A big cube is cut into 99 smaller cubes. Exactly 98 of these 99 smaller cubes are unit cubes.

Question: What is the volume of the big cube?


1 Answer 1



Let $k$ denote the side length of the non-unit cube. We know $k>1$ because if it weren't, then there couldn't be a cube, and for the same reason we know $k\in\mathbb{N}$.

Let $n$ denote the length of the full cube. This means $n^3-k^3=98$.

That factors into $(n-k)(n^2+nk+k^2)=98$; the only factors of 98 that produce an integer value for $n$ and $k$ are 2 and 49. This makes $n=5$ and $k=3$.

So the volume of the full cube is 125 cubic units.

  • 5
    $\begingroup$ You can factor $n^3-k^3$ as $(n-k)(n^2+nk+k^2)$ and then try each of the possibilities for factoring 98. Only $2\times49$ provides an integer solution. $\endgroup$
    – f''
    Commented Oct 18, 2015 at 15:52
  • $\begingroup$ @f'': Right, I always forget about the difference of cubes factorization. That would definitely make it easier. $\endgroup$
    – Deusovi
    Commented Oct 18, 2015 at 16:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.