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[Unsolved] Magic squares and their Inverse

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 = โ€ฆ = 15

Lets view it as a 3x3 Matrix, its determinant is ฮ” = -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 โ€ฆ

  • exists (i.e. every non-trivial magic square has an inverse)
  • is a magic square.
[Unsolved] Magic squares and their Inversehttps://en.wikipedia.org/wiki/Magic_squareOpen linkView original on sopuli.xyz
7
Comments5
siriusmart
lemmy.world

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

2
scratsearcher ๐Ÿ”๐Ÿ”ฎ๐Ÿ“Š๐ŸŽฒreply
sopuli.xyz

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

  • 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 โ€ฆ and it will also stay a magic square โ€ฆ

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 โ€ฆ

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 โ€ฆ

2
siriusmart
lemmy.world

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

2
siriusmart
lemmy.world

what about this square, it does not have an inverse

1 1 1
1 1 1
1 1 1
2

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[Unsolved] Magic squares and their Inverse | Spyke