Tokarev S.i.

Can natural numbers from 1 to 81 be written in the cells of a $9 \times 9$ table so that the sum of the numbers in each $3 \times 3$ square is the same

#
Table $T$, depicted in the left figure, contains each of the numbers $1,2, \ldots, 9$ nine times and has the property that the sum of the numbers in each $3 \times 3$ square is 36.

| 1 | 4 | 7 | 1 | 4 | 7 | 1 | 4 | 7 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 2 | 5 | 8 | 2 | 5 | 8 | 2 | 5 | 8 |
| 3 | 6 | 9 | 3 | 6 | 9 | 3 | 6 | 9 |
| 7 | 1 | 4 | 7 | 1 | 4 | 7 | 1 | 4 |
| 8 | 2 | 5 | 8 | 2 | 5 | 8 | 2 | 5 |
| 9 | 3 | 6 | 9 | 3 | 6 | 9 | 3 | 6 |
| 4 | 7 | 1 | 4 | 7 | 1 | 4 | 7 | 1 |
| 5 | 8 | 2 | 5 | 8 | 2 | 5 | 8 | 2 |
| 6 | 9 | 3 | 6 | 9 | 3 | 6 | 9 | 3 |


| 7 | 8 | 9 | 4 | 5 | 6 | 1 | 2 | 3 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 4 | 5 | 6 | 1 | 2 | 3 | 7 | 8 | 9 |
| 1 | 2 | 3 | 7 | 8 | 9 | 4 | 5 | 6 |
| 7 | 8 | 9 | 4 | 5 | 6 | 1 | 2 | 3 |
| 4 | 5 | 6 | 1 | 2 | 3 | 7 | 8 | 9 |
| 1 | 2 | 3 | 7 | 8 | 9 | 4 | 5 | 6 |
| 7 | 8 | 9 | 4 | 5 | 6 | 1 | 2 | 3 |
| 4 | 5 | 6 | 1 | 2 | 3 | 7 | 8 | 9 |
| 1 | 2 | 3 | 7 | 8 | 9 | 4 | 5 | 6 |

It is clear that the table $Q$, obtained from it by a $90^{\circ}$ rotation (right figure), also has the specified properties. Notice now that the same numbers in table $T$ correspond to different numbers in table $Q$, so there is a one-to-one correspondence between the cells of the $9 \times 9$ table and pairs $(a, b)$, where $a, b=$ $1,2, \ldots, 9$. If we now take a new $9 \times 9$ table and in its cell corresponding to the pair $(a, b)$, write the number $9a + b - 9$, then each of the numbers from 1 to 81 will appear in such a table exactly once, and the sum of the numbers in each $3 \times 3$ square will be the same (it is equal to $9 \cdot 45 + 45 - 9 \cdot 9 = 369$, see the figure).

| 7 | 35 | 63 | 4 | 32 | 60 | 1 | 29 | 57 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 13 | 41 | 69 | 10 | 38 | 66 | 16 | 44 | 72 |
| 19 | 47 | 75 | 25 | 53 | 81 | 27 | 50 | 78 |
| 61 | 8 | 36 | 58 | 5 | 33 | 55 | 2 | 30 |
| 67 | 14 | 42 | 64 | 11 | 39 | 70 | 17 | 45 |
| 73 | 20 | 48 | 79 | 26 | 54 | 76 | 23 | 51 |
| 34 | 62 | 9 | 31 | 59 | 6 | 28 | 56 | 3 |
| 40 | 68 | 15 | 37 | 65 | 12 | 43 | 71 | 18 |
| 46 | 74 | 21 | 52 | 80 | 22 | 49 | 77 | 24 |

## Answer

Yes.