Given 11 different integers from 1 to 20. Prove that at least two of them are exactly 5 apart.

Question

Given 11 different integers from 1 to 20. Prove that at least two of them are exactly 5 apart.

Answer 1

Hello ISC B.,

I will get you started. You make pigeonholes listing all the possibilities for numbers that are 5 apart. As soon as you use a number in a pigeonhole, you cannot use it again. For example:

1, 6 2, 7 3, 8 4, 9 5, 10 BUT not 6, 11 since 6 is already used.

See if you can finish the list ending with 15, 20. You should get 10 pigeonholes. Since there are 11 numbers being selected, at least 2 of them will end up in the same pigeonhole.

See if you can fill in the rest of the details.