In the following figure, we have a $2 \times 2 \times 2$ cube made with pieces of wire. The edge of each cube is a piece of $1 \mathrm{~cm}$ of wire and a total of 54 such pieces were used. To make a $10 \times 10 \times 10$ cube, how many pieces of wire will be used?

![](https://cdn.mathpix.com/cropped/2024_05_01_e67c0440f1fa8bc62cdcg-02.jpg?height=352&width=365&top_left_y=638&top_left_x=719)

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Solution

Each vertex of the small cubes is connected by segments that can be classified into three different directions associated with the edges of the larger cube. Thus, we can divide the task of finding the number of segments used according to these directions. From a top view of any face of the larger cube, the figure resembles a $10 \times 10$ square composed of 100 small $1 \times 1$ squares. There are 121 vertices in these small squares, and from each of them, in the direction perpendicular to the face, 10 other segments must be connected. That is, in a certain direction, there are $121 \cdot 10=1210$ segments.

![](https://cdn.mathpix.com/cropped/2024_05_01_e67c0440f1fa8bc62cdcg-02.jpg?height=528&width=534&top_left_y=1716&top_left_x=634)

Since we have 3 directions, the total number of segments is $1210 \cdot 3=3630$.