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[Unsolved] Magic squares and their Inverse (en.wikipedia.org)

submitted 2 weeks ago* (last edited 2 weeks ago) by [email protected] to c/[email protected]

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cross-posted from: https://sopuli.xyz/post/22688165

Random thought on magic squares:

If I view the smallest possible non-trivial magic square
2 7 6
9 5 1
4 3 8
since its rows and diagnoals sum up to 2+5+8 = 2+7+6 = 4+5+6 = 2+9+4 = … = 15

Lets view it as a 3x3 Matrix, its determinant is Δ = -360 . Its inverse:
-37/360 19/180 23/360
17/90 1/45 -13/90
-7/360 -11/180 53/360
note how this is a magic square, rows and diagonals sum up to 1/15.

https://matrix.reshish.com/inverse.php

Now if you are really bored (I can not do this): proof that for any non trivial magic squares the inverse …

exists (i.e. every non-trivial magic square has an inverse)

is a magic square.

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[–] [email protected] 2 points 1 week ago

this is actually interesting, im currently on bed so im not writing a proof, but the rule for a 3x3 magic square of conseqetive numbers seemed to be (tested on one other example) that its inverse is 1/(3*middle number), the 3 could be the dimension of the square?

if this is true, this begs the questions:

is it still true for squares of non conseqetive numbers?
what happens when the square is 4x4 where a centre number doesnt exist

either way im gonna have a go at it when i wake up tmr, and im pinning this for being one of the coolests finds in the community