If you Google magic squares, you'll find a big literature on
them.
What's the easiest magic square to make? In my opinion,
it's this one:
16 02 03 13
05 11 10 08
09 07 06 12
04 14 15 01
It's a type of magic square called an associative magic square because every pair of numbers equidistant from the center sum to 17. And, as per usual for a normal 4 x 4
magic square, the sum of every row, column and diagonal is 34 (which is true for all 880 4thorder magic squares).
How is that magic square derived? Start with:
01 02 03 04
05 06 07 08
09 10 11 12
13 14 15 16
Just flip the diagonals will give you the magic square.
I played around with the magic square and rotated the quadrants 180° to give me this magic square:
11 05 08 10
02 16 13 03
14 04 01 15
07 09 12 06
Another associative magic square. I studied it and made a discovery (which I believe I'm original with).
Those two magic squares have many nice properties in relationship to the number 34 plus other properties in relationship to multigrades (Google multigrade equations). However the second magic square has a distinct property in relationship to its row and columns, but not its diagonals, which is why it's termed semimagic with respect to that property which I'm leaving as a puzzle for someone to try to solve.
If you find the puzzle too hard to figure out, I may drop a few more hints to help you out. So what is that property the second magic square has?
PhilX
Magic square puzzle
Re: Magic square puzzle
Can you explain the connection to the philosophy of mathematics?

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 Joined: Sun Aug 31, 2014 7:39 am
Re: Magic square puzzle
Sure can. There is no connection to the philosophy of mathematics because there is no philosophy of mathematics.
No one has established a foundation of mathematics because there are several leading proposed foundations of math. For example there is logicism which Hilbert was a leading proponent. Then we have intuitionism of which Brouwer was a leading proponent (currently though there's been a decline in mathematicians favoring this branch of math).
We have Godel's theorem that says in any branch of math large enough to include the laws of arithmetic, there will always be unprovable theorems.
So for me, there is no philosophy of mathematics. Nothing to hang your hat on.
I'm a math explorer. I have a preference for recreational math in which there is plenty to explore and discover. I've put up a puzzle you won't be able to find on the internet.
Are you ready for those hints?
PhilX
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