SOLUTION: Place numbers 1 through 12 randomly around a circle. Show it is impossible for all sets of three consecutive numbers to sum to 19 or 20.

SOLUTION: Place numbers 1 through 12 randomly around a circle. Show it is impossible for all sets of three consecutive numbers to sum to 19 or 20.

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Question 441367: Place numbers 1 through 12 randomly around a circle. Show it is impossible for all sets of three consecutive numbers to sum to 19 or 20.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Suppose that, in clockwise order,

are the twelve numbers satisfying

.

Without loss of generality, let

. It follows that

, because if it equaled 19, then

. This implies

.

Hence,

, because if it was 20, then

, so

. By a simple inductive argument,

This implies contradiction, so it cannot occur.

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