Calvin the Frog is trying to hop across a river. The river has 10 lily pads on it, and he must hop on them in a specific order (the order is unknown to Calvin).

If Calvin hops to the wrong lily pad at any point, he will be thrown back to the wrong side of the river and will have to start over.

Assuming Calvin is infinitely intelligent, what is the minimum number of hops he will need to guarantee reaching the other side?

The answer is 176.

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Calvin needs (at most) $i(10-i)$ hops to determine the ith lily pad he should jump to, then an additional 11 hops to actually get across the river. Thus he requires $\displaystyle \sum_{i=1}^{10}(i(10-i))+11=\boxed{176}$ hops to guarantee success.