# Problem 6.

A $10 \times 10$ square was cut into rectangles, the areas of which are different and expressed as natural numbers. What is the maximum number of rectangles that can be obtained?
Solution. The area of the square is 100. If we represent 100 as the sum of natural numbers, the number of addends will be the largest if the difference between the numbers is one. Let's take rectangles with areas of $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. Their total area is 55. Therefore, the sum of the areas of the remaining rectangles is 45.

Notice that if the area of a rectangle is greater than 10, it cannot be

![](https://cdn.mathpix.com/cropped/2024_05_06_f6ec34859a1caedb9525g-3.jpg?height=660&width=651&top_left_y=612&top_left_x=57)
a prime number, otherwise such a rectangle would have a side greater than 10 and would not fit into a $10 \times 10$ square. Composite numbers greater than ten are $12, 14, 15, 16, 18$, $19, \ldots$ Any four of them sum to a number greater than 45. Sums equal to 45 can be given, for example, by such three numbers: $12, 15, 18$ or $14, 15, 16$.

Thus, the number of rectangles is less than or equal to 13. An example of a possible arrangement for 13 rectangles is shown in the figure.

## Evaluation criteria for the correspondence round tasks for 9th grade

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