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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