Magic square - Wikipedia
en.wikipedia.org

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.
  • siriusmart@lemmy.worldM
    2·
    2 days ago

    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

    • scratsearcher 🔍🔮📊🎲@sopuli.xyzOPEnglish
      1·
      1 hour 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 also dont change the inversibility of the matrix and result in a “new” magic square.

  • siriusmart@lemmy.worldM
    1·
    2 days 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