The digits form three 3-digit numbers reading left-to-right, and three 3-digit numbers reading top-to-bottom. Also consider these six 3-digit numbers reversed, that is, reading right-to-left and bottom-to-top.

For each of those nine digits, d, you count how many of the twelve 3-digit numbers are divisible by the number d. In the case of each d, that number, the count, is itself divisible by d.

How are the numbers arranged, with the understanding that rotations and reflections of that arrangement are possible alternatives?

The twelve numbers are:
164, 179, 239, 284, 356, 461, 482, 653, 758, 857, 932 and 971.

Where d is 1, each of the twelve numbers are, of course, divisible by d. The count for 1 is 12. 12, of course, is divisible by 1.

Half of the numbers are even, therefore, when d is 2, half the numbers -- that is, 6 -- are divisible by 2. 6 is also an even number, and, thus, divisible by 2.

One third of the twelve numbers are divisible by 4. Thus, where d is 4, the count of divisiblity by 4 is 4. 4 is obviously divisible by itself.

The remaining counts of divisiblity are each 0. 0, divided by any number (other than 0) is 0. Thus each of the other counts are also divisible by d.

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

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