In a set of weights, no weight exceeds 10 kg. If the set is divided arbitrarily into two groups, the combined mass of one of these groups also will not exceed 10 kg. What’s the greatest possible mass of the full set of weights?
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Three 10 kg weights will get the result we want — dividing this set in two will always leave us with one group that weighs no more than 10 kg. To see that a more massive set must fail, choose any weight in that massive set and add more weights, one by one, until the total mass M in that group exceeds 10 kg. If we are to succeed, the remaining mass m must not exceed 10 kg. But if a is the mass of the last weight we added to the first group, we know that a ≤ 10 kg and that M – a ≤ 10 kg. So the combined mass of all the weights is M + m = (M – a) + a + m ≤ 30 kg.
From the November-December 1994 issue of Quantum.
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