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Hex "generalizations"

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marks...@gmail.com

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Jan 29, 2008, 6:37:16 PM1/29/08
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I keep seeing the term "generalization of Hex" bandied about to
describe variants of Hex and families of variants of Hex. Most
recently in Board Game News:

"Mark Steere has designed . . . Atoll and Begird . . . [which] are
generalized forms of the games Hex and Y. . . . Have other designers
already covered this territory?"

"Eric, . . . Nick Bentley's Mind Ninja is another stone placement game
which is a generalization of a number of designs, including Hex and
Y."

"Good call, Larry. . . . Mind Ninja actually encompasses Atoll..."

There is only one generalization of Hex: Atoll. Atoll is a precisely
defined, infinite set of games all of whose members share the central,
defining, geometric principals of Hex: The goal is to connect opposite
segments, and on a filled board you have exactly one winner. In the
special case of Atoll in which the set member has the least allowable
number of perimeter segments, four, Atoll is exactly equivalent to
Hex.

Mind Ninja is certainly a variation of Hex, or a family of variations
of Hex. I don't mean to impugn Mind Ninja. I'm sure it's a wonderful
game, having been created by the talented designer Nick Bentley. But
it's not a generalization of Hex, or Atoll, or anything else that I'm
aware of. It certainly doesn't "encompass" Atoll.

Beyond that, Atoll is not a variant of Hex. Atoll is a superset of
Hex. Hexwiki recently categorized Atoll as a variant of Hex. It
isn't. Atoll is the natural connection game for even segmented,
hexagonally tessellated boards. Hex is merely one member of the set
precisely defined by Atoll. A set is not a variant of one of its
members.

Ed Murphy

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Jan 30, 2008, 12:54:34 AM1/30/08
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marks...@gmail.com wrote:

> There is only one generalization of Hex: Atoll. Atoll is a precisely
> defined, infinite set of games all of whose members share the central,
> defining, geometric principals of Hex: The goal is to connect opposite
> segments, and on a filled board you have exactly one winner. In the
> special case of Atoll in which the set member has the least allowable
> number of perimeter segments, four, Atoll is exactly equivalent to
> Hex.
>
> Mind Ninja is certainly a variation of Hex, or a family of variations
> of Hex. I don't mean to impugn Mind Ninja. I'm sure it's a wonderful
> game, having been created by the talented designer Nick Bentley. But
> it's not a generalization of Hex, or Atoll, or anything else that I'm
> aware of. It certainly doesn't "encompass" Atoll.

Having just now heard about and googled Mind Ninja, it certainly sounds
like a generalization of both: The goal is to get from a starting
arrangement to a type of ending arrangement (both of which are chosen by
one of the players). In the special case of Mind Ninja in which the
player chooses the starting arrangement "empty board" and the type of
ending arrangement "connect opposite segments", Mind Ninja is exactly
equivalent to Atoll.

There's an initial bidding war that appears to serve as a handicap of
some sort. Also, you may well be intending "generalization" and/or
"superset" to indicate that all cases within the larger set involve
similar strategies to those within the smaller set, in which case it
would be much more reasonable to say that Mind Ninja is not a
generalization of Atoll after all; "connect opposite segments" involves
different strategies than, say, "create a solid 3x3 block" or "create
the corners of three different squares of any size and orientation".

marks...@gmail.com

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Jan 30, 2008, 2:51:24 AM1/30/08
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On Jan 29, 9:54 pm, Ed Murphy <emurph...@socal.rr.com> wrote:
> In the special case of Mind Ninja in which the
> player chooses the starting arrangement "empty board" and the type of
> ending arrangement "connect opposite segments", Mind Ninja is exactly
> equivalent to Atoll.
>

No, that would not make Mind Ninja "exactly equivalent to Atoll".
With those starting and ending arrangements in Mind Ninja you could
very easily fill the board with stones and not have any two opposite
segments connected, resulting in a draw. Draws can't occur in Atoll.
Before you posit any more ridiculous conjectures about what Atoll is
"exactly equivalent to" you might want to actually look at the Atoll
rule sheet.

