21–30 of 36 posts
Huh, this potential technique seems fairly neat. Thank you for explaining the whole thing in a fairly accessible way. Enjoyable interactive bits as well. An aside is that the playground looked fine on my iPhone. The anticipatory objection of smallness did not materialize.
Then again, I do mostly remember this in terms of the yet unsolved magic-square-of-squares problem, not the standard magic square.
I loved this article and its interactive elements. The potential field is an elegant abstraction which really elevates this from a math puzzle into something new. I'd love to see just how Lipschitz continuous, how smooth, the potential field can be; how adding features puts it closer to or further from solutions that fit the consecutive-no-duplicate constraint, say. Adding a smooth 'hill' is probably viable; is a 'ri…
Thanks a lot for the detailed perspective! The potential fields are a curious phenomenon. I feel like - and I haven't verified this in any way yet - the smoothness is dictated by the fact that all values are within a not-so-broad range. Then as you "peel" the hexagon from the outer layer using those 6-rings, building the potential field, each next layer inwards shouldn't changed too much. I had also explored a case o…
I would guess that the smooth mountain-looking structure come from a very simple observation: a gadget consisting, in potential space, of a 2 surrounded by a ring of 6 1’s fully cancels at the center and in the ring immediately around the center and leaves a nice pattern of +1 and -1 residuals in the ring around it. I suspect that, fairly generally, as you try to build out small numbers around the outside of the magic hexagon, you end up with a large pile of things like this in the center, and a sum, even a very noisy one, of things that even vaguely Gaussians, tends to produce Gaussians. (That’s the central limit theorem.)
I loved this article and its interactive elements. The potential field is an elegant abstraction which really elevates this from a math puzzle into something new. I'd love to see just how Lipschitz continuous, how smooth, the potential field can be; how adding features puts it closer to or further from solutions that fit the consecutive-no-duplicate constraint, say. Adding a smooth 'hill' is probably viable; is a 'ri…
Earlier quoted context omitted.
Thanks a lot for the detailed perspective! The potential fields are a curious phenomenon. I feel like - and I haven't verified this in any way yet - the smoothness is dictated by the fact that all values are within a not-so-broad range. Then as you "peel" the hexagon from the outer layer using those 6-rings, building the potential field, each next layer inwards shouldn't changed too much. I had also explored a case o…
Can you clarify what the “potential field” actually is? The text is not really precise, and the little interactive tool has the very curious property that I can set all the potentials to 0 and I don’t get all zeros in the magic hexagon. I would guess that the smooth mountain-looking structure come from a very simple observation: a gadget consisting, in potential space, of a 2 surrounded by a ring of 6 1’s fully cance…
The "wide ring" gadget is an interesting idea. I think it will be linear pyramids and not gaussian. A set of these gadgets can also form a basis, and in such a different basis the potential fields may look very different - almost flat, perhaps?..
Good stuff. Maybe I missed it in the post, but I didn't see a formal proof of the conjecture?
Why is not every 45 degree line considered for the rectangular grids? (In the hexagons, all lines are considered even if they don't have the maximum length) (PS: make sure you hover your mouse over the diagrams)
The shortest 45 degree line in a magic square is a single cell, making it somewhat tricky to add up to the same total as the other lines.