Problem 3

A box whose shape is a rectangular parallelepiped can be completely filled with cubes of side

1

. If one places inside it the maximum possible number of cubes, each of volume

2

, with their sides parallel to those of the box, then exactly

40\%

of the volume of the box is occupied. Determine the possible dimensions of all such boxes.

Step 6 of 6: Determine the third edge and conclude

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In plain words

By the third edge, the same three-step pattern (shrink the upper bound for large values, then hand-check the few small remaining candidates) has been applied once per edge, systematically exhausting all cases without missing any.

\frac cz=\frac32,\qquad c\in\{3,6\}

Detailed analysis

Now $c/z=5/2\div(5/3)=3/2$ . If $c\ge8$ then $z\ge6$ and $c/z<\sqrt[3]2(1+1/6)<3/2$ (since $2\cdot7^3<3^3\cdot6^3$ ), a contradiction. Checking $c=1,\ldots,7$ against $c/z=3/2$ leaves exactly $c=3$ (with $z=2$ ) and $c=6$ (with $z=4$ ). Together with $a=2,b=5$ from Steps 4–5, the only possible boxes are $\{a,b,c\}=\{2,3,5\}$ and $\{2,5,6\}$ (up to permutation of the edges).