The edges and the space diagonal of a rectangular cuboid are integers, the measures of its surface area and volume are equal. A square with an area equal to the surface area can be cut into 6 pieces in a suitable way, and from each piece, we can obtain a face of the cuboid. What are the lengths of the edges?

 

The edges and the space diagonal of a rectangular cuboid are integers, the measures of its surface area and volume are equal. A square with an area equal to the surface area can be cut into 6 pieces in a suitable way, and from each piece, we can obtain a face of the cuboid. What are the lengths of the edges?
Let the edges meeting at a vertex of the cuboid be $a, b, c$, and the space diagonal be $d$. Since with the given method of cutting, the faces can be obtained in one piece, the cutting lines are parallel to the sides of the square, so the side $e$ of the square is also an integer. Thus, the surface area of the body is a perfect square. According to the given problem,

$$
a b c=2(a b+a c+b c) \quad\left(=e^{2}\right)
$$

from this equation, by division,

$$
\frac{1}{2}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}
$$

We can assume that $a \leqq b \leqq c$. Thus, on the one hand,

$$
\frac{1}{2} \leqq \frac{1}{a}+\frac{1}{a}+\frac{1}{a}=\frac{3}{a}, \quad \text { and on the other hand } \quad \frac{1}{2}>\frac{1}{a}
$$

from these, $a \leqq 6$ and $a>2$, so the value of $a$ is $3, 4, 5$, or 6.

With $a=3$, similarly, we get $66$, but only $3 b / e=1, b-6=2$ can satisfy, so $b=8, c=24, e=24$. However, in this case, $d^{2}=a^{2}+b^{2}+c^{2}=649$ is not a perfect square, so this set of values does not satisfy the conditions.

With $a=4$, similarly, $4<b \leqq 8$, and according to (3),

$$
a b c=\frac{2 a^{2} b^{2}}{(a-2) b-2 a}=\frac{16 b^{2}}{b-4}=e^{2}, \quad b-4=\left(\frac{4 b}{e}\right)^{2}
$$

thus $b-4$ is a perfect square. Since $0<b-4 \leqq 4$, the value of $b-4$ is 1 or 4. With these, the following possibilities arise for $a, b, c, e$:

(I.)

(II.) $4,5,20,20, \quad$ or

$4,8,8,16$

In both cases, the space diagonal is an integer: $d=21$, or $d=12$.

With $a=5$ and $a=6$, we do not get a suitable triplet of values for $a, b, c$. In both cases, $b \leqq 6$, and further, with $a=5$, $b=5$, we get $c=10$, and thus $a b c$ is not a perfect square. Similarly, with $a=b=c=6$, it is not a perfect square, and with $a=5, b=6$, $c$ is not an integer.

In case I, from a square with side $e=20$ units, two $a \times c=4 \times 20$, two $b \times c=5 \times 20$, and two $a \times b=4 \times 5$ unit-sized rectangles need to be cut out in one piece. The first four cuts are unique because $c=e$, but from the remaining $2 \times 20$ sized rectangle, it is impossible to cut out the $4 \times 5$ sized rectangles.

![](https://cdn.mathpix.com/cropped/2024_05_02_ff33b8978eefb653cb89g-1.jpg?height=384&width=417&top_left_y=1998&top_left_x=843)

In case II, the diagram shows one possible way to cut, according to which the only solution to the problem is if the lengths of the edges of the cuboid are $4, 8, 8$ units (a half-cut 8-unit edge cube).

Tistyán Péter (Békéscsaba, Rózsa F. g. IV. o. t.)

Note. The above result can also be reached by trial and error starting from the cutting requirement, - only it does not become clear that this is the only solution. Indeed, from the paper model of the cuboid, setting aside the two $b \times c$ sized faces, and cutting the remaining net along two opposite edges and placing them side by side, we can cover a $(b+c) \times(2 a)$ sized rectangle. By combining the $b \times c$ sized faces into a $(2 b) \times c$ sized rectangle, in any case, it can form a rectangle if $2 b=b+c$, i.e., $b=c$. This rectangle will be a square if $2 b=c+2 a$, i.e., $b=c=2 a$. From these, $d=3 a$. If $a$ is an integer, then $b, c$, and $d$ are integers. However, the surface area and volume will only be equal if $16 a^{2}=4 a^{3}$, i.e., if $a=4$, meaning the unit of length is a quarter of the shorter edge.

Máté Attila (Szeged, Dózsa Gy. ált. isk. VIII. o. t.)

Lipcsey Zsolt (Budapest, Petőfi S. g. II. o. t.)

[^0]:    ${ }^{1}$ For a more detailed explanation of the applied thought process, see K. M. L. 19 (1959) 83. o.