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## Summary:

- Are there additional theorems one can show from an incomplete 5x5 magic square that completing it is possible or impossible?

A magic square is a NxN array of numbers from 1 to N

15 19 4 7 20

6 14 24 16 5

9 21 1 22 12

10 3 23 18 11

25 8 13 2 17

There are a number of theorems one can prove about incomplete 5x5 squares. For example,

14 X 4 X 20

X 15 24 16 X

9 21 1 22 12

X 3 23 18 X

25 X 13 X 17

will never complete to form a magic square because it fails what I call the "nine minus four" theorem. All magic squares have the difference between the sum of the inner nine cells minus the four corners equal to 65. For the second square, this is 67.

There is also what I call the parity theorem. The 5 cells that form a + sign centered on the center cell (24,21,1,22,23) must have an odd sum.

Are there other theorems, independent from these two, that can be used to determine if an incomplete 5x5 square can be completed as magic? For example,

14 X X X 20

X 15 X 10 X

X X 1 X X

X 13 X 16 X

21 X X X 19

cannot be completed. (Exhaustive test). Can one tell this from the 9 numbers already placed?

^{2}such that the sum of elements of each row, column and diagonal adds up to the same number, 65 in the case of 5x5's. An example would be:15 19 4 7 20

6 14 24 16 5

9 21 1 22 12

10 3 23 18 11

25 8 13 2 17

There are a number of theorems one can prove about incomplete 5x5 squares. For example,

14 X 4 X 20

X 15 24 16 X

9 21 1 22 12

X 3 23 18 X

25 X 13 X 17

will never complete to form a magic square because it fails what I call the "nine minus four" theorem. All magic squares have the difference between the sum of the inner nine cells minus the four corners equal to 65. For the second square, this is 67.

There is also what I call the parity theorem. The 5 cells that form a + sign centered on the center cell (24,21,1,22,23) must have an odd sum.

Are there other theorems, independent from these two, that can be used to determine if an incomplete 5x5 square can be completed as magic? For example,

14 X X X 20

X 15 X 10 X

X X 1 X X

X 13 X 16 X

21 X X X 19

cannot be completed. (Exhaustive test). Can one tell this from the 9 numbers already placed?