In the old manor, the house is surrounded in a circle by tall trees - pines, spruces, and birches. There are a total of 96 trees. These trees have a strange property: of the two trees growing one apart from any coniferous tree - one is coniferous, and the other is deciduous, and of the two trees growing three apart from any coniferous tree - one is also coniferous, and the other is deciduous. How many birches are planted around the house?
Let's mentally remove half of the trees - those planted every other one. Then 48 trees will remain, and the condition will be as follows: of the two trees growing next to a coniferous one, one is coniferous and the other is a birch, and of the two trees growing one apart from the coniferous one, one is coniferous and the other is a birch.

Consider one of the planted coniferous trees. Let's call it tree 1 and number all the trees in order. If tree 1 is coniferous, then of trees 48 and 2, one is coniferous and the other is a birch. For definiteness, let's assume that tree 2 is a birch and tree 48 is coniferous. Consider tree 48. Next to it are tree 1 (coniferous) and 47 (so 47 is a birch). One tree apart from 1-47 (birch) and 3 (so 3 is coniferous). Tree 3 has two neighbors - 2 (birch) and 4 (coniferous). Now it is clear that the group of three trees - BHH (birch and two coniferous) always repeats. There are 48 trees in total, so this group will repeat 16 times.

Similarly, the number of birches in the remaining half of the trees is also 16. Therefore, 32 birches are planted around the castle.

## Answer

32 birches.