8.6. Can a rectangular block of size $3 \times 4 \times 5$

a) be cut into 10 rectangular blocks of pairwise different volumes and with integer edges;

b) be cut into 11 rectangular blocks of pairwise different volumes and with integer edges?
Justify the answer.

Solution: The minimum 11 different volumes of blocks are the natural numbers from 1 to 11. Their sum is 66, which is greater than the volume of the block to be cut. Therefore, it is impossible to cut the original block into 11 blocks of different integer volumes, and the answer to part b) is negative.

The minimum 10 different volumes of blocks are the natural numbers from 1 to 10, but a block with a volume of 7 has dimensions \(1 \times 1 \times 7\) and cannot be obtained. The same applies to a block with a volume of 11. Therefore, the 10 permissible minimum volumes are the numbers \(1, 2, 3, 4, 5, 6, 8, 9, 10\), and 12, and their sum is exactly \(60 = 3 \times 4 \times 5\). We will show how to cut the desired block into blocks of the specified volumes. For example, we can first cut the block "layer by layer" into three parallelepipeds of size \(1 \times 4 \times 5\), and then cut each layer as shown in the figure.

| 12 | 12 | 12 | 4 | 3 |
| :--- | :--- | :--- | :--- | :--- |
| 12 | 12 | 12 | 4 | 3 |
| 12 | 12 | 12 | 4 | 3 |
| 12 | 12 | 12 | 4 | 1 |

![](https://cdn.mathpix.com/cropped/2024_05_06_5f022ae4898e2d94657eg-7.jpg?height=302&width=371&top_left_y=2242&top_left_x=845)

| 10 | 10 | 10 | 10 | 10 |
| :---: | :---: | :---: | :---: | :---: |
| 10 | 10 | 10 | 10 | 10 |
| 8 | 8 | 8 | 8 | 2 |
| 8 | 8 | 8 | 8 | 2 |

To the solution of problem 8.6

Answer: a) can b) cannot.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct solution of both parts a) and b) | 7 points |
| Proven that it is impossible to cut into 11 blocks, and correctly found the volumes of 10 blocks to cut in part a); no example of cutting into 10 blocks | 5 points |
| Proven that it is impossible to cut into 11 blocks; no progress in solving part a) | 4 points |
| A correct example of cutting into 10 blocks is given, but part b) is not solved | 3 points |
| Correctly determined the volumes of 10 blocks to cut in part a); no other progress | 2 points |
| Correct answer without justification or incorrect answer | 0 points |