Problem 7.7. A circle of 130 trees has been planted: birches and lindens (both species are present). On each tree, there is a sign that reads: "Two different trees grow next to me." It is known that among all the trees, the statement is false on all lindens and on exactly one birch. How many birches could have been planted? List all possible options.
Answer: 87.

Solution. Let's divide all the trees into alternating groups of consecutive birches and consecutive lindens (by the condition, there are groups of both types).

Suppose there exists a group of at least 2 lindens. Then the truth would be written on the outermost of them (since it is between a birch and a linden) - a contradiction. Therefore, all groups of lindens consist of 1 tree. Then each linden is between two birches, and the lie is indeed written on it.

Now let's consider how many groups of birches there can be in terms of quantity.

- If a group has at least 4 birches, then on all non-edge (at least two) the lie is written (since each of them is between two birches) - a contradiction, since there is only one such birch.
- If a group has 3 birches, then only the middle one has the lie written on it.
- If a group has 2 birches, then the truth is written on each of them.
- If a group has 1 birch, then the lie is written on it (since it is between two lindens).

Therefore, there are several groups of 2 birches and exactly one group of 1 or 3 birches.

If there is a group of 1 birch, temporarily forget about it and one neighboring linden. Then the remaining $130-2=128$ trees will be divided into blocks of 3 trees: two birches and one linden. But 128 is not divisible by 3 - a contradiction.

If there is a group of 3 birches, temporarily forget about one of the birches in this group. Then the remaining $130-1=129$ trees will be divided into blocks of 3 trees: two birches and one linden. There are $\frac{129}{3}=43$ such blocks, and the total number of birches in them is $43 \cdot 2=86$. Returning the forgotten birch, we get that the total number of birches is $86+1=87$.

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