The Twelve Coin Puzzle: Solution

The rules

Tap a coin to put it on the left pan, again for the right pan and again to take it off. Each weighing needs the same number of coins on each side. When you know the fake, press Answer, tap it and, where asked, say whether it is heavier or lighter.

The game plays fair but tough. The fake is never chosen in advance: after each weighing the scale gives whichever result leaves you the most to do, as long as some coin could still be the fake. So a lucky guess never works, and an answer only counts when the weighings prove it. A wrong answer counts as one more weighing.

Think in threes

A weighing has three results: left side down, right side down, or balanced. So the best weighing splits the possibilities into three equal groups: some coins on the left, the same number on the right, and the rest off the scale. If the fake is heavier, a down side holds it; if the scale balances, it is among the coins you left off.

That gives the famous powers of three: one weighing can find a heavier fake among 3 coins, two weighings among 9 and three among 27.

• Three coins: weigh coin 1 against coin 2. The heavier side holds the fake, and if they balance it is coin 3.
• Nine coins: weigh 1 2 3 against 4 5 6. The heavier group, or 7 8 9 if they balance, holds the fake. Then weigh two coins of that group against each other.
• Twenty-seven coins: weigh 9 against 9, then 3 against 3 from the right group of nine, then 1 against 1.

When the fake may be lighter

If the fake could be heavier or lighter, every coin brings two possibilities, and you have to keep track of which ones are left. Two things help. A coin that has been on the down side once and on the up side once must be good. And coins already proven good are very useful: put them on the scale to make the groups the size you need. The notes in the game let you mark coins as good, maybe heavier or maybe lighter.

Three coins, heavier or lighter: weigh 1 against 2. If they balance, coin 3 is fake: weigh it against coin 1 to see which way. If 1 goes down, either 1 is heavier or 2 is lighter: weigh 1 against 3. A balance means 2 is lighter; otherwise 1 is heavier. If 2 goes down, do the same the other way round.

Four coins, just find it: weigh 1 against 2. If they balance, weigh 3 against 1: a tip means coin 3, a balance means coin 4. If they don’t balance, weigh 1 against 3: a tip means coin 1, a balance means coin 2.

The twelve coins in three weighings

Twelve coins, one fake, heavier or lighter, and you must say which. That is 24 possibilities, and three weighings have 27 results, so there is almost no room to waste.

Weighing 1: 1 2 3 4 against 5 6 7 8.

If it balances, coins 1 to 8 are good and the fake is among 9 to 12.

• Weighing 2: 9 10 11 against 1 2 3. If it balances, coin 12 is fake: weigh 12 against 1 to see whether it is heavier or lighter.
• If 9 10 11 go down, one of them is heavier: weigh 9 against 10. The heavier one is fake, or coin 11 if they balance.
• If 9 10 11 go up, one of them is lighter: weigh 9 against 10. The lighter one is fake, or coin 11 if they balance.

If the left side goes down, either one of 1 to 4 is heavier or one of 5 to 8 is lighter. Coins 9 to 12 are good.

• Weighing 2: 1 2 5 against 3 6 9.
• If the left goes down again, the fake is 1 or 2 (heavier) or 6 (lighter). Weigh 1 against 2: the heavier one is fake, or 6 is lighter if they balance.
• If the right goes down, the fake is 3 (heavier) or 5 (lighter). Weigh 3 against 9: if 3 goes down it is heavier, if they balance 5 is lighter.
• If it balances, the fake is 4 (heavier), 7 or 8 (lighter). Weigh 7 against 8: the lighter one is fake, or 4 is heavier if they balance.

If the right side goes down, the same plan works with heavier and lighter swapped: weigh 1 2 5 against 3 6 9 again. Left down means 3 is lighter or 5 is heavier; right down means 1 or 2 is lighter or 6 is heavier; a balance means 4 is lighter or 7 or 8 is heavier. One more weighing settles each case, exactly as above.

Thirteen coins and a spare

With thirteen coins there are 26 possibilities, and without help three weighings can’t always say heavier or lighter. A spare coin known to be good fixes that.

Weighing 1: 1 2 3 4 5 against 6 7 8 9 and the spare. Nine coins are on the scale, which splits the possibilities 9, 9 and 8.

• If it balances, the fake is among 10 to 13. Weigh 10 11 12 against 1 2 3, then finish as with the twelve coins: one of three is heavier or lighter, or coin 13 is fake.
• If the left goes down, one of 1 to 5 is heavier or one of 6 to 9 is lighter. Weigh 1 2 6 7 against 3 8, the spare and coin 10. Left down: 1 or 2 heavier, or 8 lighter (weigh 1 against 2). Right down: 3 heavier, or 6 or 7 lighter (weigh 6 against 7). Balance: 4 or 5 heavier, or 9 lighter (weigh 4 against 5).
• If the right goes down, use the same second weighing with heavier and lighter swapped.

A habit for every puzzle

Before each weighing, ask what you would do after each of the three results. If one result would leave more possibilities than the weighings you have left can handle, change the weighing. That one question is the whole secret of the twelve coin puzzle.