Working Backwards: Solving a Problem From Its Answer

A chain hidden inside a story

Problems rarely arrive labelled "here is a chain of four operations". More often the chain is described as events.

A jug is emptied in three pourings. Each pouring removes half of whatever is in the jug at that moment, plus a further 10 ml. After the third pouring the jug is empty. How much did it hold?

Work from the end. Before the third pouring the jug held some amount whose half, plus 10 ml, emptied it. If half of it plus 10 equals all of it, then 10 ml equals the other half, so it held 20 ml. Before the second pouring, the jug held an amount that fell to 20 ml: half of it minus 10 equals 20, so half of it is 30, so it held 60 ml. Before the first pouring, half minus 10 equals 60, so half is 70, and the jug held 140 ml.

Check forwards: from 140, remove 70 + 10 = 80, leaving 60. From 60, remove 30 + 10 = 40, leaving 20. From 20, remove 10 + 10 = 20, leaving nothing. The chain closes.

Note what made the backward route easy and the forward route hard. Forwards, the unknown starting amount propagates through three stages and you end up managing an expression. Backwards, every stage starts from a number you already know.

Backward induction: working back from the end of a game

The same move solves a class of two-player games. Twenty-one counters sit on a table. Players alternate, removing one or two counters each turn, and whoever takes the last counter wins. Who should win?

Reason from the end. Facing 1 or 2 counters, you take them all and win. Facing 3, every move you make leaves 1 or 2 for your opponent, who then wins — so 3 is a losing position for whoever must move. Facing 4 or 5, you can take enough to leave exactly 3, handing your opponent the losing position. Facing 6, you cannot: any move leaves 4 or 5. The pattern is that the multiples of 3 are the losing positions, and 21 is a multiple of 3. The player who moves first therefore loses against accurate play, and the second player's strategy is simply to restore a multiple of 3 every turn.

Nothing about that argument required playing the game forwards. Working back from the terminal position — the definition of backward induction — is what makes the whole strategy visible at once.

Where it helps, and where it does not

Working backwards is worth trying when the end state is known and the start is not, when every step is individually reversible, and when there is a single unknown. It offers nothing when both ends are open, when a step loses information (squaring, rounding and taking an absolute value all destroy the sign or the detail you would need), or when the process branches. Those are honest limits, and recognising them quickly is most of what separates a usable heuristic from a superstition.

For practice, IQ Test & Brain Training Games — our own app, built by Reign Creative — ships a Reverse Calculation mode that presents exactly this task on a 75-second clock at Easy, Medium and Hard settings, so the chains lengthen as you improve. Because the items are generated rather than drawn from a fixed worksheet, you cannot learn the answers, which is the property that makes repetition useful here. The app records accuracy and response time per run, so you can see whether the order reversal has actually become automatic or whether you are still reasoning it out each time. The modes and the progress record are described on the IQ Test & Brain Training Games app page.

For graded problems rather than generated ones, the CEMC at the University of Waterloo publishes a free weekly problem set across five grade bands, many of which reward exactly this approach. More of our writing on solving methods sits in the education and brain training section, the broader taxonomy of puzzle types is in the guide to number sequence puzzles, and the full set of reasoning articles is collected in the logic and pattern puzzles hub.