Five Weights on Both Pans
You have a two-pan balance and a set of exactly five weights, whose masses you may choose freely. To weigh an object, you put it on the left pan and may place any of the weights on either pan, and the object's mass is read off when the pans balance exactly. Find the largest integer N such that a set of five weights exists that can weigh every integer mass 1, 2, ..., N.
Related problems & prerequisites
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Source: Bachet's weights problem (1612, 'Problemes plaisants'); the ternary solution is classical. Statement written for AxiomIQ.