3. In the box, there are geometric shapes: triangles, squares, and circles. In total, there are 24 shapes. There are 7 times as many triangles as squares. How many of each shape could be in the box?
3. The first method of solving. Let's represent the condition of the problem in the form of a drawing.

squares

![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-049.jpg?height=83&width=151&top_left_y=1466&top_left_x=701)

triangles

![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-049.jpg?height=82&width=903&top_left_y=1558&top_left_x=702)

circles

![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-049.jpg?height=95&width=971&top_left_y=1643&top_left_x=702)

From this, it is clear that the sum of the triangles and squares should be divisible by eight. There are only two numbers less than 24 and divisible by eight - these are 16 and 8. Let's check each of them.

1) \(16 \div 8 = 2\) (squares); \(\quad\) 1) \(8 \div 8 = 1\) (squares);
2) \(2 \times 7 = 14\) (triangles)
3) \(1 \times 7 = 7\) (triangles);
4) \(24 - 16 = 8\) (circles).
5) \(24 - 8 = 16\) (circles).

The second method of solving. Since, according to the condition, there are 7 times more triangles than squares, there could not be more than three squares, otherwise, the total number of figures would exceed 24.

Let's assume there was 1 square, then there were 7 times more triangles, that is, 7. The number of circles can be determined by calculating the difference between 24 and 8. There were 16 circles.

Let's assume there were 2 squares, then there were \(14 = 2 \times 7\) triangles, and \(8 (24 - (14 + 2)) = 8\) circles.

Let's assume there were 3 squares, then there would be 21 triangles, and their sum would be 24. In this case, there would be no circles, which contradicts the condition of the problem.

The third method of solving (by Yelena Viazmina, a student of School No. 46).

There are 7 times more triangles than squares. In total, there are 8 such parts. 24 is divisible by 8, but we still have circles. We take the nearest number that is divisible by 8. This number is 16.

1) \(16 \div 8 = 2\) (squares)
2) \(2 \times 7 = 14\) (triangles)
3) \(24 - 16 = 8\) (circles).

Answer: 2 squares, 14 triangles, 8 circles or 1 square, 7 triangles, 16 circles.