Super square
81 numbers are placed in a 9x9 grid such that the sum of the 9 numbers in any 3x3 square is .
What is the sum of the numbers in the 9 squares where and take on the values 1, 5, and 9.
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1 solution
In the above figure we are required to find the sum total of all the red squares.
Since the 9 × 9 grid can be partitioned into 9 - 3 × 3 squares ,the total sum of all the squares in the grid will be 9 ( e + i π )
There are 4 - 3 × 3 squares in the middle as shown.
A horizontal block of 3 elements will preserve its sum if its row is incremented/decremented by a multiple of 3
A vertical block of 3 elements will preserve its sum if its column is incremented/decremented by a multiple of 3
Thus the blue blocks and the green blocks form 4 - 3 × 3 squares in total.
Thus we can find the sum of the red squares by subtracting the sum of middle squares & the sum of blue and green blocks from the total.
sum of red squares= 9 ( e + i π ) − 4 ( e + i π ) − 4 ( e + i π ) = ( e + i π )