133. a) The castle has the shape (in plan) of an equilateral triangle with a side of 100 meters. It is divided into 100 triangular halls. All the walls of the halls have the same length - 10 meters, and a door is made in the middle of each wall between the halls. Prove that if a person wants to walk through the castle, visiting each hall no more than once, then he will be able to visit no more than 91 halls.

b) Each side of an equilateral triangle is divided into $k$ equal parts. Lines are drawn through the division points parallel to the sides. As a result, the triangle is divided into $k^{2}$ small triangles. We call a "chain" a sequence of triangles in which no triangle appears more than once, and each subsequent triangle shares a side with the previous one. What is the maximum possible number of triangles in a "chain"?
133. We will color the triangles in two colors as shown in Fig. $62, a_{2}$

The number of black triangles is $k$ more than the number of white triangles. Therefore, the total number of white triangles is $\frac{1}{2}\left(k^{2}-k\right)$, and the number of black triangles is $\frac{1}{2}\left(k^{2}+k\right)$. In the chain of triangles, the colors alternate. Therefore, the number of black triangles in the chain is no more than $\frac{1}{2}\left(k^{2}-k\right)+1$. The total number of triangles in it is no more than

$$
\frac{1}{2}\left(k^{2}-k\right)+\frac{1}{2}\left(k^{2}-k\right)+1=k^{2}-k+1
$$

An example of a chain where the number of triangles is exactly $k^{2}-k+1$ is shown in Fig. 62, $a$.