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**boldbrandywine** Let S = {1,2,...12,}. Suppose the elements of S are scattered at random around a circle. Show that there exists some string of three consecutive numbers whose sum is at least 20.

My attempt: I know that 1+...+12 = 78, and there are 12 ways to add three consecutive numbers around the circle, the smallest three adding to 6 (1+2+3), and the highest adding to 33 (10+11+12). Note however the circle doesn't have to be "nice" in the sense of a clock. So we could have 7+12+2 = 21 and so on. I'm just stumped on where to continue from here.