## Task 4 - 340634

Vera tells her friend Ute that she measured the edge lengths of a cuboid and noticed the following:

(1) One of the edges is twice as long as another of the edges.

(2) The sum of the lengths of all twelve edges of the cuboid is $320 \mathrm{~cm}$.

(3) Exactly two of the six faces of the cuboid are squares.

Ute thinks that these statements uniquely determine the edge lengths of the cuboid.

Investigate whether Ute's opinion is correct!

If it is correct, give the edge lengths; if it is not correct, give all possibilities for the edge lengths of a cuboid that meet Vera's statements!

Hint: When giving the edge lengths of a cuboid, you do not need to specify twelve edge lengths, but it is sufficient to give the lengths of the three edges that meet at a corner of the cuboid.
I. If $a, b, c$ are the numerical values of the lengths of three edges of a cuboid meeting at a corner, measured in cm, and Vera's statements (1), (2), (3) apply to them, then:

From (3), exactly two of $a, b, c$ are equal; the labels can be chosen such that $b=c$.

From (2) and because each of the edge lengths $a, b, c$ occurs exactly 4 times, it follows that $a+2b=320:4=80$. (4)

From (1) it further follows: Either $a=2b$ (5) or $b=2a$. (6)

If (4) and (5) hold, then $a+a=80$, so $a=40, b=20, c=20$. (7)

If (4) and (6) hold, then $a+4a=80, 5a=80$, so $a=16, b=32, c=32$. (8)

II. The numbers mentioned in (7) satisfy (4) and (5), and the numbers mentioned in (8) satisfy (4) and (6); in both cases, $b=c$ also holds.

Therefore, in both cases, Vera's statements (1), (2), (3) are true.

With I. and II., it is shown: The edge lengths of a cuboid are not uniquely determined by Vera's statements (1), (2), (3) (i.e., Ute's opinion is not true), but there are exactly two possibilities for the edge lengths of a cuboid for which Vera's statements apply, namely that these edge lengths are either $40 \mathrm{~cm}, 20 \mathrm{~cm}, 20 \mathrm{~cm}$ or $16 \mathrm{~cm}, 32 \mathrm{~cm}, 32 \mathrm{~cm}$.