In the $4 \times 4$ square shown, each row, column and diagonal should contain each of the numbers $1,2,3$, and 4 . Find the value of $K+N$.
(A) 4
(D) 6
(B) 3
(C) 5
(E) 7

| 1 | $F$ | $G$ | $H$ |
| :---: | :---: | :---: | :---: |
| $T$ | 2 | $J$ | $K$ |
| $L$ | $M$ | 3 | $N$ |
| $P$ | $Q$ | 1 | $R$ |
In the $4 \times 4$ square shown, each row, column and diagonal should contain each of the numbers $1,2,3$, and 4 . Find the value of $K+N$.
(A) 4
(B) 3
(C) 5
(D) 6
(E) 7

| 1 | $F$ | $G$ | $H$ |
| :---: | :---: | :---: | :---: |
| $T$ | 2 | $J$ | $K$ |
| $L$ | $M$ | 3 | $N$ |
| $P$ | $Q$ | 1 | $R$ |

Solution

Since $R$ is on a main diagonal and the numbers 1,2 and 3 have already been used on this diagonal, then $R=4$.

The easiest way to look at how to arrange the numbers is to look at boxes $P$ and $Q$.

$Q$ must be either 2 or 3 but since there is already a 2 in the same column as $Q$, we conclude that $Q=3$ and $P=2$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-470.jpg?height=358&width=290&top_left_y=301&top_left_x=1403)

From this point we simply fill in the boxes according to the rule that each row, column and diagonal contains each of the numbers $1,2,3$, and 4 .

Doing this, we arrive at the following arrangement of numbers.

| 1 | 4 | 2 | 3 |
| :--- | :--- | :--- | :--- |
| 3 | 2 | 4 | 1 |
| 4 | 1 | 3 | 2 |
| 2 | 3 | 1 | 4 |

We see that $K+N=3$.

NOTE 1: It is not necessary that we complete all the boxes but it is a useful way to verify the overall correctness of our work.

NOTE 2: We could have started by considering $H, K$ and $N$ but this takes a little longer to complete.

ANSWER: (B)