52. Magic Squares - Line Total | |||
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'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.