## What Digit?

A digit is placed in each empty square in the grid so that each row and each column contains the digits $1,2,3,4,5$. What digit is placed in the square at the bottom right corner?

Source: NCTM Mathematics Teacher, November 2006

SOLUTION
Let’s start with the first column by placing $3$ in the upper left square because $1,2$ are already placed in the first column and $4,5$ are already placed in the first row. After that, we place $4$ in the third row because $5$ is already placed in it. We then place $5$ in the last row because it is the only place left.

We now work on the third column. We place $2$ right below $4$ because $1,3$ are already placed in the second row and $4,5$ are already placed in the third column. We place $1$ in the last row because that is the only place left. Next we look at the fourth row. We place $4$ right next to $2$ because $1,2,3$ are already placed in it and $5$ is already placed in the second column.

At last we work on the last row. We place $2$ right below $4$ because $3,4,5$ are already placed in the second column and $1$ is already placed in the last row. We are left with $3$ and $4$ to place in the last row. We cannot place $3$ next to $1$ because $3$ is already placed the fourth column. So we place $4$ there and $3$ in the bottom right corner.

Answer: $3$