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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* (last edited 1 week ago) (1 children)

was about to fall asleep but then PROPOSITION:

all magic squares are constant multitles (all numbers multiplied by a certain value), or shifted (all numbers added to by a certain value) of a very few variations of "base" magic squares this seems intuitively true by a sleepy deprived person at 1am, but if we can prove for these "base" squares, would that be the way to prove for the general case?

i.e. there r and infinite number of 1x1 magic squares, but there is only one case to deal with if we want to prove anytning about it

[–] [email protected] 2 points 1 week ago* (last edited 1 week ago)

yeah once we have a non-invertible base we can construct many more magic squares using construction principles … Some of these for uneven size are outlined in mathloggers youtube videos …

if we swap rows or columns inside a Matrix the absolute value of its determinant will not change, it will stay invertible/non singular using these operations … and it will also stay a magic square …

For example the following will lead to a magic square, if we start from a non singular magic square we will end with one:

swap row 1 and 2
swap column 1 and 2.

By doing this we transform …

ABC
DEF
GHI

into

EDF
BAC
HGI

Due to commutativity of addition operation these row/col swaps also dont change the inversibility of the matrix and result in a "new" magic square.