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Subject: Mathematical Recreation: How many tiles in a set of Hexadoms? rss

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K Septyn
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I happened to bump into Hexadoms while looking for another game, but domino-style games always seem to catch my mathematical eye, and this one was no different. At BGG I didn't see any indication of how many tiles were in the set, so the puzzle became one of finding that number.

The pictures I found show the hexes are divided into 6 triangular areas (sextants?), and opposite areas have the same number. This seemed like a good rule to observe, as it would keep the number of tiles reasonable. The pictures don't show anything that would indicate a "starting" or otherwise unique area, so rotational symmetry would also keep the number low.

If there were a unique area, the number of tiles in a 3-high set (like the one pictured) would be (3+1)^3 = 64. This seems too large for the size of the package, and there's no sign of that unique area anyways.

I wanted to see if there were any special rules visible in the tiles that would restrict their ordering. For example, Tri-Ominoes requires the numbers increase as they are listed clockwise around the tile, same for Quad-Ominoes. Some Google searches found a picture of a full face-up set--there are 24 tiles total. The puzzle was solved....

But 24 tiles seems like a small number for a domino-like game. Sure, a traditional double-6 set of dominoes has 28 tiles, but a double-9 set has 55, and the Tri-Ominoes set ("triple-5") has 56. How much bigger could a "triple-4" set of Hexadoms be?

The picture showed there wasn't any apparent restriction on number placement: the set contains both "012" and "021". That would make the math a little easier.

I'll skip the math discovery phase to say there is a formula for determining the number of tiles in a triple-n set of Hexadoms. It's identical to the "number of ways to color vertices of a triangle using
 
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K Septyn
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...and of course, the rest of the post was deleted in transit due to a less-than sign. Yay.


The Online Encyclopedia of Integer Sequences (OEIS) lists this particular sequence as "A006527". This is "Number of ways to color vertices of a triangle using <= n colors, allowing only rotations." That makes sense given we're dealing with the rotation of a duplicated 3-number set.

My earlier post was a little more wordy, but I'm going to sum up:

The formula for tiles in a Hexadom set is (x^3 + 2*x)/3, where n = highest number in the set and x = n + 1. For example, the published game is a triple-3 set, 3+1 = 4, and 4 into the formula gives 24 as the answer.

A triple-4 set of Hexadoms doesn't seem too unwieldy, it's only 45 tiles total. The triple-5 set isn't bad either at 76 tiles (less than the 91 of a double-12 domino set), but a triple-6 set would need 119 tiles (one less than a double-14 domino set), and I think that's going to be way too many hexagons for the average domino player.
 
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