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 1 of 6: Set up integer edges and the packing count

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In plain words

Packing cubes of an irrational side-length $\sqrt[3]2$ along an integer edge always leaves a leftover gap, and the floor function is exactly the right tool to count how many whole cubes fit before that gap appears.

x=\left\lfloor\frac{a}{\sqrt[3]2}\right\rfloor,\quad y=\left\lfloor\frac{b}{\sqrt[3]2}\right\rfloor,\quad z=\left\lfloor\frac{c}{\sqrt[3]2}\right\rfloor

Detailed analysis

Since the box can be tiled exactly by unit cubes, its edge lengths $a,b,c$ are positive integers, with volume $V_1=abc$ . A cube of volume $2$ has side $\sqrt[3]2$ , and the maximum number that fits along an edge of length $a$ (sides parallel) is $x=\left\lfloor a/\sqrt[3]2\right\rfloor$ , and similarly $y,z$ for $b,c$ . The maximal number of volume- $2$ cubes fitting in the box is $xyz$ , giving occupied volume $V_2=2xyz$ .