Four pieces of a clock face with equal sums?

Question

It is a well-known trick to cut a clock face into three pieces, each containing numbers that sum to 26.

Is it possible to cut the face into four pieces so that the numbers in each piece have the same sum? You may use any straight or curved cuts you like, as long as every number ends up in exactly one piece and no number is cut in half.

Accepted Answer

No, it is impossible, and no cleverness with the cuts can change that. The arithmetic obstacle. The numbers 1 to 12 add up to 78. Four pieces with equal totals would each need 78 / 4 = 19.5. But every piece contains whole numbers, so its total is a whole number, and a whole number cannot equal 19.5. Therefore no arrangement of cuts, however elaborate, produces four equal-sum pieces. Which counts are possible? The same test tells you which numbers of pieces are worth attempting. The target per piece is 78 divided by the number of pieces: - 2 pieces, 39 each: possible. A single chord entering b…

Four pieces of a clock face with equal sums?

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