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Comparing Totals

Age 11 to 14 Short Challenge Level:

The positive integer $n$ is between $1$ and $20$.

Milly adds up all the integers from $1$ to $n$ inclusive.
Billy adds up all the integers from $n+1$ to $20$ inclusive.

Their totalsĀ are the same.

What is the value of $n$?

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.