Mat was digging a hole. Pat asked him how deep the hole would be. Mat answered with a riddle: "I am $90 \mathrm{~cm}$ tall and I have just dug half of the hole. When I finish digging the hole, the top of my head will be as deep below the ground as it is now above the ground." How deep was the hole Mat was digging?
From Mat's explanation, it follows that his height is exactly between the depth of the entire pit and half of it. That is, $90 \mathrm{~cm}$ is equal to three quarters of the depth of the entire pit. Mat dug a pit that was $\frac{4}{3} \cdot 90=120(\mathrm{~cm})$ deep.

Suggested grading. 3 points for recognizing that $90 \mathrm{~cm}$ corresponds to $\frac{3}{4}$ of the depth of the entire pit; 3 points for calculating the depth of the pit.

Alternative solution. If we denote $j$ as the depth of the entire pit, then according to Mat's explanation, the distance from the top of Mat's head to the surface of the ground can be expressed as

$$
j-90=90-\frac{1}{2} j .
$$

From this, we get $\frac{3}{2} j=180$, so $j=120$. Mat dug a pit that was $120 \mathrm{~cm}$ deep.

Suggested grading. 3 points for expressing the information from the problem using the unknown $j; 3$ points for calculating the depth of the pit.

Note. If we denote $v$ as the distance from the top of Mat's head to the surface of the ground and $j$ as the depth of the entire pit, then Mat's explanation can be written as

$$
90=\frac{1}{2} j+v, \quad 90=j-v .
$$

By usual manipulations, the equivalent equation to (1) can be obtained from this system.