In a block of $1 \times 2 \times 3$ centimeters, we mark three faces with the letters X, Y, and Z, as shown in the figure. The block is placed on an $8 \times 8$ cm board with the face X facing down, in contact with the board, as shown in the figure. We rotate the block $90^{\circ}$ around one of its edges so that the face Y is facing down (that is, completely in contact with the board). Then, we rotate the block again $90^{\circ}$ around one of its edges, but this time so that the face Z is facing down.

We rotate the block three more times $90^{\circ}$ around one of its edges, making the faces X, Y, and Z face down, in that order. How many different squares of the board have been in contact with the block?
(a) 18
(b) 19
(c) 20
(d) 21
(e) 22

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-010.jpg?height=394&width=494&top_left_y=257&top_left_x=1341)
The correct option is (b).

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-053.jpg?height=399&width=580&top_left_y=1211&top_left_x=361)

According to the figure, we can conclude that the dimensions of the faces X, Y, and Z are 2, 3, and $6 \mathrm{~cm}^{2}$, respectively. Next, we will indicate the movements made by the block and the faces that come into contact with the squares at each step. Remember that we rotate the block five times.

The following figures show the squares of the board that come into contact with each of the three faces of the block, from the initial position to the final position after the last rotation.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-053.jpg?height=365&width=420&top_left_y=1825&top_left_x=381)

Initial Position

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-053.jpg?height=371&width=403&top_left_y=2279&top_left_x=387)

3rd Rotation

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-053.jpg?height=363&width=394&top_left_y=1826&top_left_x=891)

1st Rotation

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-053.jpg?height=366&width=417&top_left_y=2284&top_left_x=885)

4th Rotation

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-053.jpg?height=363&width=394&top_left_y=1826&top_left_x=1408)

2nd Rotation

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-053.jpg?height=360&width=388&top_left_y=2284&top_left_x=1422)

5th Rotation

Some squares come into contact with the faces more than once, as shown in the following figure, which shows all the squares that had contact with the faces of the block from the initial position to the last rotation.

|  |  |  |  |  |  |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  |  |  | **X** |  |  |  |  |
|  | **Y** | **Y** | **X / Y** | **X** | **X** |  |  |
|  | **Z** | **Z** | **Y / Z** | **Z** | **Z** |  |  |
|  | **Z** | **Z** | **Y / Z** | **Z** | **Z** |  |  |
|  |  |  | **Y** | **Z** | **Z** |  |  |

Counting in this last figure, we see that the block came into contact with 19 squares of the board.