This question is one of the logical riddles that interviewers love to ask during job interviews:
There is a 3×3×3 cube. You are asked to cut it into 27 1×1×1 cubes. Every cut is straight and plain, but between the cuts the pieces can be rearranged freely. You are allowed to cut a part, put it on top of the other part and cut them together with the next cut. The question is: what is the minimal amount of cuts that is required for the task?
The trivial solution:
We can perform two cuts parallel to every coordinate plain (X00, 0Y0 and 00Z), just like shown on the illustration above. That would be 6 cuts, and the parts of the cube may be left as they were between the cuts. No rearrangements are needed.
Is there a possibility to achieve the same result with fewer cuts?
Solution 1:
It’s easy to see that every cut at most doubles the number of pieces, unless we put a piece in the way that it will be entirely omitted by the cut. We are going to get 27 cubes, which is less than 32 = 25, but more than 16 = 24. Therefore even theoretically we can’t achieve the goal with less than 5 cuts.
However, when we arrive to the 5th cut, we have to have 18 pieces. It’s more than we can achieve with 4 cuts, therefore we have to use 6 cuts.
Solution 2:
Let’s look at the central part of the big cube. When we finish the task, it becomes a 1×1×1 cube, but currently it is surrounded with cube’s material from all sides. This 1×1×1 cube has 6 faces. Among them, there are no two faces that can be created with the same cut. Therefore, clearing this cube from all other material requires 6 cuts.
Conclusion
The trivial solution is also the most effective one. 😜