Oct 21 - Magic Square Solution

1-9, all rows, columns and diagonals add to 15

Starting with the number 1, because it’s the smallest, how many ways can we add to 15?

1+5+9=15

1+6+8=15

Since there are only two ways to use 1 to add to 15, it has to go in one of the squares that only has 2 additions (put another way, a square without a diagonal)

	1?
1?		1?
	1?

How many ways can we use 9, since it’s the biggest?

9+5+1 = 15

9+4+2 = 15

So, 9 will be opposite from 1, since it also only has 2 ways to sum to 15, which places 5 in the middle square

	1? 9?
1? 9?	5	1? 9?
	1? 9?

 

Let’s try a specific alignment based on these 4 sums

6? 8?	1	6? 8?
3? 7?	5	3? 7?
4? 2?	9	4? 2?

 

Note that 8+3+4 = 15 and 6+7+2 = 15, which satisfies the edge columns

And 8+5+2 = 15, and 6+4+5 = 15 which satisfies the diagonals

So, one solution is

8	1	6
3	5	7
4	9	2