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Subject: Euler Getter rss

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Takehiko Yasuda
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Here is another hex variant I introduced, called Euler Getter.

http://www.sci.kagoshima-u.ac.jp/~yasuda/Takehiko_Yasudas_ho...

The game is more mathematical. Two cells along the edge in symmetric positions are identified. Mathematically speaking, this means that the board is the real projective plane. (I don't think this is a new idea.) The game ends when all cells are filled. The winner is the player who has larger Euler characteristic.

I expect this game is useful for the education of the geometry.
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David Bush
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By "identified," I assume you mean "are regarded as the same cell." So, when any cell on the perimeter is filled, its opposite is automatically also filled as part of the same move.

Wikipedia defines the Euler characteristic for planar graphs. Would it be more appropriate to represent the Hex board as a grid of triangles instead of hexagons? That way, each play would be on a vertex, and the references in the Wikipedia definition to vertices, edges, and faces would directly correspond to the visual representation of the Hex board.

http://en.wikipedia.org/wiki/Euler_characteristic
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Takehiko Yasuda
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Yes, I meant it by "identified."

Maybe, it is easier to compute Euler characteristics with triangles, if you do this by counting vertices, edges and faces. But I think that the hexagonal board is better, because it enables us to regard the game as dividing the projective plane into two regions.
Then by some reason, the sum of the Euler characteristics of the two regions is one, and the game never ends in a tie like Hex.
There is no problem in computing Euler characteristics on the hexagonal board. Just compute
#{vertices} - #{edges} + #{cells}
or more easily
#{connected components} - #{loops}.
 
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David Bush
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Okay, I agree a grid of Hexagons is probably better, but if the GUI shows only the 5x5 region, it is not easy to determine how many connected groups and how many loops each side has formed. I believe that a 5x5 grid can be extended to an infinite grid, where there are just 17 distinct cells for the players to fill, and the following pattern repeats itself:


..|.......|.......|
--A-B-C-D-E-D-C-B-A-
..F.G.H.I.J.O.P.Q.F
..K.L.M.N.K.L.M.N.K
..J.O.P.Q.F.G.H.I.J
--E-D-C-B-A-B-C-D-E-
..J.I.H.G.F.Q.P.O.J
..K.N.M.L.K.N.M.L.K
..F.Q.P.O.J.I.H.G.F
--A-B-C-D-E-D-C-B-A-
..|.......|.......|

Applying this to the final position shown in the video, we get:


The central 5x5 region is from the video.

Now it's a bit easier to see that blue has two distinct groups and two loops, for a score of 0, and red has three distinct groups and two loops (one of which goes around blue's large group) for a score of 1. Did I do this right?

Maybe it would be easier still if the display were extended even further, but then the cells might become too small to read the position.

As far as the game itself is concerned, I don't have a clue if it's a good game or not. It seems very opaque to me. Has a computer program that plays the game been written?
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Takehiko Yasuda
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Yes, one can extend the board to the infinite grid.
But it does not seem to make the computation easier.
Indeed in your figure, red has only one connected component, while the infinite figure appears to show three red components.

The red area in the central 5x5 grid is as follows:

. a . b c
d . . e f
. . g h .
f . i . d
c b . a .

The alphabets express the red cells and "."'s blue ones.
The red has no loop and Euler characteristic 1-0=1.
On the other hand, the blue area has one connected component and one loop.
The loop runs from the top to the bottom, then returns (connects) to the top. So the Euler characteristic is 1-1=0.

I have written a program (Windows and Mac OSX). You can download it from
http://www.sci.kagoshima-u.ac.jp/~yasuda/Takehiko_Yasudas_ho...
Unfortunately, I have not implemented AI. So you can not play against the computer.

The web application is also available here:
http://misc.tokyoenvious.net/euler-getter/
When you visit the site, choose the size of the board and click "start".
Then you see a new page. You will play red.
Send the url of the page to the opponent.
The second person visiting the page will be blue.

 
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David Molnar
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highernash wrote:
Here is another hex variant I introduced, called Euler Getter.

http://www.sci.kagoshima-u.ac.jp/~yasuda/Takehiko_Yasudas_ho...

The game is more mathematical. Two cells along the edge in symmetric positions are identified. Mathematically speaking, this means that the board is the real projective plane. (I don't think this is a new idea.) The game ends when all cells are filled. The winner is the player who has larger Euler characteristic.

I expect this game is useful for the education of the geometry.

Indeed, the game Projex takes place on the projective plane.
It's not at first glance obvious that this game is not in fact equivalent to Projex, although it would seem that it is different, since the winning condition for Projex involves a single group, and yours involves the whole board. I would conjecture that this is more like Star (which is not in the database). I am very intrigued.

I'm wondering also if you didn't mean the smaller Euler characteristic wins - the point of connection games, after all, is ordinarily to have fewer connected groups.
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Takehiko Yasuda
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Thank you for your useful info.

I have checked Projex. Euler Getter and Projex are similar because both of them are played on the projective plane. But not equivalent. Projex concerns only a loop which crosses the board, while Euler Getter concerns every loop (even topologically trivial loop) and every components with or without a loop. Moreover Euler Getter has scores. The most probable score is 1 to 0, but 2 to -1 and 3 to -2 are also possible. So if ones play the game several times, they can take scores into account.

I am interested in what the game Star is like.

Yes, the version of the game where the smaller Euler characteristic wins is possible. Other possible versions are the one where the odd Euler characteristic wins and the one where the even Euler characteristic wins. I do not know at all which version is the most amusing.
 
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Takehiko Yasuda
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One more comment.

Euler Getter is, in a sense, a mixture of the (dis)connection game like Hex and the counting (measuring territories) game like Go and Reversi.
 
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