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Magic Love - February 13, 2011
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Spiked Math Comic - Magic Love


The (geo)magic square above is based on one originally created by Lee Sallows and modified by using a heart instead of a fish.



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18 Comments

I don't like the redundancy in the "sums" on the left AND right hand sides and the top AND bottom.

I also thought Love was a bit of a puzzle. Nice to see how the parts all fit ...

Magic squares were the first things I learned while teaching myself MATLAB last year, so this is particularly appealing to me - thanks for all the excellent comics, Mike!

Maths is my valentine. <3

Same here......

best valentine's day card EVER

I'm glad he didn't have to work out a 6x6 magic square...

I think my favorite two sums are the ones that are made through the diagonals (one each) of the hearts "offset" in the squares (the bottom two hearts). Even though those two solutions aren't displayed around the perimeter, is that a guarantee for this kind of magic square? I can't seem to mentally reorder the images to make that not happen and still satisfy the rest of the solutions.

I don't understand which two sums you're referring to. :( Which squares are involved?

I think he means the four corners and the center 2x2 square. They all make solid hearts, but they don't appear with the other sums. Interesting ...

Ah, okay. Well, I don't know if it's a guarantee that they will, but iff the four corners do make a solid heart, then the center 2x2 square must as well (since for the diagonals to work, the corners plus the center square must equal 2 <3 [that is, 2 hearts; this notation could get confusing.])

If you take the centre of the edges and pair them horizontally and vertically, you get another two sums. {(1,2),(1,3),(4,2),(4,3)} and {(2,1),(3,1),(2,4),(3,4)}.

Actually, let's label the grid with letters for the X axis and numbers for the Y (chess-like, bottom-left to top-right).

The sums I was referring to are from the hearts in A2 and C1. From A2, progress diagonally up and left (meaning you'll loop to the other side of the board, and again loop over the board). That sum adds up to fill the heart appropriately, but doesn't appear along the outside.

From C1, progress diagonally up and right, and the same thing will occur. (For each case, A2 and C1, if you progress diagonally the other way, you'll cross two blue pieces and another heart).

But it's equally interesting that the outsides and insides add up correctly, as well, especially considering that gives an additional solution to each of the two hearts I wasn't referring to.

Interestingly, that did make me notice that the two hearts that I have been referring to have even more additional solutions that mirror the outside and inside hearts solutions that you just brought up.

Without changing the relative positioning of the squares, consider how many ways you can adjust the absolute boundary of the overall grid, and these additional solutions emerge.

Because the grid boundary seems to become arbitrary, I'm beginning to lean towards the idea that these initial solutions must always be there.

I reread that and I am terrible at explaining things with words. So screw it, hopefully my superawesomeexpialidocious powerpoint skills are easier to understand :) Solutions:

http://i541.photobucket.com/albums/gg391/PlaneShaper_2008/Spiked_Math/Hearts.png

The two extra solutions I was originally referring to are the extra two diagonals in the top right. But as you can see, there are lots of others.

My question is, if the conditions for the bottom right solutions are satisfied (the classic grid), are the conditions for all other solutions satisfied, as a rule?

Not all magic squares are algebraic. (There are 432 algebraic magic squares, depending on how you count them.) So I guess not all geomagic squares will be toroidally magic.

OK, last one cause I'm tired and don't want Mike to think I'm spamming or anything :)

http://i541.photobucket.com/albums/gg391/PlaneShaper_2008/Spiked_Math/Hearts2.png

There's a bunch of potential solutions that aren't satisfied with the current layout as well.

Overall (counting duplicates) I count 64 potential solutions, with 36 of them being satisfied (mainly because the second picture I posted "counts" as 16 solutions due to duplicates).

Are there more?

There's so many combinations that aren't shown. There's rows, columns diagonals, then there's also four corners, four center, 4 2 by 2 boxes, the corners of each 3 by 3 box, and then if you switch the right and left halves, there's a new set of diagonals and four corners and four in the center.

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