A $120 \mathrm{~m}^{3}$ swimming pool needs to be built using $1 \times 1 \mathrm{~m}$ cement sheets. The sheets must not be cut. What should be the dimensions of the swimming pool so that the least number of sheets is required?
If the length of a pool is $a \mathrm{~m}$, the width is $b \mathrm{~m}$, and the depth is $c \mathrm{~m}$, then its volume is $a b c \mathrm{~m}^{3}$, and the area of the sheets to be tiled is $(a b+2 c(a+b)) \mathrm{m}^{2}$. Since the sheets cannot be broken, the values of $a, b, c$ can only be integers. Due to the choice of the roles of the letters, $a \geqq b$. We will show that it is enough to deal with the case $b \geqq c$. If this were not the case, by swapping the roles of $b$ and $c$, we would get a pool for which $a(c-b)$ fewer sheets are needed.

Accordingly, we are looking for positive integers $a, b, c$ such that $a \geqq b \geqq c, a b c=120$, and $T=a b+2 c(a+b)$ is as small as possible. Since for $c \geqq 5$, the value of $a b c$ is at least 125 due to $a \geqq b \geqq c$, $c$ can only take four values: $1,2,3$ and 4. For any value that $c$ takes, $T$ will be the smallest when $a+b$ is minimal, since for a fixed $c$, the value of the product $a b=\frac{120}{c}=s$ is also constant.

The sum $a+b$ is minimal when its square is. However,

$$
(a+b)^{2}=\left(\frac{s}{b}-b\right)^{2}+4 s
$$

is the smallest for a given $s$ when $b$ is the largest possible value for which $b \leqq \frac{s}{b}$, i.e., $b \leqq \sqrt{s}$. For each value of $s$, therefore, we must choose for $b$ the largest divisor of $s$ that is not greater than $\sqrt{s}$. Thus, we get the following cases:

| $c$ | $s$ | $b$ | $a$ | $T$ |
| :---: | :---: | :---: | :---: | :---: |
| 1 | 120 | 10 | 12 | 164 |
| 2 | 60 | 6 | 10 | 124 |
| 3 | 40 | 5 | 8 | 118 |
| 4 | 30 | 5 | 6 | 118 |

Therefore, we need at least 118 cement sheets. With this many, we can build two types of pools, one 4 meters deep, 5 meters wide, and 6 meters long, and the other 3 meters deep, 5 meters wide, and 8 meters long.

Note. If we take seriously what the text says, of course we would choose the latter dimensions, as the depth of a swimming pool beyond a certain limit does not matter much, but the length does. However - as in any such case - the reality of the problem can be questioned from the outset.