Thanks to Sam Shah for introducing me to this fascinating online game: Entanglement.
The rules are simple. You are given hexagonal tiles, one at a time, each adorned with six short segments of rope. Use them to construct the longest possible knot (measured in segments) before running into a wall. Entanglement is fun and addicting!
How high can you go? A quick analysis shows that the highest possible score is 169. How do we come up with that value?
Each tile has six segments on it—two ends on each side of the hex. There are spaces for 36 hexagons. Thus a full board will have 6*36=216 strands. However, some of these strands will end at a wall. To be precise, there are 48 boundary sides. One of those boundaries is the starting wall (and the ending wall of a “perfect game”). So a perfect game must contain at least 47 unused strands (such as the strand shown above that starts and ends at the central hex). Thus it is impossible to get a score higher than 216-47=169.
Sure, that is a theoretical upper bound. Is it attainable? It turns out that it is! A player named “atomic” got a perfect 169 and there is a screenshot to prove it.
Now the only question is, what knot is that?
Update: Thank you to commenter “Evan” for pointing out a very similar board game, Tantrix, which came out in the late ’80′s. Also, I want to mention KnoTiles which was given to me by my friend Gene Chase—also very fun.
Well I can answer the last one: it’s no knot at all. It might be a tangle, though.
By: John Armstrong on August 24, 2010
at 9:52 pm
Of course you’re right (but I was thinking of them being connected at the center to become a knot).
By: Dave Richeson on August 24, 2010
at 10:00 pm
Looks like a knock-off of Tantrix: http://en.wikipedia.org/wiki/Tantrix
I’ve got the Tantrix Game Pack, which allows for up to four players. It’s quite fun to play with the tiles.
By: Evan on August 24, 2010
at 11:50 pm
Thanks, Evan. I don’t think I’ve ever seen Tantrix.
By: Dave Richeson on August 25, 2010
at 9:45 am
[...] http://divisbyzero.com/2010/08/24/a-game-for-budding-knot-theorists/http://divisbyzero.com/2010/08/24/a-game-for-budding-knot-theorists/ [...]
By: 一个小游戏：Entanglement « ELLY66 on August 25, 2010
at 6:58 am
I would say it’s some 13 crossings knot. But there’s hella of them :)
By: Czeckie on August 26, 2010
at 7:33 pm
I agree that it seems to have 13 non-trivial crossings. I guess it’s also a composite knot made of three trefoil knots and a few crossings.
By: Rick Meese on August 30, 2010
at 8:35 pm
[...] by kylejburns — Leave a comment September 24, 2010 In Division By Zero’s blog, A Game for Budding Knot Theorists, he introduces the game Entanglement (CAUTION: Entanglement is rather addicting. Proceed with [...]
By: Knot-ty Games « Kyle Wants to be a Mathematician on September 24, 2010
at 5:44 pm