3Among the unseen beasts that left tracks on unknown paths, there was a herd of one-headed Thirty-four-legged creatures and three-headed Dragons. In total, the herd has 286 legs and 31 heads. How many legs does a three-headed Dragon have?

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Think about how many Thirty-four-legged creatures can be in a herd.

## Solution

In the herd, there are 286 legs, which means that the number of Thirty-four-legged creatures cannot be more than 8, since $34 \cdot 9 > 286$. In the herd, there are 31 heads, and each Dragon has three heads, so the number of Thirty-four-legged creatures, when divided by 3, should leave a remainder of 1. Therefore, the number of Thirty-four-legged creatures can be either 1, 4, or 7. Knowing the total number of heads, we can determine the number of Dragons - which is either 10, 9, or 8, respectively.

In the first case, 10 Dragons have 252 legs, in the second case, 9 Dragons have 150 legs, and only in the third case does each Dragon have an integer number of legs, specifically 6 (48 legs for 8 Dragons). Thus, each Dragon has 6 legs.