We have two compact wooden cubes, their edge lengths are 14 and $10 \mathrm{~cm}$. We want to cut these cubes into pieces such that the pieces can be assembled into three cubes with edge lengths of 13, 11, and $6 \mathrm{~cm}$. The cuts are only allowed along planes parallel to the faces of the cubes, and the distance between the cutting planes and the cube faces must be an integer in cm. The dissection should result in as few pieces as possible. Provide such a dissection. (Label the pieces appropriately, and show the assembly in cross-sections or in any other suitable manner.) State the number of pieces created as well.
The dissection can be carried out without remainder because the sum of the volumes of the desired cubes is equal to the sum of the volumes of the given cubes:

$$
6^{3}+11^{3}+13^{3}=14^{3}+10^{3}
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_84a2070d40f7e42a150eg-1.jpg?height=844&width=1278&top_left_y=320&top_left_x=412)

Cut out an $11 \mathrm{~cm}$ edge cube (A-piece) from the $14 \mathrm{~cm}$ edge cube so that their front upper right vertices coincide (part a) of the figure). The cutting surface consists of three $11 \mathrm{~cm}$-sided squares, perpendicular to which the thickness of the remaining body is $3 \mathrm{~cm}$, which is half the edge of the smallest cube to be made. By cutting out a $12 \times 6 \times 3 \mathrm{~cm}$ rectangular prism along the front lower edge and then cutting it perpendicularly to its length (B- and C-pieces), and aligning their square faces, we obtain the $6 \mathrm{~cm}$ edge cube (part b) of the figure).

Fit the $10 \mathrm{~cm}$ edge cube (D-piece) into the remaining body so that its rear, left, and lower vertex coincides with the corresponding vertex of the A-cube. This way, its front, right, and top faces are $13 \mathrm{~cm}$ away from the intersection of the back, left, and bottom faces of the $14 \mathrm{~cm}$ edge cube. By cutting off the parts of the remaining body that extend beyond these three faces, a $13 \mathrm{~cm}$ edge cube remains, but there is a rectangular prism-shaped gap where the B- and C-pieces were, with dimensions $11 \times 5 \times 3 \mathrm{~cm}$. This gap can be filled with 5 plates and 1 L-shaped piece, resulting from cutting the excess part into 6 pieces as shown in part c) of the figure, and arranging them as shown in part d) of the figure.

Including the remaining body, we have created 11 pieces from the two given cubes.

László Domokos (Tatabánya, Árpád g. III. o. t.)