538 Riddler: Hungry Bears
05 August ’16
My first intuition after reading this problem was "why would the bear ever reject the first fish?" If the first fish is big, then it makes sense to eat it; there are no guarantees the next fish will be as big. If the first fish is small, then eat it because the next fish is likely to be bigger, meaning we can eat it too. Some simple math confirms this. Let's say the first fish is of size. The probability that we eat the next fish is and the expected size of this fish is . The total meal size if we eat the first fish is is always greater than or equal , the expected meal size if we wait for the second fish. So if we're the bear, our optimal strategy is simple: eat whatever we see.
Given our rather simple strategy, let's come up with a generalized expression for the expected number of kilograms that the bear eats in a hour tour:
This is the harmonic series minus 1! In 2 hours, we expect the bear to eat kilograms. In 3 hours, we expect the bear to eat kilograms. The harmonic series is divergent, so the amount eaten by the bear does not converge on a value as N goes to infinity, even though the amount consumed by the bear in the Nth hour converges to 0.