How to Solve Coin Weighing Puzzles, Step by Step

To solve a coin weighing puzzle, split the coins into three groups, not two. A balance scale has three possible results, so each weighing can rule out two thirds of the coins, and with that one idea you can find a single lighter coin among 9 coins in 2 weighings, or among 27 coins in 3.

This guide explains why three groups work, walks through the easy version step by step, then solves the famous 12-coin version where the odd coin may be heavier or lighter. A practice puzzle waits at the end.

What is a coin weighing puzzle?

In the simplest version you have a pile of coins that all look the same. One of them is fake and lighter than the others, and all the real coins weigh the same. You have a balance scale with two pans and no weights. The question is the smallest number of weighings that will always find the fake coin ^[1].

This puzzle first appeared in 1945, and many variations have followed since then ^[1]. In some the fake coin may be heavier or lighter. In others the fake coin can even change its weight from one weighing to the next ^[1].

Why split the coins into three groups?

A balance can only tell you one of three things: the left pan is lighter, the right pan is lighter, or the two pans stay level ^[1]. A good plan uses all three results.

Suppose you put one group on the left pan, one group of the same size on the right pan, and keep a third group off the scale:

• If the left pan rises, the fake coin is in the left group.
• If the right pan rises, the fake coin is in the right group.
• If the pans stay level, the fake coin is in the group you kept off the scale.

Whatever happens, you are left with one group. If the three groups are the same size, each weighing cuts the number of suspects to a third.

Splitting into two groups wastes a result. If you put half the coins on each pan, the pans can never stay level, so one of the three possible answers never gets used.

How many coins can you handle?

There is a neat way to see the limit. Write down the result of each weighing as a symbol: one for level, one for left lighter and one for right lighter. After all your weighings, the string of symbols must point to exactly one coin, because two different fake coins cannot give the same string ^[1]. With 2 weighings there are only 9 possible strings, and with 3 weighings there are 27. So no plan can ever handle more than 3 × 3 = 9 coins in 2 weighings, 27 coins in 3 or 81 coins in 4: one more factor of 3 for each extra weighing ^[1].

The three-group method reaches that limit exactly, which is why it is the best you can do.

Step by step: 9 coins, one lighter, 2 weighings

Number the coins 1 to 9.

Weighing 1. Put coins 1, 2 and 3 on the left pan and coins 4, 5 and 6 on the right. Keep 7, 8 and 9 off the scale.

• Left pan rises: the fake coin is 1, 2 or 3.
• Right pan rises: the fake coin is 4, 5 or 6.
• Level: the fake coin is 7, 8 or 9.

Weighing 2. Take the three suspects. Put one on each pan and keep the third off.

• The pan that rises holds the fake coin.
• If the pans stay level, the coin you kept off is the fake.

That is all. Two weighings, and every case ends with one coin.

What if the numbers do not divide by three?

Make the groups as equal as you can, with the two groups on the scale the same size. With 8 coins, weigh 3 against 3 and keep 2 off. If the pans stay level, weigh the last two against each other. If one pan rises, you have three suspects, and one more weighing finds the fake.

The groups on the pans must always match in size. If they do not, a level result means nothing, because the pans were not fair to begin with.

Planning ahead or deciding as you go

There are two kinds of plans. In an adaptive plan, each weighing can depend on the results so far. In an oblivious plan, every weighing is written down in advance, before you see any result ^[1]. The steps above are adaptive, since the second weighing depends on the first.

Oblivious plans are harder to design, but they are tidy: you could hand the full list of weighings to someone else and decide the answer later from the results. Researchers study both kinds, and for some versions of the puzzle an adaptive plan can do more than any fixed plan ^[1].

The harder version: 12 coins, heavier or lighter

Now one coin of 12 is fake, but you do not know whether it is heavier or lighter. You have 3 weighings to find the fake coin and say whether it is heavy or light. This version can be solved in 3 weighings ^[1].

This time each coin has two possible stories, heavy or light, so you have to keep track of both. Number the coins 1 to 12.

