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The Tale of the Dead Princess and the Seven Bogatyrs. Once, returning home in the evening, the bogatyrs gave the princess their prey - 29 gray ducks. Each brother shot at least one duck. All of them hunted a different number of ducks: the older the brother, the more game he shot. What was the catch of the eldest brother?

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Let's consider the seven smallest natural numbers.

## Solution

The youngest brother cannot shoot fewer than 1 duck, the next one fewer than 2, the next one fewer than 3, and finally, the oldest brother cannot shoot fewer than 7 ducks. This means that the minimum total number of ducks shot by the brothers is \(1+2+\ldots+7=28\) ducks. According to the problem, the brothers shot 29 ducks. Therefore, one of the brothers must have shot exactly one more duck. (If he had shot more than one, then someone else would have to shoot fewer than they currently did, which is impossible.) But only the oldest brother can shoot one more duck, because if anyone else did, there would be two brothers who shot the same number of ducks. Therefore, the oldest brother's catch is 8 ducks.