Puzzle 1: Use four weights of 1, 2, 4 and 8 pounds. Any value from 0 to 15 can be formed by a suitable combination.
Puzzle 2: Use weights of 1, 2, 4, ..., 2k-1. Every value from 0 to 2k-1 can be formed. For instance, if the maximum weight is 63 = 2^6 - 1, then we use six weights of 1, 2, 4, 8, 16 and 32 pounds.
Puzzle 3: Sorry, no solution provided.
Puzzle 4: Use weights of 1, 3 and 9 pounds. Consider the following:
0 = 0 1 = + 1 2 = + 3 - 1 3 = + 3 4 = + 3 - 1 5 = 9 - 3 - 1 6 = 9 - 3 7 = 9 - 3 + 1 8 = 9 - 1 9 = 9 10 = 9 + 1 11 = 9 + 3 - 1 12 = 9 + 3 13 = 9 + 3 + 1
Puzzle 5: Sorry, no solution provided.
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