Find the Fake Coin: Balance Scale Puzzles Explained

Fake coin puzzles give you a pile of identical-looking objects, tell you one is the wrong weight, and hand you a two-pan balance. The question is always the same: what is the fewest number of weighings that is guaranteed to find the odd one out, no matter which one it is?

NRICH, the mathematics enrichment project at the University of Cambridge, has two classic versions. Spot the Fake asks how many coins you can handle in two weighings when the fake is known to be heavier. Find the Fake asks about 12 coins where the fake could be heavier or lighter. Below is the one idea that unlocks this whole family, followed by three original puzzles. The answers were checked by testing every possible weighing plan and every possible position of the fake.

1. The key idea: a balance has three outcomes

Most people think of a weighing as a yes or no question. It is not. A two-pan balance can:

1. tip left,
2. tip right, or
3. stay level.

So each weighing can split your suspects into three groups: the coins on the left pan, the coins on the right pan, and the coins left off the scale. If you know the fake is heavier, a tip tells you it is on the heavy side, and a level balance tells you it is in the pile you set aside.

That means one weighing can narrow 3 suspects to 1, two weighings can narrow 9 to 1, and three weighings can narrow 27 to 1. In general, k weighings can handle up to 3 to the power k coins when you already know whether the fake is heavier or lighter, and no plan can do better, because k weighings only produce 3 to the power k possible result patterns.

2. How to split the coins

Divide the suspects into three groups that are as equal as possible, with the two groups on the pans exactly equal in count. The group sizes matter only in one way: after the weighing, the largest group you might be left with should be small enough to finish in the weighings you have left.

3. Puzzle 1: eight marbles

You have 8 marbles. One is slightly lighter than the others, which all weigh the same. Using a two-pan balance, what is the fewest number of weighings that guarantees you find the light marble?

4. Puzzle 2: twenty coins

You have 20 coins. One is slightly heavier. What is the fewest number of weighings that always finds it?

5. Puzzle 3: seven bags and a digital scale

You have 7 bags, each holding plenty of coins. Every genuine coin weighs 10 grams. All the coins in exactly one bag are fakes that weigh 9 grams each. This time you have a digital scale that shows an exact weight, not a balance. What is the fewest number of weighings that tells you which bag holds the fakes?

6. Answers

Puzzle 1: 2 weighings. One weighing is not enough, because it can only separate 3 groups and you have 8 marbles. Two weighings can handle up to 9, so 2 is the target. First weighing: put 3 marbles on each pan and keep 2 aside.

• If one pan rises, the light marble is among those 3. Weigh 1 against 1 of them. If one rises, that is the marble; if they balance, it is the third one.
• If the pans balance, the light marble is one of the 2 set aside. Weigh them against each other and the rising pan holds it. Every case ends after 2 weighings.

Puzzle 2: 3 weighings. Two weighings can separate at most 9 coins, and you have 20, so 2 is impossible. Three weighings can separate up to 27, so 3 is enough. First weighing: 7 coins against 7, with 6 aside. Whichever group holds the heavy coin has at most 7 coins, and 7 is no more than 9, so two more weighings finish the job using the method from Puzzle 1 (3 against 3 with 1 aside, then 1 against 1).

Puzzle 3: 1 weighing. A digital scale gives a number, not just left, right, or level, so the three-outcome limit no longer applies. Take 1 coin from bag 1, 2 coins from bag 2, 3 from bag 3, and so on up to 7 coins from bag 7. That is 28 coins in total. If they were all genuine they would weigh 280 grams. Each fake is 1 gram light, so the scale comes up short by exactly as many grams as the number of fake coins on it, which is the bag number. A reading of 276 grams, for example, is 4 grams short, so bag 4 holds the fakes.

7. A harder challenge

NRICH's Find the Fake uses 12 coins where the fake might be heavier or lighter, and you do not know which. That doubles the possibilities to 24 (12 coins times 2 directions), which is still under 27, and NRICH's published student solutions confirm that 3 weighings are enough. The plan is trickier, because a tip no longer tells you directly which side is suspect. A good first move is 4 against 4 with 4 aside.

8. Common mistakes

• Splitting in half. Weighing 4 against 4 for Puzzle 1 wastes the third outcome. You learn which half is light but never use the "set aside" pile.
• Unequal pans. If the pans hold different numbers of coins, a tip tells you nothing, because the fuller pan would be heavier anyway.
• Counting the lucky case. "Fewest weighings that guarantees" means the worst case. Getting lucky on the first weighing does not count.

9. Key takeaways

• A two-pan balance has three outcomes, so each weighing can split suspects into three groups.
• With the fake's direction known, k weighings can find it among up to 3 to the power k coins, and no plan can do better.
• Split into three nearly equal groups, with the same number of coins on each pan.
• A scale that reads an exact weight changes the rules: taking a different number of coins from each bag can solve the problem in a single weighing.
• If the fake could be heavier or lighter, count both possibilities for every coin before deciding how many weighings you need.