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  1. 074A gold fund's vault holds nine coins that look identical, but one is slightly lighter than the others. Using a balance scale only twice, how do you find the light coin?Logic and numeracy brainteasersCoreIndian AMCsGlobal asset managers

    Try it first

    What should the first weighing be?

    Show the worked solution

    Weigh three coins against three. The pan that rises holds the light coin; if they balance, it is among the three left aside. Then weigh two of those three suspects against each other: the rising pan holds it, and a balance means it is the third. Each weighing has three possible outcomes, so two weighings can separate 3 x 3 = 9 coins.

    Why split into three groups rather than two?

    A shopkeeper asking a customer "yes, no, or not sure?" learns more from one answer than one asking only "yes or no?". A balance scale gives three answers, left pan rises, right pan rises, or the pans balance, so the best weighing splits the suspects into three equal groups and lets the scale tell you which group holds the light coin. Put coins 1 to 3 on the left and 4 to 6 on the right. If the left pan rises, the light coin is 1, 2 or 3. If the right pan rises, it is 4, 5 or 6. If they balance, it is 7, 8 or 9.

    Each weighing has three outcomes, so it splits the suspects into thirdsWeighing 1coins 1-3 vs coins 4-6Left pan risesLight coin is 1, 2 or 3Weigh 1 vs 2132Pans balanceLight coin is 7, 8 or 9Weigh 7 vs 8798Right pan risesLight coin is 4, 5 or 6Weigh 4 vs 5465Nine end points for nine coins: every coin is found in exactly two weighings.Why not halves? Weighing 4 against 4 wastes the third outcome, the balance.Two weighings tell apart at most 3 x 3 = 9 cases; with halves you need 3 weighings for 9 coins.
    Weighing three coins against three leaves three suspects whatever the scale shows, and weighing two of those three against each other identifies the light coin, so two weighings cover all nine coins.

    How do you know two weighings is the minimum, and how far does this go?

    Count outcomes. One weighing has 3 outcomes, two have 3 x 3 = 9, and there are 9 possible light coins, so two weighings are just enough. The number of weighings needed is the smallest w with 3 to the power w at least the number of coins: 9 coins need 2, 27 coins need 3. One weighing can only handle 3 coins, so you cannot do 9 in one. Splitting into halves wastes the balance outcome and needs three weighings for nine coins.

    The relationship
    3w≥N  ⇒  w=⌈log⁡3N⌉,N=9:  w=23^w \ge N \;\Rightarrow\; w = \lceil \log_3 N \rceil, \qquad N = 9:\; w = 2
    wthe number of weighings
    Nthe number of coins, one of them light
    3the outcomes of one weighing: left rises, right rises, balance
    What it says in wordsEach weighing can at most divide the suspects by three, so the weighings needed grow with the logarithm to base three of the number of coins.

    The desk lesson is about information, not coins: a test with three outcomes is worth more than one with two, if you design it to use all three. The limit of the puzzle is that you are told the odd coin is lighter. If it could be heavier or lighter, each coin has two possible states, there are more cases to separate, and the strategy needs more care, which is the classic twelve-coin follow-up.

    Where candidates lose it

    The trap is splitting in halves: four against four, then two against two, then one against one. It works but takes three weighings, and the interviewer asked for two. Candidates who think in halves have missed that a balance is an answer too.

    The second miss is getting the method but not the reason. Say that a weighing has three outcomes and two weighings give nine, which is why nine coins is the most two weighings can handle.

    What the interviewer asks next

    • How many coins can you handle with three weighings?
    • Twelve coins, one odd, and you do not know whether it is heavier or lighter. Can you find it in three weighings?
    • You have a digital scale that shows exact weights instead. How many weighings do you need for nine coins?