Each outer pair of numbers will add up to the same total. As there are four pairs of numbers, we know that the combined total of these outer number pairs must divide by four to give a solution. We can also see that the numbers 1 to 9 to add to 45 (a quick way to check this is to look at the middle answer above: four pairs adding to ten, plus the five gives us 45). So if we take one number away from 45, as it will be in the centre of the puzzle, what must the other eight numbers add up to?

With 1 in the middle, the other eight numbers add to 45 – 1 = 44, the highest possible total for the eight numbers.  This divides by four to give 11, and so we had four pairs that added to 11. With 5 in the middle, the rest of the numbers add to 45 – 5 = 40, which give four pairs that add to 10. With 9 in the middle, the other numbers add to 45 – 9 = 36, which divides by four to give nine, and is the lowest possible total for the numbers. There are no numbers in the range from 36 to 44 that divide by four with no remainder; so these are the only possible answers.