I found that as long as you have an odd number in all the corners and the middle then the pattern will always be odd. Because the difference between ANY odd number and ANY even number will always be odd.
E.G 6 and 9, the difference is 3
21 and 32, the difference is 11
69873 and 69882 is 9
You can not make a pattern of even numbers because the difference between an odd number and an odd number is and even number, and the difference between an even number and an even number is an even number is an even number. But somewhere an odd number and an even number will cross on the grid, making a difference of and odd number.
This is shown below:
6 8 2
4 5 3
9 7 1
Can you see where the odd and even number meet? It makes a difference of and odd number, so it makes it clear that it won't work.
Look below to see the odd number difference solutions that I came up with.