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 4 of 6: Determine the first edge

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In plain words

Ordering the three ratios lets us focus the largest one against the cube root of $5$ ; checking only the small integers $a=2,3,4$ by hand (larger ones being ruled out by the shrinking-bound argument) quickly forces the exact value.

a=2,\ \frac ax=2

Detailed analysis

Assume WLOG $a/x\ge b/y\ge c/z$ , so $a/x\ge\sqrt[3]5$ . If $a\ge4$ then $x\ge3$ and $a/x<\sqrt[3]2(1+1/3)=\sqrt[3]2\cdot\frac43<\sqrt[3]5$ (since $2\cdot4^3<5\cdot3^3$ ), a contradiction; so $a\in\{2,3\}$ . For $a=3$ , $x=2$ and $a/x=3/2<\sqrt[3]5$ , also a contradiction. Hence $a=2$ , $x=1$ , and $a/x=2$ .