King Ivan
There was a king with 4 sons. Each of these sons had 0 or more sons, and so did each of his grandsons, and so on. We know that, going down the family tree, among all his descendants
- 10 had exactly 3 sons each
- 10 had exactly 2 sons each
- 10 had exactly 1 son each.
However, without knowing which of his sons (or grandsons, and so on) had 1, 2, or 3 sons, we can uniquely determine the total number of sons, grandsons, great-grandsons, and so on this king had!
Find the total number.
The answer is 64.
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1 solution
There are ten descendants of the king with three sons each, so 3 ⋅ 1 0 = 3 0 . Similarly, there are ten descendants of the king with two sons each, so 2 ⋅ 1 0 = 2 0 . By the same logic, there are ten descendants of the king with one son each, so 1 ⋅ 1 0 = 1 0 . Add up the numbers: 3 0 + 2 0 + 1 0 = 6 0 . Furthermore, the king himself has 4 sons, so add that number in: 6 0 + 4 = 6 4 . The number of descendants of this king is 6 4 .