The Least Number of Cuts Required for Transforming a Cube into 24 Identical Cuboids

When faced with the challenge of transforming a cube into 24 identical cuboids without rearranging any pieces, determining the minimum number of cuts required can be quite intriguing. This task involves understanding the relationship between the original cube's volume and the volume of the resulting cuboids, as well as strategically making the required cuts.

Step 1: Understanding the Dimensions of the Cuboids

To achieve 24 identical cuboids from a single cube, we must first express the volume of each cuboid. The volume of the cube is given by (V s^3), where (s) is the side length of the cube. If we aim to create 24 identical cuboids, the volume of each cuboid is:

[V_{text{cuboid}} frac{s^3}{24}]

Our objective then becomes finding the dimensions (a), (b), and (c) such that:

[a middot; b middot; c frac{s^3}{24}]

Step 2: Analyzing Factorization Possibilities

Since our goal is to create 24 cuboids, we need to factor 24 into three integers. The prime factorization of 24 is:

[24 2^3 middot; 3^1]

The following combinations of factors that multiply to 24 are possible:

1 middot; 1 middot; 24 1 middot; 2 middot; 12 1 middot; 3 middot; 8 1 middot; 4 middot; 6 2 middot; 2 middot; 6 2 middot; 3 middot; 4

Among these factors, the most balanced option is 2 middot; 3 middot; 4, suggesting the dimensions of the cuboids should be:

[frac{s}{2} text{, } frac{s}{3} text{, and } frac{s}{4}]

Step 3: Executing the Cuts

To achieve the desired dimensions, we must make the following cuts:

First Dimension

To divide the cube into 2 equal parts along one dimension:

1 cut is required.

Second Dimension

To further divide each of those 2 parts into 3 equal parts along the second dimension:

2 cuts are required, one at frac{s}{3} and another at frac{2s}{3}.

Third Dimension

Next, to divide each of the resulting 6 parts into 4 equal parts along the third dimension:

3 cuts are required, one at frac{s}{4}, another at frac{2s}{4}text{ (or }frac{s}{2}), and one at frac{3s}{4}text{ (or }frac{3s}{4}).

Total Number of Cuts

Adding up the cuts:

1 cut for the first dimension 2 cuts for the second dimension 3 cuts for the third dimension

Total number of cuts: 1 2 3 6 cuts.

Conclusion

The least number of cuts required to cut a cube into 24 identical cuboids, ensuring no rearrangement of the pieces, is 6 cuts.