If we place the numbers 1,2,3,...,10 in a circle (as in a clock) in any order, then at least three consecutive numbers in our circle will sum to 16 or higher. Present an argument on why this is true; don’t quote specific examples as proof, since they won’t cover all 9!/2 = 181400 possible configurations. (Hint: One direction to approach might be by contradiction. Assume there is a configuration where no three consecutive numbers sum to 16 or higher; e.g. The maximum any three consecutive numbers in this configuration could sum to would be 15. Hence the sum of all triples of consecutive numbers can’t exceed a certain number. Then reach a contradiction by arguing that the sum of all triples of consecutive numbers must be larger than what is allowed.)
I understand how a normal proof by contradiction works but this has made me very confused. Thank you!