## Difference Payoff

You have 100 stones and ten buckets. After you place the stones in the buckets, two buckets will be chosen at random. You will then be paid a number of dollars equal to the difference in stones between the two buckets.

1. How do you arrange the stones to maximize your expected winnings?
2. If you want to guarantee you will win at least something (no two buckets hold the same number of stones), how can you maximize your winnings?  And how can you maximize your guaranteed amount (the smallest amount you can win)?

Your stones are now colored. You have 50 blue and 50 green stones. After placing them in the ten buckets, two buckets will be chosen at random, and you will be paid a number of dollars equal to the difference between the blue and green stones combined in the two buckets.

1. How do you arrange the stones to maximize your expected winnings?
2. Is it possible to distribute the blue and green stones to guarantee you will win at least something?  How can you maximize this?

Source: Original.

Solution
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