Problem 6. Peter has 5 rabbit cages (the cages are in a row). It is known that there is at least one rabbit in each cage. We will call two rabbits neighbors if they sit either in the same cage or in adjacent ones. It turned out that each rabbit has either 3 or 7 neighbors. How many rabbits are sitting in the central cage?
Answer: 4 rabbits.

Solution. Let's number the cells from 1 to 5 from left to right.

Notice that the neighbors of the rabbit in the first cell are all the rabbits living in the first two cells. The neighbors of the rabbit in the second cell are all the rabbits living in the first three cells. The third cell cannot be empty, so the rabbit in the second cell has more neighbors than the rabbit in the first cell. Therefore, the rabbit in the first cell has three neighbors, and the rabbit in the second cell has seven neighbors. The difference between the number of neighbors of the rabbit in the second cell and the number of neighbors of the rabbit in the first cell is equal to the number of rabbits in the central cell. Therefore, there are $7-3=4$ rabbits in the central cell.

Such an arrangement of rabbits exists. It is sufficient to place 2 rabbits in the first, second, fourth, and fifth cells, and 4 rabbits in the third cell.