this post was submitted on 15 Feb 2025
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[Unsolved] Magic squares and their Inverse (en.wikipedia.org)
submitted 1 month ago* (last edited 1 month ago) by glowing_hans@sopuli.xyz to c/dailymaths@lemmy.world
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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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[–] siriusmart@lemmy.world 2 points 1 month 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

[–] siriusmart@lemmy.world 2 points 1 month ago* (last edited 1 month 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

[–] glowing_hans@sopuli.xyz 2 points 1 month 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:

  1. swap row 1 and 2
  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 result in a "new" magic square. Row/coll swaps also dont change the inversibility of the matrix …

[–] siriusmart@lemmy.world 2 points 1 month ago (1 children)

what about this square, it does not have an inverse

1 1 1
1 1 1
1 1 1
[–] glowing_hans@sopuli.xyz 2 points 1 month ago

That magic square would be called trivial, since it contains the same entry multiple times.

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