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The Maths Group from Devonshire Primary sent in some solutions that they found so that the differences between joined squares is odd:

9 2 7   967   361   769   567   967   967   965   967   967   947   367   123   321
8 5 4   854   854   854   894   814   834   874   254   852   856   854   458   854
3 6 1   321   927   123   321   325   521   321   381   341   321   921   769   967

Children from Merton Park said: You have to have an odd number next to an even number or it (the difference) won't be odd.
You can have any numbers in any place as long as the odd numbers are in the corners and middle.

Well done - you're spot on and in fact that means there are many solutions - thousands! Taranjot and Amrita at Alexandra Junior School point out that:

It is not possible to make even differences using each number once. Even numbers cannot be next to even numbers and odd numbers cannot be next to odd, because their differences would be even.

Harmohan, Akash and Ayman also from Alexandra School came up with some general rules about odd and even numbers:

Even is e and odd is o
e-o=o
o-e=o
e+o=o
o+e=o
e-e=e
o-o=e
o+o=e
e+e=e

Thank you to all those of you who sent in solutions.