Financial Analysis puzzles, solved step by step

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  1. 067You have twelve coins that look identical. One is either heavier or lighter than the rest, you do not know which. Using a balance scale only three times, find the odd coin and say whether it is heavy or light.Logic and counting brainteasersHardConsulting-style caseCorporate finance

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    What is the key to fitting the puzzle into three weighings?

    Show the worked solution

    Weigh coins 1 to 4 against 5 to 8. If they balance, the odd coin is among 9 to 12 and two weighings against known good coins find it. If one side is heavy, eight cases remain, and a second weighing that swaps some coins across the pans cuts them to three or fewer. It works because three weighings give 27 outcomes and there are only 24 cases.

    Why can three weighings be enough?

    Picture a guessing game where every answer is higher, lower or spot on. A question that leaves you with one of three equal groups does three times the work of a yes or no. A balance has three outcomes, left heavy, right heavy or level, so three weighings can tell apart 3 x 3 x 3 = 27 situations, and this puzzle has only 24: any of 12 coins, heavy or light. The plan works only if every weighing splits the remaining cases as evenly as possible into three.

    That rules out six against six: one side always tips, so the balance outcome is wasted and twelve cases are left for two weighings that can separate nine. Four against four is right. If the left pan is heavy, the odd coin is one of 1 to 4 and heavy, or one of 5 to 8 and light: eight cases. If the pans balance, coins 1 to 8 are good and the odd coin is among 9 to 12, heavy or light: eight cases again.

    Use all three outcomes of every weighing: 24 cases fit inside 27Weighing 1: coins 1 2 3 4 against 5 6 7 824 cases: 12 coins x heavy or lightLeft pan heavy: 8 cases1-4 heavy or 5-8 lightW2: 1 2 5 against 3 6 9Balance: 8 cases9-12, heavy or lightW2: 9 10 11 against 1 2 3Right pan heavy: 8 cases5-8 heavy or 1-4 lightMirror of the left planleft heavybalanceright heavyL heavy1H 2H 6LW3: 1 vs 2R heavy3H 5LW3: 3 vs 9balance4H 7L 8LW3: 7 vs 8L heavy9-11 heavyW3: 9 vs 10balance12 H or LW3: 12 vs 1L light9-11 lightW3: 9 vs 10Same moves, heavy andlight swappedEach leaf holds at most 3 cases, and one weighing has 3 outcomes3 x 3 x 3 = 27 outcomes for 24 cases
    Four against four splits the 24 cases into three groups of eight, the second weighing splits each eight into groups of three or fewer, and the third weighing settles each group, because three weighings give 27 outcomes for 24 cases.

    How does the unbalanced branch get from eight cases to three?

    Suppose the left pan was heavy. Weigh 1, 2, 5 against 3, 6 and coin 9, which is known to be good. Coins 4, 7 and 8 sit out. If the left is heavy again, the culprit is 1 or 2 heavy, or 6 light; weigh 1 against 2, and if they balance it is 6. If the right is now heavy, it is 3 heavy or 5 light; weigh 3 against a good coin. If they balance, it is 4 heavy or 7 or 8 light; weigh 7 against 8, and the lighter one is the culprit, or 4 if they balance. The trick is that some coins stay, some swap pans and some leave, so each outcome points to a different group.

    If the first weighing balanced, weigh 9, 10, 11 against good coins 1, 2, 3. A heavy left means one of 9 to 11 is heavy, and 9 against 10 finds it. A light left means one is light, found the same way. A balance means coin 12 is odd, and weighing it against any good coin says whether it is heavy or light.

    Why would a finance interviewer ask this?

    It tests whether you count the possibilities before you start moving coins, which is the habit behind good diligence: design each question so that every possible answer tells you something new. The interviewer is happy with a clear first move and the counting argument, even if you need a minute for the unbalanced branch. The limit worth adding: with thirteen coins you can still find the odd coin in three weighings, but not always say whether it is heavy or light.

    Where candidates lose it

    Most candidates start with six against six, because halving feels efficient. It throws away the balance outcome, and from twelve cases left with two weighings there is no way back.

    The second loss comes in the unbalanced branch: weighing the suspect coins against each other without mixing in known good coins, so two outcomes point to the same group. Use the good coins from the first weighing as references and move coins between pans on purpose.

    What the interviewer asks next

    • What is the largest number of coins you can handle in three weighings if you only need to find the odd one?
    • You are told the odd coin is heavier. How many coins can three weighings handle?
    • How would you explain the counting argument to someone without using the word information?