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Subject: Non-Discrete Hex and Y rss

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Nick Reymann
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I just thought of something amazing (to me at least) that has probably been thought of before, but I haven't seen anything similar online yet: Non-Discrete Hex and Y.

To play ND Hex, use a quadrilateral and designate opposite sides as usual. For ND Y, use a triangle. Instead of there being discrete places to put a piece down, one simply places a circle of their color with a radius less than or equal to some constant length (specified at the beginning of the game) in any non-occupied area of the board. Connections are established between tangent circles. Standard goals apply (try to connect your designated sides in Hex; try to connect all three sides with one chain in Y).

Now, one can play these with a swap rule as a handicap tool. However, there is a really interesting alternative: force the first move to have a smaller radius than normal. Why is this interesting? Let's use Y as an example. Take an equilateral triangle. Now, suppose that the first player is allowed to make a circle right in the middle, big enough to touch each side. Easy win. Now, suppose that they have to make it slightly smaller, so that one in the center is a tiny bit away from touching each side. Still an easy first-player win, as you can try out for yourself.

Now, suppose the first move has an incredibly small size restriction, so that the circle's radius can only be 1/1,000,000th of the triangle's side length. Now, with a little scratchwork, you will find that it is now a second player win, provided that the normal length restriction is big enough. If you haven't figured out what the big deal is by now, here it is: somewhere, as the first move size limit decreases from a size where the first player wins, to zero, there is a shift in who will win given perfect play.

Here is the mystery I am trying to solve: does the switch happen without anything in between, or is there a value or range of values where, given perfect play, the game will go forever, with the circles the players are using getting smaller and smaller to fit in the increasingly tinier spaces?
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David Molnar
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Mingy Jongo wrote:

Here is the mystery I am trying to solve: does the switch happen without anything in between, or is there a value or range of values where, given perfect play, the game will go forever, with the circles the players are using getting smaller and smaller to fit in the increasingly tinier spaces?


Not sure i understand the question. I thought that the protocol you were suggesting was that the first move would be "of size 1/n" and all subsequent moves would be of size 1. If that's the case, the game has to end.

See also Calculus.
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Nick Reymann
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molnar wrote:
Mingy Jongo wrote:

Here is the mystery I am trying to solve: does the switch happen without anything in between, or is there a value or range of values where, given perfect play, the game will go forever, with the circles the players are using getting smaller and smaller to fit in the increasingly tinier spaces?


Not sure i understand the question. I thought that the protocol you were suggesting was that the first move would be "of size 1/n" and all subsequent moves would be of size 1. If that's the case, the game has to end.

See also Calculus.


Not quite. The protocol is that the first move must be less than or equal to size m, and all subsequent moves must be less than or equal to size n, where m is less than or equal to n.
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Nick Reymann
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Here is an example of a finished game (red wins):
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Russ Williams
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I've tried a few continuous abstract games, e.g. the already-mentioned Calculus as well as Coin Clusters. Continuity is interesting/amusing in theory, but in practice is sometimes an annoyance as it's sometimes hard to tell if 2 objects are touching or not. It breaks one of the fundamental appealing (to me) properties of games as (discrete) formal systems, namely that the game state is unambiguous and easy to see (e.g. in Chess, I see that you just moved a rook onto e5; there is no ambiguity that "Hmm, it's kind of on the edge, maybe your rook is on e6" - and if you moved it clumsily so that it is ambiguous, we would simply correct its positioning to be clearer.) Given this real practical problem, I find I prefer "light/filler" continuous abstract games (e.g. Coin Clusters) because the fundamental problem of ambiguity of game state is a turnoff for a more serious thinky game. The continuity makes the game feel like a lighter game not to be taken seriously because at some point the players have to just agree by hesitant mutual consent (or disagree and flip a coin) about the game state, which seems contradictory with serious strategizing and competition, for me.

Given all that, the idea of varying size circles seems clever. And raises an additional implementation problem since you wouldn't even be able to play with physical pieces, unlike Calculus and Coin Clusters.

I guess you're imagining the game as computer-only, played with arbitrary-precision arithmetic, or something? But then it's not truly continuous, as you would explicitly be disallowing placing the circles at most points of the 2d real plane and disallowing most possible radii - only permitting finitely representable real values.

Or would it be played by math geeks who note the exact values of each circle placed, via whatever mathematical definition they find appropriate, e.g. "the x coordinate is pi, and the y coordinate is the maximum value root of the following polynomial... and the radius is 1 / the ackermann function of ..." and solve the equations manually (possibly requiring complex proofs) to decide if 2 circles touch?
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Nick Reymann
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I don't really imagine anyone actually playing this game; it is pretty much a purely theoretical exercise. If it was actually played, it would have to be like your math geek example.

However, I do not think it would be as complex as you make it out to be. I'm fairly certain that in any case, it is optimal to make your circles as big as the rules allow (i.e. fitting the space where you are placing it as much as possible), though I would welcome a counterexample.
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Nick Bentley
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There was an iphone app for non-discrete hex for a while, but now I can't find it.

[edit] of course because iphones are digital it was only pseudo-non-discrete, but in practice close enough to non-discrete for human players.
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Chris Huntoon
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russ wrote:
I've tried a few continuous abstract games, e.g. the already-mentioned Calculus as well as Coin Clusters. Continuity is interesting/amusing in theory, but in practice is sometimes an annoyance as it's sometimes hard to tell if 2 objects are touching or not. It breaks one of the fundamental appealing (to me) properties of games as (discrete) formal systems, namely that the game state is unambiguous and easy to see (e.g. in Chess, I see that you just moved a rook onto e5; there is no ambiguity that "Hmm, it's kind of on the edge, maybe your rook is on e6" - and if you moved it clumsily so that it is ambiguous, we would simply correct its positioning to be clearer.) Given this real practical problem, I find I prefer "light/filler" continuous abstract games (e.g. Coin Clusters) because the fundamental problem of ambiguity of game state is a turnoff for a more serious thinky game. The continuity makes the game feel like a lighter game not to be taken seriously because at some point the players have to just agree by hesitant mutual consent (or disagree and flip a coin) about the game state, which seems contradictory with serious strategizing and competition, for me.

Given all that, the idea of varying size circles seems clever. And raises an additional implementation problem since you wouldn't even be able to play with physical pieces, unlike Calculus and Coin Clusters.


This reminds me of something similar I saw a number of years ago. The game involved arranging different sized coins together - but I can't remember anything else about it beyond that. I remember feeling the same annoyance regarding the ambiguity of rather two of the circles were really touching or not. A thought occurred to me at the time that the game would have been better implemented using octagons. They would all be of the same sizes and roughly the same shape, but because they had flat edges, you'd be able to tell for sure if they pieces were really touching.
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Bennett Gardiner
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russ wrote:

Or would it be played by math geeks who note the exact values of each circle placed, via whatever mathematical definition they find appropriate, e.g. "the x coordinate is pi, and the y coordinate is the maximum value root of the following polynomial... and the radius is 1 / the ackermann function of ..." and solve the equations manually (possibly requiring complex proofs) to decide if 2 circles touch?


As a math geek, first of all... hilarious. Second of all, most of those proofs would at some point require numerical solution anyway
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Matteo Perlini
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Fascinating theoretical concept, but unfortunately, as said by other posters, it is not practical for mortals like us.

Anyway, here a similar game: Clara. It has an app too.
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