12 Coins Puzzle Answer: Find the Odd Coin in 3 Weighings [Solved]

The 12 coins puzzle answer is to split the coins into three groups of four and weigh four against four. The result tells you which group holds the odd coin, and two more weighings name it and its direction.

Here is the question, as it appears in interview reports on Glassdoor, including a ZS Associates interview report. You have 12 identical-looking coins. Eleven weigh the same. One weighs a different amount, but you do not know whether it is heavier or lighter. With a balance scale and three weighings, find the odd coin and say whether it is heavy or light.

The Question, Stated Plainly

A balance scale gives no numbers. It only says which pan is heavier, or that the pans match. Each weighing gives one of three answers: left down, right down, or level. Three weighings give three answers in a row. Those answers must pick one coin out of twelve and name its direction.

The unknown direction is the hard part. If you knew the odd coin was heavier, the puzzle would be simple. Weigh four against four, keep the heavier group, weigh two against two, and finish one against one. But a dropped pan has two meanings. It can mean a heavy coin on that side or a light coin on the other side.

The 12 Coins Puzzle Answer in 3 Weighings

Number the coins 1 to 12. Each weighing is chosen from the result before it. Every coin you prove genuine becomes a tool: only a known good coin can be weighed against a suspect.

Weighing 1: four against four

Put coins 1, 2, 3 and 4 on the left pan. Put coins 5, 6, 7 and 8 on the right. Leave 9, 10, 11 and 12 off the scale. Three results are possible, each sending you down a different path.

Case A: the scale balances

If the pans match, all eight coins on the scale are genuine. The odd coin is one of the four set aside: 9, 10, 11 or 12. For weighing 2, put 9, 10 and 11 on the left pan. Put known good coins 1, 2 and 3 on the right. If it balances, the odd coin is 12. Weighing 3 is 12 against 1: a drop means heavy, a rise means light.

If the left pan drops in weighing 2, the odd coin is 9, 10 or 11, and it is heavy. Weighing 3 is 9 against 10: the heavier coin is the odd one, or 11 if they balance. If the left pan rises, the same weighing finds the light coin instead.

Case B: the left pan drops

If the left pan drops, the odd coin is heavy among 1, 2, 3 and 4. Or it is light among 5, 6, 7 and 8. Coins 9 to 12 are now known good. For weighing 2, put 1, 2 and 5 on the left pan. Put 3, 6 and 9 on the right. Coin 9 is known good. Each side holds coins that could push it down and coins that could push it up.

If weighing 2 balances, the odd coin is one of the three left off the scale. It is 4, which would be heavy, or 7 or 8, which would be light. Weighing 3 is 7 against 8: the lighter coin is the odd one, or 4 if they balance.

If the left pan drops in weighing 2, the odd coin is 1 or 2 and heavy. Or it is 6 and light. Weighing 3 is 1 against 2: the heavier coin is the odd one, or 6 if they balance.

If the left pan rises in weighing 2, the odd coin is 5 and light, or 3 and heavy. Weighing 3 is 5 against 9. A rise names 5 as the light coin, and a balance names 3 as the heavy one.

Case C: the left pan rises

If the left pan rises, the picture is the mirror image. The odd coin is light among 1, 2, 3 and 4, or heavy among 5, 6, 7 and 8. Use the same second weighing: 1, 2 and 5 on the left, 3, 6 and 9 on the right.

Run the same second weighing, and read it as the mirror image of Case B. If weighing 2 balances, the suspects are 4, which would be light, or 7 or 8, which would be heavy. Weighing 3 is 7 against 8. If the left pan drops, the odd coin is 5 and heavy, or 3 and light. Weighing 3 is 5 against 9. A rise in the left pan means the odd coin is 1 or 2 and light, or 6 and heavy. Weighing 3 is 1 against 2. In each finish, the coin that tips the scale in the direction its case predicts is the odd one. A balance points at the coin left out.

That covers every path. Three results at the first weighing, three at the second in each case, and one final weighing each time.

Why the 12 Coins Puzzle Answer Needs 3 Weighings

Three weighings are enough. Each weighing has three possible results, so three weighings give 27 different result patterns. The odd coin could be any of 12 coins, heavy or light: 24 possibilities. Since 24 fits inside 27, three weighings carry enough information. Two weighings give only 9 patterns, so they are impossible. The same counting shows 12 is the largest number that works. A balance leaves at most 4 suspects aside, and a tip leaves at most 4 per side. So 4 plus 4 plus 4 is the limit.

The Trap Most Candidates Fall Into

The first trap is assuming the odd coin is heavy. Many candidates weigh as if a dropped pan always points at the odd coin. But half the time the odd coin is light. Say out loud that a dropped pan has two explanations before you touch a coin.

The second trap is splitting the coins into two groups of six. Six against six finds which half holds the odd coin, but leaves six suspects and two weighings. Groups of four split twelve into three equal parts, the natural split for a scale with three outcomes.

The third trap is forgetting the known good coins. After the first weighing, the cleared coins are the most useful coins you own. This solution uses coin 9, a proven good coin, as a reference. It uses the same coin again in the third weighing. Candidates who leave them on the table end up comparing two suspects, which tells them nothing.

What the Interviewer Is Actually Testing

This question is not a test of coin knowledge. It is a test of how you handle information that comes in threes. First, the interviewer watches whether you notice that a balance scale has three outcomes, not two. The instinct is to split things in half, but the strong candidate splits into three groups.

Second, they watch whether you plan for the unknown direction instead of assuming it away. A good interviewer lets a candidate build a heavy-only plan, then asks what happens if the odd coin is light. The candidate who planned for it answers at once; the other starts again from zero.

Third, they watch whether you work the cases to the end. The first weighing is the easy part. The interview is won in the second, where you mix suspects and known good coins. Each result leaves at most three suspects for one final weighing. That is systematic case work, done out loud.

Finally, the follow-up matters as much as the answer. A good interviewer asks whether 13 coins could be done in three weighings. They may also ask how many coins four weighings could handle. The counting above gives both answers: 13 cannot be done, and four weighings handle up to 39 coins.

Try Another Puzzle From the Series

If you liked the balance scale, try the burning rope puzzle. Two 60-minute ropes have to measure exactly 45 minutes: Burning Rope Puzzle: Measure Exactly 45 Minutes (Solved).

For the same case-by-case discipline in a different setting, try this river crossing. Three missionaries and three cannibals must cross in a two-person boat: Missionaries and Cannibals Puzzle Answer: 11 Trips [Solved].

Frequently Asked Questions

What if you are told the odd coin is heavier?

Then the 12 coins puzzle answer gets much simpler, but it still takes three weighings. Weigh four against four, keep the heavier group, weigh two against two, and weigh one against one. The direction is unknown in the hard version, and that is why interviewers ask it that way.

Where does this question come from?

The 12 coins puzzle is a classic balance-scale problem. It appears in interview question collections, including a ZS Associates interview report on Glassdoor. The solution above is original, worked out from the weighing rules.