[Math] Proving an interesting feature of any $1000$ different numbers chosen from $\{1, 2, \dots,1997\}$

pigeonhole-principle

Assume you choose $1000$ different numbers from the group $\{1, 2,
\dots,1997\}$.

Prove that within the $1000$ chosen numbers, there is a couple which
sum is $1998$.

I defined:

  • pigeonholes: possible sums.
  • pigeons: the $1000$ different numbers.

Is this definition good or there is something better?

Best Answer

Look at the pairs $(1,1997)$, $(2, 1996)$, and so on up to $(998,1000)$, together with the singleton $999$. These are the pigeonholes. Every number belongs to exactly one pigeonhole. If we choose $1000$ numbers, then since there are only $998$ pairs and $1$ singleton, at least $2$ of our numbers end up being in the same pigeonhole, that is, adding up to $1998$.

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