29. 10 elves and trolls were each given a token with a different number from 1 to 10 written on it. They were each asked what number was on their token and all answered with a number from 1 to 10. The sum of the answers was 36. Each troll told a lie and each elf told the truth. What is the smallest number of trolls there could be in the group? Note that if there were 0 trolls, then our sum is 55. That's just simply a sum of numbers from 1 to 10. Well, that's a little bit too high, so there must have been at least a few trolls. So what if there are two trolls? Well, if there are two trolls, these two trolls take the 9 and the 10 tokens. That makes up for the difference that we see. 55 minus 36 is equal to 19. But the trolls must have answered at least one, right? They all answered with a number from 1 to 10. None of them answered 0. So if we have elves that say the number is 1 to 8, and then trolls that make up the 9 and 10, the minimum answer has to be 38. So unfortunately, that doesn't work for us. Now, what if there are three trolls? Now these three trolls have the 8, 9, and 10 tokens. Their respective lies would need to sum to 8 because the elves would have the numbers 1 to 7. The sum of the numbers from 1 to 7 is 28. So these three trolls need to have lies that sum up to 8. They could say numbers, you know, maybe like 2, 3, and 3, and that would work. That would solve our situation. So therefore, the answer has to be B.

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