Sum of the numbers from 37 to 52 = (n/2)[2a + (n-1)d] = (16/2) [74+15] = 8 * 89 = 712

Therefore the sum of all numbers in the grid = 712

If x is the sum of all numbers in a row/column/major diagonal, then

4x = 712

=> x = 178 = sum of all numbers in any row = sum of all numbers in any diagonal = sum of all numbers in either major diagonal

Now, consider the grid as follows:

1' 2' 3' 4'

5' 6' 7' 8'

9' 10' 11' 12'

13' 14' 15' 16'

We know that 1' + 6' + 11' + 16' = 178

4' + 7' + 10' + 13' = 178

=> 1' + 6' + 11' + 16' + 4' + 7' + 10' + 13' = 356

=> 6' + 7' + 10' + 11' + 1' + 13' + 4' + 16' = 356

Also 5' + 6' + 7' + 8' + 9' + 10' + 11' + 12' = 356

and 1' + 5' + 9' + 13' + 4' + 8' + 12' + 16' = 356

=> 6' + 7' + 10' + 11' - 1' - 13' - 4' - 16' = 0

Therefore 6' + 7' + 10' + 11' = 356/2 = 178 = sum of the middle four numbers

Option (C).

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