6-7. In a confectionery store, the saleswoman laid out 91 candies of several varieties in a row on the counter. It turned out that between any two candies of the same variety, there was an even number of candies. What is the smallest number of varieties there could have been?
Answer: 46.

Solution. We will prove that there could not have been three or more candies of the same type. Indeed, let candies of the same type $A, B$, and $C$ lie in that exact order. Suppose there are $2x$ candies between $A$ and $B$, and $2y$ candies between $B$ and $C$, then there are $2x + 2y + 1$ candies between $A$ and $C$, which is an odd number of candies, contradicting the condition.

Since there are no more than 2 candies of each type, there must be at least 46 types. Let's provide an example where there could be 46 types. For instance, the candies can be arranged in pairs: two of the first type, two of the second type, and so on for 45 pairs, with one more candy of the 46th type at the end. There are many other examples.