A $4 \times 4$ grid is to be covered with 16 square tiles. There are four tiles in each of the colours red, black, green, and yellow. Each row must contain one tile of each colour. Each pair of tiles that touch along a side or at a corner must have different colours. In how many different ways can these tiles be arranged?
(A) 256
(B) 24
(C) 120
(D) 576
(E) 64

## Part C: Each correct answer is worth 8.
Begin by placing four tiles in the squares of the first row.

The only restriction is that the row must contain one tile of each colour.

Thus in the first row, there are 4 choices of tile colour for the first column, 3 choices for the second, 2 for the third, and 1 choice for the fourth.

That is, there are $4 \times 3 \times 2 \times 1=24$ different ways to cover the squares in the first row using one tile of each of the four colours.

For example, using $R$ for red, $B$ for black, $G$ for green, and $Y$ for yellow, the first row could contain tiles coloured $\begin{array}{llll}G & Y & R & B\end{array}$, in that order.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-016.jpg?height=54&width=1453&top_left_y=1019&top_left_x=279)

We will show that there is only one way to arrange the remaining 12 tiles in the grid.

Consider the first square in the second row, that is, the square directly below the tile coloured $G$. The tile placed in this square cannot be coloured $G$ since it shares an edge with the tile coloured $G$ in row 1 .

Also, the tile placed in this square cannot be coloured $Y$ since it touches the corner of the square containing the tile coloured $Y$ in row 1 .

Assume that the tile in the first square of row 2 is coloured $B$, as shown.

Next, consider the colour of the tiles that could be placed in the second square of row 2.

The tile in this square cannot be coloured $Y$ since it shares an edge with the tile coloured $Y$ in row 1 .

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-016.jpg?height=219&width=236&top_left_y=1460&top_left_x=1560)

Also, the tile in this square cannot be coloured $G$ since it touches the corner of the square containing the tile coloured $G$ in row 1.

Further, the tile in this square cannot be coloured $R$ since it touches the corner of the square containing the tile coloured $R$ in row 1 .

Since the first square in this row contains a tile coloured $B$, then we have no possible tile that can be placed in the second square of row 2 .

This means that the tile in the first square of row 2 cannot be coloured $B$, and thus it must be coloured $R$, as shown.

The tile in the second square of row 2 cannot be coloured $Y$ or $G$ (as noted earlier), and thus must be coloured $B$.

Continuing to move right along row 2 , the next tile cannot be coloured $Y$ since it touches the corner of the square containing

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-016.jpg?height=225&width=233&top_left_y=2061&top_left_x=1564)
the tile coloured $Y$ in row 1, and so the tile in this square must be coloured $G$, with the final tile in the row being coloured $Y$.

That is, the positions of the 4 tiles in row 2 are completely determined by the tiles in row 1 . Thus for each of the 24 different ways to place the tiles in row 1 , there is exactly one way to place the tiles in row 2.

Repeating the argument, the same is then true for the tiles in row 3 and row 4; that is, there
is exactly one choice for the location of each of the coloured tiles within each of these two rows as well.

The $4 \times 4$ in our example above is completed here, as shown.

You should justify for yourself that each of rows 3 and 4 must contain tiles exactly as shown.

For each of the 24 different ways to cover the squares in the first row using one tile of each of the four colours, there is exactly one

| $G$ | $Y$ | $R$ | $B$ |
| :---: | :---: | :---: | :---: |
| $R$ | $B$ | $G$ | $Y$ |
| $G$ | $Y$ | $R$ | $B$ |
| $R$ | $B$ | $G$ | $Y$ |

way to cover all remaining squares in the grid.

Thus, there are 24 different ways that the tiles can be arranged.

ANswer: (B)