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Ten students are sitting around a campfire. A teacher randomly assigns each student a different number from 1-10. Another teacher assigns a new number to each student with the requirement that the new number assigned to a student is equal to the students' old number plus the sum of the two numbers of his neighbors. Prove or disprove: one of the students must have a new number STRICTLY GREATER than 17.

What I have done so far: It is very easy to show that one of the students must have a new number strictly greater than 16. This can be done by assuming that each of the numbers is less than or equal to 16. A quick sketch is as follows:

Label the students $a_1, \dots a_{10}$. By contradiction, we have:

$a_1+a_2+a_3 \le 16$

$\dots$

$a_{10} +a_1+a_2 \le 16$

Summing up:

$3a_1+\dots +3a_{10} \le 16\cdot 10$

This is a contradiction as $a_1+\dots + a_{10} = 55$

But this argument breaks down when we try it for 17. Im not even sure that the proposition that a student must have a number more than 17 is even correct! Can someone prove it or find a counterexample?

3 Answers 3