Twenty-six coins, one heavy, still only three weighings

Question

A bag holds 26 coins. Exactly one is heavier than the rest, and you know it is heavier. You have a two-pan balance and no other weights. Most people know the tidy version of this puzzle with 27 coins, where you split into three groups of nine. Twenty-six does not split evenly into thirds. How many weighings do you need to guarantee finding the heavy coin, and how do you split the piles when the numbers do not divide nicely?

Accepted Answer

Three weighings are enough, and two are not. Two weighings can tell apart at most 9 possibilities, and 26 suspects are more than 9, so at least three weighings are needed. Three weighings distinguish up to 27 possibilities, so 26 should fit, provided we split cleverly. The rule for uneven splits. At each step, put equal numbers on the two pans and leave the rest aside. If the pans tip, the culprit is in the heavy pan; if they balance, the culprit is in the leftover pile. To keep the worst case small, make all three piles as close to equal as possible, with the leftover pile never bigger than …

Twenty-six coins, one heavy, still only three weighings

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