Mean Square

The mean of 2 numbers will always be between the 2 numbers.

Whichever number goes in the middle will need to be the mean of the most different pairs of numbers, so it must be the 5, because the 5 is in the middle of all of the numbers.

	5

Then the numbers which go either side of the 5 must be pairs of numbers that add up to 10 (because to find the mean of 2 numbers, they must be added together and then divided by 2). So they must be:
1 and 9
2 and 8
3 and 7
4 and 6

9 cannot be the mean of two of the other numbers, because it is the largest number. So it must go in one of the corners.

1
	5
		9

9 is odd, and if it is added to an even number, then the result will be odd. When this odd number is divided by 2, the result will not be a whole number. So the numbers in the other corners cannot be even numbers. So they must be 3 and 7.

1		3
	5
7		9

Filling in the remaining boxes gives the completed square.

1	2	3
4	5	6
7	8	9

Other solutions can be obtained by rotating and reflecting this one.