This week the #puzzle is: Can You Tile the Hexagon? #counting #hexagon #tiling #macmahon #coding #memoization

I’m redoing my kitchen floor using rhombus-shaped tiles composed of two congruent equilateral triangles. One such tile is shown in blue below. How many distinct ways can I use these to tile the outlined region below, which consists of 24 equilateral triangles arranged in a regular hexagon?

And for extra credit:

I’m also redoing my patio, using similar rhombus-shaped stones. How many distinct ways can I tile the outlined region below, which consists of 54 equilateral triangles?

Can You Tile the Hexagon?

Solution, possibly incorrect:

Program

Method 1: I quickly whip up a program. And I add memoization to it. And it arrives at a state, where it can find the answer in 3 minutes.

Result: 20.

I also produce a video, documenting all the possible tilings. (Beware: this post has a lot of videos and images.)

And just to go nuts, I produce screenshots from that video. Yes, I know, you don’t have to thank me.

I go through my notes and end up producing small families of solutions, where each family is connected by rotations. Like this one, with 6 rotations of the same solution.

Or this one:

A smaller family.

And an even smaller.

Like this one:

And finally a singleton.

Mirrors just end of producing the same results as rotations.

I will get back to the fiddler again later.

And for extra credit:

Wikipedia YouTube

Method 1: Run the program again. Ehm. That will take too long. Sigh.

Method 2: Search internet. Hey, what’s this?

“the total number ${\displaystyle N_1(r,s,t)}$ of plane partitions that fit in the ${\displaystyle r\times s\times t}$ box ${\displaystyle \mathcal{B}(r,s,t)}$:”

I think the box is if the hexagon tiling is viewed in a 3d way. The “box” has the same side lengths as the hexagon. The plane partitions correspond to the tilings. Yeah, this one confirms it:

And just to make sure, the fiddler has a 2x2x2 box.

Or, from the other formulation of the count:

Awesome! And the video already did the calculation we want. $\underline{N=980}$.

More extra credit later.

The fiddler, again

Studying the sources mentioned above, I find a different method to produce the same solutions, with slightly squinked rhombi:

This set is numbered like xy: x cubes, the yth way.

The 2 sets of solutions correspond. To make me (and probably nobody else) happy, here’s the full correspondence description.

01	1	11	2	21	7	22	4
23	3	31	9	32	8	33	5
41	15	42	10	43	12	44	6
51	16	52	13	53	11	61	18
62	17	63	14	71	19	81	20

And just to show that these are the same solutions, here’s a comparison of 2 versions of the same, 9 and 31:

Or maybe it’s easier to compare these 2 prettier versions of the same solution:

I then proceed to create a new program, to make really pretty versions of these tilings. Here’s you go, fiddler:

And extra credit: