Proving 3 consecutive random numbers (from 1-10) can add up over 16 using contradiction?

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!

• Your title is misleading. You aren't trying to prove that "3 consecutive numbers from $1$ to $10$ can add up over 16"; that's obvious ($10+9+8$ is more than 16). The point is that the integers from $1$ to $10$ are arranged randomly, "in any order," in a circle, and yet still there must be three consecutive (in the sense of adjacent) numbers that sum to at least $16$. That is what you're trying to prove. – symplectomorphic Nov 14 '17 at 22:29
• @symplectomorphic noted and edited title. Thank you – Mario Liden Nov 14 '17 at 22:34

1 Answer

Consider your arrangement. There are $10$ triples of consecutive numbers so if each sum were at most $16$ then the sum of all ten sums would be at most $160$. On the other hand, in that sum each of the ten digits is counted exactly three times. Thus the sum of the ten sums has to equal $$3\times (1+\cdots + 10)=3\times 55=165>160$$ A contradiction.

• thank you, my issue was realizing that the numbers were in a random order. Originally I was looking at 10+9+8 which is obviously over 16 so I was having difficulty thinking about what to prove. – Mario Liden Nov 14 '17 at 22:35