Let $t$ be TNYWR.

In a magic square, the sum of the numbers in each column, the sum of the numbers in each row, and the sum of the numbers on each diagonal are all the same. In the magic square shown, what is the value of $N$ ?

|  | $3 t-2$ | $4 t-6$ |  |
| :---: | :---: | :---: | :---: |
| $4 t-1$ | $2 t+12$ | $t+16$ | $3 t+1$ |
| $N$ | $4 t-2$ |  | $t+15$ |
|  |  |  | $4 t-5$ |
The sum of the entries in the second row is

$$
(4 t-1)+(2 t+12)+(t+16)+(3 t+1)=10 t+28
$$

This means that the sum of the four entries in any row, column or diagonal will also be $10 t+28$.

Looking at the fourth column, the top right entry is thus

$$
10 t+28-(3 t+1)-(t+15)-(4 t-5)=2 t+17
$$

Looking at the top row, the top left entry is thus

$$
10 t+28-(3 t-2)-(4 t-6)-(2 t+17)=t+19
$$

Looking at the southeast diagonal, the third entry is thus

$$
10 t+28-(t+19)-(2 t+12)-(4 t-5)=3 t+2
$$

Looking at the third row,

$$
N=10 t+28-(4 t-2)-(3 t+2)-(t+15)=2 t+13
$$

Since the answer to (b) is 5 , then $t=5$. Thus, $N=23$.

We note that we can complete the grid, both in terms of $t$ and using $t=5$ as follows:

| $t+19$ | $3 t-2$ | $4 t-6$ | $2 t+17$ |
| :---: | :---: | :---: | :---: |
| $4 t-1$ | $2 t+12$ | $t+16$ | $3 t+1$ |
| $2 t+13$ | $4 t-2$ | $3 t+2$ | $t+15$ |
| $3 t-3$ | $t+20$ | $2 t+16$ | $4 t-5$ |


| 24 | 13 | 14 | 27 |
| :--- | :--- | :--- | :--- |
| 19 | 22 | 21 | 16 |
| 23 | 18 | 17 | 20 |
| 12 | 25 | 26 | 15 |

ANSWER: $8,5,23$