The square of the sum of the first n integers is equal to the sum of the first n cubes. Also, 91*19=Ramanujan number

Anonymous

To visualize the first of these, first notice that the square of the sum of the first k positive integers is exactly the sum of all products nm, with n and m between 1 and k. Then, if each nm corresponds to an n by m array of cubes, it’s just a matter of rearranging them to form cubes of side 1 thru k. This isn’t hard, but my saying that is actually code for I know how to do it but I’m too distracted to make the effort to make a graphic right now.

(re: the second: that is true.)

jackmusclescarier replied to your post “differentialprincess replied to your photo:This is the magic hexagon….”

that “tensor product”, i think, is the correct operation to make it “interesting”, and i suppose it works if you allow formal sums of magic hexagons of different sizes

yeah! the thing that I wonder about though is that in principle there’s size(B)*symmetries-of-a-hexagon ways to define the product A*B (if B is homogeneous (i.e., just one magic hexagon))

also this obviously generalizes to any other kind of magic thing which is fun

twocubes:

the identity is super magical what are you talking about

how is that the identity tho

hmmmmmmmmmm

so like, if we allow the numbers to be whatever integers, rather than just 1,2…n, we can combine magic hexagons to form other magic hexagons. You can multiply all the elements by a given integer, and if they’re the same size, you can add corresponding entries, and you can also replace all the entries of one hexagon by (that entry)*some other hexagon (which is an operation that doesn’t commute, but w/e).

which I think makes it… some kind of graded algebra? or something?

and the hexagon with 1 in it is the identity for that last operation

or

something

This is the magic hexagon. The numbers add up to 38 along each diagonal/vertical line.
It can be called the magic hexagon rather than a magic hexagon because there are no other hexagons numbered 1,2…n with this property, no matter how many layers the arrangement has.
(well, except for the one which is just one hexagon with ‘1’ written in it, and that’s hardly magical…)
(credit: Mathematical Gems I by Ross Honsberger)

This is the magic hexagon. The numbers add up to 38 along each diagonal/vertical line.

It can be called the magic hexagon rather than a magic hexagon because there are no other hexagons numbered 1,2…n with this property, no matter how many layers the arrangement has.

(well, except for the one which is just one hexagon with ‘1’ written in it, and that’s hardly magical…)

(credit: Mathematical Gems I by Ross Honsberger)