Always the Same

Thus replacing the ascending numbers with 8.5 in every cell and circling four cells gives a total of 34. Or as Natalie did, she realised that "you pick numbers from each column and row" and took the average between the sum of the four columns:

i.e. (28 + 32 + 36 + 40)/4 = 34

A good solution with this method came from Melanie and Rachel of Flegg High School.

A proof of this problem could be as follows.

Let the first number be a.

Then when choosing numbers from rows and column that do not coincide we have: a + (a + 4) + (a + 8) + (a + 12) + 1 + 2 + 3 = 'Magic number' We add 1 because there is one number in the second column, 2 because there is one number in the third column and 3 because there is one number in the fourth column.

Hence:

4a + (4 + 8 + 12) + ( 1 + 2 + 3) = 34
i.e. 4a + 30 = 34
i.e a = 1
and the array is 1 through 16 as set.

But suppose the 'magic number' had been 62 then

4a + 30 = 62
i.e a = 8
and the array would have been 8 through 23.

Hope that the explanation above helps especially Josh at Russell Lower School to "work out where we went wrong".

We could have used a 5 by 5 array of ascending numbers!