The final stick wins. Choose whether to move first or second, then play your way through: your removals are marked in blue, the tutor’s in brown, and the dark stick at the right is the one worth taking.
Work backwards from the end. Ask yourself what happens when exactly four sticks remain and it is your turn, then keep reasoning towards the opening move.
What to noticeIn this game the number of sticks left tells you almost everything you need to know. If you can control that number on your way down, you can control the ending.
The same twenty-one sticks, arranged in seven rows of three. On your turn, take one, two or three sticks, and they must all come from the same row. You may choose any row, in any order, and the final stick anywhere on the board wins. Tap a row to select it, then choose how many sticks to remove from that row alone. The tutor again plays optimally.
Two identical rows is an uncomfortable position for whoever has to move next. See whether you can hand that shape to your opponent, and then answer whatever they do.
What to noticeThe total left is no longer enough. Seven sticks sitting in one row and seven sticks spread across several rows are very different positions, so the arrangement is what matters here. If seven rows feel unwieldy, start smaller: two rows of three, and ask whether you would rather move or wait.
A large maze looks daunting from its entrance, yet the route often becomes clearer once you start from the exit. Sequential games behave the same way: identify the positions that are already lost, and the winning line appears. That habit of mind is called backward induction, and it is the same reasoning our tutors use every day with students who are learning to think rather than to memorise.
In “What 21 Coins Taught Me About Game Theory”, Dr. Revi Panidha traces these games from a five pound prize on an LSE classroom table to an Oxford interview, and sets out the full strategy for both boards above.
Written by Dr. Revi Panidha, award-winning LSE, Oxford and Sciences Po lecturer