## 6. Cube from sticks

![](https://cdn.mathpix.com/cropped/2024_05_30_12924d306592e3a90c60g-05.jpg?height=392&width=371&top_left_y=1186&top_left_x=180)

Result:

Cubes are made from sticks. For a cube $1 \times 1 \times 1$, 12 sticks are needed. To complete this cube to a cube $2 \times 2 \times 2$, 42 additional sticks are required (see the image). How many additional sticks are needed to complete a cube $8 \times 8 \times 8$ to a cube $9 \times 9 \times 9$?
## First Solution.

Consider a cube of dimensions $n \times n \times n$ and determine how many sticks it is made of. Notice that some sticks are placed vertically, while others are arranged in horizontal planes.
![](https://cdn.mathpix.com/cropped/2024_05_30_12924d306592e3a90c60g-05.jpg?height=1074&width=1766&top_left_y=1402&top_left_x=136)

There are $n+1$ such planes. In each of them, the sticks form an $n \times n$ square. We observe sticks in two directions, in each direction there are $n+1$ rows of $n$ sticks each. Thus, each $n \times n$ square is made up of $2 n \cdot (n+1)$ sticks. In total, there are $2 n \cdot (n+1)^2$ sticks in the horizontal planes.

We still need to count the vertical sticks. Notice that above each point of the square in the bottom plane, there is a vertical line of $n$ sticks. This means there are a total of $(n+1)^2 \cdot n$ vertical sticks.

Therefore, the cube $n \times n \times n$ is made of $3 n \cdot (n+1)^2$ sticks.

A cube $8 \times 8 \times 8$ consists of $3 \cdot 8 \cdot 9^2 = 1944$ sticks, and a cube $9 \times 9 \times 9$ consists of $3 \cdot 9 \cdot 10^2 = 2700$ sticks.

To complete the cube, $2700 - 1944 = 756$ additional sticks are needed.