# Problem 7. (4 points)

Can the numbers from 0 to 999 (each used exactly once) be arranged in a $100 \times 10$ rectangular table so that the sum of the numbers in each $2 \times 2$ square is the same?
Answer: Yes, it can.

Solution:

The desired table is obtained as the sum of two tables.

The first table looks like this:

| 0 | 99 | 0 | 99 | 0 | 99 | $\ldots$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 0 | 99 | 0 | 99 | 0 | 99 | $\ldots$ |
| 1 | 98 | 1 | 98 | 1 | 98 | $\ldots$ |
| 1 | 98 | 1 | 98 | 1 | 98 | $\ldots$ |
| 2 | 97 | 2 | 97 | 2 | 97 | $\ldots$ |
| 2 | 97 | 2 | 97 | 2 | 97 | $\ldots$ |
| $\ldots$ |  |  |  |  |  |  |
| 49 | 50 | 49 | 50 | 49 | 50 | $\ldots$ |

In this table, any number, together with its horizontal neighbor, sums to 99, so the sum in any $2 \times 2$ square is 198.

| 900 | 0 | 800 | 100 | 700 | 200 | $\ldots$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 0 | 900 | 100 | 800 | 200 | 700 | $\ldots$ |
| 900 | 0 | 800 | 100 | 700 | 200 | $\ldots$ |
| 0 | 900 | 100 | 800 | 200 | 700 | $\ldots$ |
| 900 | 0 | 800 | 100 | 700 | 200 | $\ldots$ |
| 0 | 900 | 100 | 800 | 200 | 700 | $\ldots$ |
| $\ldots$ |  |  |  |  |  |  |

In this table, any number, together with its vertical neighbor, sums to 900, so the sum in any $2 \times 2$ square is 1800.

When these tables are summed, the sum in each $2 \times 2$ square will obviously be 1998.

It remains to understand that all numbers will appear in the table.

Consider the placement of a number of the form $100k$ in the second table. This number is located in two adjacent columns on cells that, when colored in a checkerboard pattern, are of the same color. In the first table, in these two columns, all numbers from 0 to 99 appear, with each number appearing once on a black cell and once on a white cell. Therefore, the number $100k$ will be summed with each number from 0 to 99 exactly once. Thus, we will get all numbers from $100k$ to $100k+99$ each exactly once, and therefore, in the entire table, we will get all numbers from 0 to 999.