There are 3 coins in a pile. Players alternate taking 1 or 2 coins. Whoever takes the last coin loses. You go first. Can you figure out the winning move by thinking backward from the end of the game?
Your turn — take 1 or 2 coins
Backward induction is a powerful technique for solving games that happen in stages, where players take turns and each player can see what happened before making their own move. Instead of trying to guess the best first move directly — which can be confusing when there are many possible paths the game could take — you start by imagining the very last decision in the game and figuring out the obviously correct choice there. Then you back up one step, and since you already know what happens at the end, that second-to-last decision becomes easy too. You keep working backward, stage by stage, until you reach the very first move, which now has an obvious best answer.
In the coin game above, the key insight is realizing that whoever is forced to take the very last coin loses. That means "leaving exactly one coin for your opponent" is a winning position. Once you know that, you can figure out what pile size you need to leave on each earlier turn to eventually force your opponent into that losing spot. This chain of reasoning — from the end of the game back to the beginning — is exactly what backward induction is.
Game theorists use backward induction to solve much more complicated sequential games, including chess endgames, business negotiations that happen over several rounds, and auctions with multiple stages. It's also the foundation for a concept called "subgame perfect equilibrium," which describes a strategy that is optimal not just from the very beginning, but at every single point in the game — even points that might never actually be reached. The technique reveals a surprising truth: to plan your best first move, you often need to think about the game's ending first.
Play the coin take-away game with a friend using real coins or buttons, starting with different pile sizes like 4, 5, or 7. For each pile size, work backward from "1 coin left equals a loss" to figure out the winning first move. See if you can find the pattern (hint: it depends on the pile size divided by 3) before checking with trial and error!
What is backward induction?
Backward induction solves multi-stage games by starting at the last move, finding the best choice there, then working backward to the first move.
Why start at the end instead of the beginning?
Because the best early move depends on what happens later — solving the end first makes each earlier decision easy to evaluate.
What games can backward induction solve?
It works best on sequential, finite-stage games where players take turns and see prior moves, like chess endgames and take-away games.
Does backward induction always predict what real people do?
Not always — people sometimes act emotionally or fairly rather than perfectly rationally, so real behavior can differ from the theoretical prediction.