Three boxes, all labelled wrong, one weighing

Question

Three sealed boxes weigh 1 kg, 2 kg and 3 kg. Each box carries a label saying "1 kg", "2 kg" or "3 kg", but a prankster has arranged the labels so that every box is labelled wrongly. The boxes feel identical in your hands. You have a two-pan balance and no reference weights. Determine the true weight of every box using the balance only once.

Accepted Answer

Weigh the box labelled "1 kg" against the box labelled "2 kg". Why only two arrangements are possible. Call the boxes by their labels: L1, L2, L3. Each label is wrong, so L1 is really 2 or 3, L2 is really 1 or 3, L3 is really 1 or 2, and all three true weights are different. If L1 = 2, then L3 must be 1 (its only other option, 2, is taken), which forces L2 = 3. If L1 = 3, then L2 must be 1 (3 is taken), which forces L3 = 2. So the two possible worlds are: - World X: L1 = 2 kg, L2 = 3 kg, L3 = 1 kg - World Y: L1 = 3 kg, L2 = 1 kg, L3 = 2 kg Arrangements where every item is in the wrong place a…

Three boxes, all labelled wrong, one weighing

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