The next time you think about which aspects of the human mind are learned and which are inherent, I want you to remember that babies are not born with object permanence.
(I’ll make a picture tomorrow)
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
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.)
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
the identity is super magical what are you talking about
how is that the identity tho
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