In the diagram, the numbers from 1 to 25 are to be arranged in the 5 by 5 grid so that each number, except 1 and 2 , is the sum of two of its neighbours. (Numbers in the grid are neighbours if their squares touch along a side or at a corner. For example, the " 1 " has 8 neighbours.) Some of the numbers have already been filled in. Which number must replace the "?" when the grid is completed?

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-073.jpg?height=336&width=422&top_left_y=1702&top_left_x=1315)
(A) 15
(B) 12
(C) 14
(D) 11
(E) 13

|  |  |  | 20 | 21 |
| :---: | :---: | :---: | :---: | :---: |
|  | 6 | 5 | 4 |  |
| 23 | 7 | 1 | 3 | $?$ |
|  | 9 | 8 | 2 |  |
| 25 | 24 |  |  | 22 |
The numbers which have yet to be placed in the grid are 10, 11, 12, 13, 14, 15, 16, 17, 18, and 19 .

We look at the numbers in the corners first, because they have the fewest neighbours.

Look first at the 25 in the bottom left corner. It must be the sum of two of its neighbours, so it must be $24+1$ or $9+16$. Since the 1 has already been placed in the grid, then the empty space next to the 25 must be filled with the 16 .

Look next at the 21 in the top right corner. It must be equal to $20+1$ (which it cannot be since the 1 has already been placed) or $4+17$, so the 17 must be placed in the space next to the 21 .

|  |  |  | 20 | 21 |
| :---: | :---: | :---: | :---: | :---: |
|  | 6 | 5 | 4 | 17 |
| 23 | 7 | 1 | 3 | $?$ |
| 16 | 9 | 8 | 2 |  |
| 25 | 24 |  |  | 22 |

Since the 17 must be the sum of two of its neighbours, then the 17 must be $4+13$ or $3+14$, so the "?" must be replaced by either the 13 or the 14 .

Consider the 22 in the corner. Since the 20 is already placed, 22 cannot be $2+20$. So the 22 is the sum of its two missing neighbours.

Since the possible missing neighbours are 10,11, 12, 13, 14, 15, and 18, then the two neighbours must be 10 and 12 in some order.

However, the 10 cannot go above the 22, since there it could not be the sum of two of its neighbours (since the 7 and 8 have already been placed).

Therefore, the 12 goes above the 22 , so we have

|  |  |  | 20 | 21 |
| :---: | :---: | :---: | :---: | :---: |
|  | 6 | 5 | 4 | 17 |
| 23 | 7 | 1 | 3 | $?$ |
| 16 | 9 | 8 | 2 | 12 |
| 25 | 24 |  | 10 | 22 |

The 13 now cannot replace the "?" since 13 is not the sum of any two of $17,4,3,2$, and 12 . Therefore, the "?" is replaced by the 14 .

At this stage, we are already finished, but we could check that the grid does complete as follows:

| 19 | 11 | 15 | 20 | 21 |
| :---: | :---: | :---: | :---: | :---: |
| 13 | 6 | 5 | 4 | 17 |
| 23 | 7 | 1 | 3 | 14 |
| 16 | 9 | 8 | 2 | 12 |
| 25 | 24 | 18 | 10 | 22 |

ANSWER: (C)