Consider a pyramid with a square base where all edges are equal. Place a right circular cylinder ($m=2r$) inside this pyramid so that its base lies on the pyramid's base and its top touches the pyramid's side faces. What is the height of the cylinder?
If one leg of the right triangle is $a$, then the height is

$$
m=\sqrt{a^{2}-\left(\frac{a}{\sqrt{2}}\right)^{2}}=\frac{a}{\sqrt{2}}
$$

Based on the corresponding similar triangles:

$$
\frac{a}{\sqrt{2}}: \frac{a}{2}=2 r:\left(\frac{a}{2}-r\right)
$$

from which

$$
m=2 r=a(\sqrt{2}-1)
$$

Number of solutions: 23.