Weighing 1. Put 1, 2, 3 and 4 on the left. Put 5, 6, 7 and 8 on the right. Keep 9 to 12 off.

If the pans stay level, coins 1 to 8 are real, and the fake is among 9 to 12.

• Weighing 2: put 9, 10 and 11 on the left and three real coins, 1, 2 and 3, on the right.
• If they stay level, the fake is 12. Weighing 3: weigh 12 against a real coin to see whether it is heavy or light.
• If the left pan goes down, the fake is 9, 10 or 11, and it is heavy. Weighing 3: weigh 9 against 10. The heavier one is fake, and if they balance, it is 11.
• If the left pan rises, the fake is 9, 10 or 11, and it is light. Weigh 9 against 10 in the same way, but look for the lighter coin.

If the left pan goes down in weighing 1, either one of 1 to 4 is heavy or one of 5 to 8 is light. Coins 9 to 12 are real.

• Weighing 2: put 1, 2 and 5 on the left. Put 3, 6 and the real coin 9 on the right. Keep 4, 7 and 8 off.
• If they stay level, the suspects are 4 (heavy), 7 (light) or 8 (light). Weighing 3: weigh 7 against 8. The lighter one is fake. If they balance, coin 4 is the heavy fake.
• If the left pan goes down again, the suspects are 1 (heavy), 2 (heavy) or 6 (light). Weighing 3: weigh 1 against 2. The heavier one is fake. If they balance, coin 6 is the light fake.
• If the left pan rises, the suspects are 3 (heavy) or 5 (light). Weighing 3: weigh 3 against a real coin. If it goes down, 3 is the heavy fake. If it balances, 5 is the light fake.

If the right pan goes down in weighing 1, use the same plan with the roles of the two groups swapped: coins 5 to 8 are now the possible heavy ones and 1 to 4 the possible light ones.

Notice the trick in weighing 2 of the unbalanced case. Some coins stay where they were, some swap pans and some come off. Each kind of move gives a different result if that coin is the fake, so one weighing can tell many suspects apart.

How do you check your own plan?

Make a list of every possible fake: each coin, and each direction if the coin could be heavy or light. For the 12-coin puzzle that is 24 cases. Then run each case through your plan and write down the three results.

Two rules must hold. Every case must end with one answer, and no two cases may end with the same answer. A 2020 paper on logic puzzles calls a puzzle good when it has exactly one solution ^[2], and a weighing plan needs the same property: each set of results must point to one coin and only one.

If two cases share the same results, change one weighing so that those two coins sit in different places: one on the scale and one off, or one on each pan.

Common mistakes to avoid

• Splitting into two piles. It feels natural, but it throws away the level result.
• Uneven pans. Put the same number of coins on each pan, or a level result tells you nothing.
• Forgetting real coins. After a weighing, coins you have cleared are useful as known real coins. They let you fill a pan without adding new suspects.
• Stopping at the fake. If the puzzle asks whether the fake is heavy or light, make sure your last weighing also answers that.

Where do these puzzles go next?

Mathematicians keep inventing new twists. A 2025 paper describes how a 2015 study introduced a coin that can pretend to be real or fake at will, and it turns out such a coin can never be told apart from real coins, because it can always behave like one ^[1]. A later idea, the alternator, switches between real and light each time it goes on the scale, and unlike the coin that chooses freely, it can always be found ^[1]. A 2025 paper studied coins that switch between light and heavy, or cycle through light, heavy and real, and worked out how many coins a given number of weighings can handle ^[1].

Practice puzzle

You have 8 coins. One is lighter than the others. Can you find it in 2 weighings?

Try it before you read on. Weigh 3 against 3 and keep 2 off. If the pans stay level, weigh the last 2 against each other, and the lighter one is the fake. If one pan rises, take its three coins, weigh one against another, and the lighter one is fake. If those two balance, the third coin is the fake.

Then try the balance riddle below. It uses the same idea of level pans, but with shapes instead of coins.