Problem 10.3. Lёsha cut a cube $n \times n \times n$ into 153 smaller cubes. Moreover, all the cubes, except one, have an edge length of 1. Find $n$.
Answer: 6.

Solution. Let the remaining cube have an edge of $s$. Obviously, the numbers $n$ and $s$ are natural, and $n>s$.

The difference in the volumes of the cubes with edges $n$ and $s$ is equal to the total volume of the unit cubes. Therefore, $n^{3}-s^{3}=152$.

Since $n^{3}>152>5^{3}$, then $n>5$. Note that $n=6$ satisfies the equation above, corresponding to $s=4$, since $6^{3}-4^{3}=152$. However, $n=7$ does not satisfy the equation, as the number $7^{3}-152=191$ is not a perfect cube. Neither do all $n \geqslant 8$ satisfy the equation, since otherwise

$$
152=n^{3}-s^{3} \geqslant n^{3}-(n-1)^{3}=3 n^{2}-3 n+1=3 n(n-1)+1 \geqslant 3 \cdot 8 \cdot 7+1>152
$$

Clearly, the only possible value of $n=6$ is valid: a cube $6 \times 6 \times 6$ can be cut into a cube $4 \times 4 \times 4$ and 152 unit cubes $1 \times 1 \times 1$.