Comment by matsemann
2 days ago
> In two dimensions, the answer is clearly six: Put a penny on a table, and you’ll find that when you arrange another six pennies around it, they fit snugly into a daisylike pattern.
Is there an intuitive reason for why 6 fits so perfectly? Like, it could be a small gap somewhere, like in 3d when it's 12, but it isn't. Something to do with tessellation and hexagons, perhaps?
> They look for ways to arrange spheres as symmetrically as possible. But there’s still a possibility that the best arrangements might look a lot weirder.
Like square packing for 11 looks just crazy (not same problem, but similar): https://en.wikipedia.org/wiki/Square_packing
Three pennies form an equilateral triangle with (of course) 60 degree angles.
Six of those equilateral triangles will perfectly add to 360 degrees. Intuitive enough? (I'm being a little hand-wavey by skipping over the part where each penny triangle shares two pennies with a neighbor — why the answer is not 18 for example.)
For my mind though, the intuitiveness ends in dimension 2 though. ;-)
It would be fun to make that square packing for 11 from wood and give it to puzzle enthusiasts with this task: Rearrange the squares so you can add an additional 12th square. And then watch them struggle putting even those 11 squares back in.