We have 1000 unit-edged small cubes. From these, we want to create a hollow cube with a wall thickness of one unit. The small cubes can be glued together, but they cannot be cut. What is the maximum volume (external) cube we can make this way?
Let's denote the edge of the cube to be constructed by $x$, and calculate how many unit-edged cubes we used to fill the hollow cube.

On the base and top, we placed a total of $2 x^{2}$ cubes, on the front and back $2 x(x-2)$ cubes, and on the remaining two opposite sides $2(x-2)(x-2)$ cubes.

![](https://cdn.mathpix.com/cropped/2024_05_02_d9b95d8ed7c801d3aaf8g-1.jpg?height=412&width=482&top_left_y=347&top_left_x=819)

The number of small cubes used can be at most 1000, so we have:

$$
2 x^{2}+2 x(x-2)+2(x-2)(x-2) \leq 1000
$$

Rearranging, we get the following quadratic inequality:

$$
3 x^{2}-6 x-496 \leq 0
$$

We see that the corresponding quadratic equation has two distinct roots of opposite signs, so the positive root is the largest value of $x$ for which the inequality holds:

$$
x=\frac{6+\sqrt{5988}}{6} \approx 13.89
$$

Since the solution must be an integer, $x=13$. In this case, we used 866 cubes, and the desired volume is $V=x^{3}=2197$ cubic units.