52. Magic Squares - Line Total
 
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magicsquares 400x154_filled



'Line' Total
We shall use 'line total' to mean a line or a row or a diagonal or the four corners' total.

1. What are the line totals?
Easy! 15, 34 and 65.
But this is possible only after making-up a square, fill the numbers and then add a line or column etc.

2. Mentally
Imagine the boxes are filled with the average of all the numbers.
3 x average or 4 x average or 5 x average is the line total.

3. Average
The 3x3 boxes are filled with 1, 2, 3, ..., 9.
Average is 5. How?

Basic method is to add 1, 2, 3 ..., 9 and divide by 9.
But you can also take the middle number of 1, 2, 3, ..., 9!

Average is also the average of the first and last numbers ( 1 and 9)!
Average is also the average of a pair of symmetrically placed numbers,
like 2 and 8 or 3 and 7 or 4 and 6 or 5 and 5!

4. Line Total
Suppose you fill a 3x3 square starting with 23. What will be the line total?
Last number will be 22+9=31;
Line total is 3 x (23+31)/2 = 3 x 27 = 81.

If you fill a 4x4 square starting with 15:
You wil end in 14+16=30 and
Line total is 4 x (15+30)/2 = 2 x 45 = 90.

If the line total of a 5x5 square is 175:
Average is 175/5 = 35;
Filling numbers start 12 below 35, i.e. 23;
Ending number is 12 above 35, i.e. 47.
So the numbers are 23, 24, 25, ..., 47.

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