We drew a 13 cm and a $35 \mathrm{~cm}$ side square on a grid paper, with each side lying on a grid line of the paper, and the distance between the lines is $1 \mathrm{~cm}$ in both directions. We want to cut these two squares into pieces so that we can form three new squares with side lengths of 17, 23, and $24 \mathrm{~cm}$. The cutting must be done along the grid lines. Provide a dissection that results in as few pieces as possible. (For example, it is sufficient to submit a $1: 4$ scale drawing of the dissection and the reassembly, indicating the number of pieces and briefly explaining why the given solution is better than other attempts.)





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The dissection is possible because the total area of the desired squares is equal to the total area of the given squares: $17^{2}+23^{2}+24^{2}=1394=13^{2}+35^{2}$. Moreover, it is clearly feasible, for example, by cutting the given squares into 1 cm wide strips and covering the desired squares with appropriately cut pieces from these strips, ensuring that the lengths of the used and remaining pieces are always integers.

The mentioned dissection can be carried out with significantly fewer pieces and in various ways. However, the number of pieces cannot be less than 5, because the desired squares can only be cut from the 35 cm square in one piece, and only one of them, while the other two squares will consist of at least 2 pieces each.

Below, we present two slightly different dissections on a common diagram; each produces 7 pieces from the given two squares, leaving the 13 cm square intact and using it in the assembly of the 17 cm square, while the 23 cm square is entirely cut out. The dividing and connecting lines of the first dissection are marked with thick lines on the diagram, while those of the second are marked with double thin lines.
![](https://cdn.mathpix.com/cropped/2024_05_02_3ad6619a3b70d1b25228g-1.jpg?height=1154&width=1062&top_left_y=632&top_left_x=520)

In the first dissection, the pieces are labeled $A, B, C, D, E, F, G$, and in the second, they are labeled $x, y, z, v, w, u, t$.

Fábry László (Miskolc, Bláthy O. vill. ip. t. II. o. t.)  
Huhn Ágnes (Szeged, Tömörkény I. g. II. o. t.)