6,7 $[\quad$ Examples and Counterexamples. Constructions

Authors: Shaovalov A.V., Raskina I.V.

From an equal number of squares with sides 1, 2, and 3, form a square of the smallest possible size.

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Evaluation. Let the desired square be composed of $n$ squares of each type. Then its area is $n \cdot\left(1^{2}+2^{2}+\right.$ $\left.3^{2}\right)=14 n=2 \cdot 7 \cdot n$. Since the side length of the desired square must be an integer, the obtained number must be a perfect square. Therefore, the number $n$ must contain the factors 2 and 7, that is, $n \geq 14$.

Example. See the figure, where 14 squares of each type are used.

![](https://cdn.mathpix.com/cropped/2024_05_06_c07b2c4be0d3dcbcc4e5g-19.jpg?height=746&width=738&top_left_y=576&top_left_x=660)

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