Harald Korneliussen

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Jan 30, 2008, 5:25:21 AM1/30/08
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On Jan 30, 8:51 am, "markste...@gmail.com" <markste...@gmail.com>
wrote:

> With those starting and ending arrangements in Mind Ninja you could
> very easily fill the board with stones and not have any two opposite
> segments connected, resulting in a draw. Draws can't occur in Atoll.
> Before you posit any more ridiculous conjectures about what Atoll is
> "exactly equivalent to" you might want to actually look at the Atoll
> rule sheet.

And you might want to look at Mind Ninja... it does not have draws.
Goals are expressed in terms of patterns the builder will try to
build, and the blocker will try to prevent from being built. If the
game ends and the pattern isn't built, the blocker has won.

It so happens that for the "Hex" pattern (and for the various n-Atoll
patterns, all of which are expressible within Mind Ninja constraints),
a failure of the builder to complete his goal means the blocker must
have completed the converse goal... but most patterns aren't symmetric
in that way. For instance, the ring, fork, bridge pattern that is in
Havannah can be used in Mind Ninja, but then those situations that
would be ties in regular Havannah become wins for the blocker (moving
second) in Mind Ninja.

It's just wrong that Atoll is the only possible generalisation of Hex.
Depending on what aspects of Hex one sees as crucial, there are surely
many other ways of generalizing it.

Torben Ægidius Mogensen

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Jan 30, 2008, 5:53:49 AM1/30/08
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"marks...@gmail.com" <marks...@gmail.com> writes:

> I keep seeing the term "generalization of Hex" bandied about to
> describe variants of Hex and families of variants of Hex. Most
> recently in Board Game News:
>

> There is only one generalization of Hex: Atoll. Atoll is a precisely
> defined, infinite set of games all of whose members share the central,
> defining, geometric principals of Hex: The goal is to connect opposite
> segments, and on a filled board you have exactly one winner. In the
> special case of Atoll in which the set member has the least allowable
> number of perimeter segments, four, Atoll is exactly equivalent to
> Hex.

Atoll is only one of many generalisations of Hex. Atoll fixes the
shape of the board and the positions and sizes of the pre-filled
groups, but you can generalise even further. My analysis found that
you can get the property of exactly one winner on a full board if the
following two criteria are fulfilled:

1. If a group of tokens of one colour is a winning connection, then
even if all the other hexes are filled by the other colour, this
can not be a winning connection for that colour.

2. If a group of tokens of one colour is a winning connection, then
you can replace any part of this by a group of the opposite colour
as long as this new group doesn't touch anything else of its
colour (including precoloured pieces or borders), and the original
group will still be a winning connection.

The first criterion guarantees that there can't be two winners, and
the second that a filled board will have a winner.

You can use this to make a toroid variant of Hex:

The board is a torus with a hexagonal tiling. You can simulate this
by a rectangular or parallelogram board that wraps around both top
to bottom and left to right. There are no edges on this board.

In order to win, Black must make a closed loop that encircles the rim
of the torus and White must make a closed loop that encircles the
center of the torus.

Criteria 1 is clearly fulfilled, as a loop around the rim and a loop
around the center must cross, so both can't exist at the same time.
Criteria 2 is fulfilled because replacing part of a loop by the other
colour can't make it into a non-loop unless the new pieces touch old
pieces of their own colour.

On a rectangular or parallelogram board with wrap, this means that
Black must make a loop that crosses the top/bottom wrap an odd number
of times and the left/right wrap and even number of times (including
0) and White must make a loop that crosses the left/right wrap an odd
number of times and the top/bottom wrap an even number of times
(including 0).

This is a generalisation of Hex, as if you precolour the top/bottom
wrap black and the left/right wrap white, you get Hex.

We can call this toroidal Hex variant "Torex".

Torben

marks...@gmail.com

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Jan 30, 2008, 2:36:40 PM1/30/08
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On Jan 30, 2:53 am, torb...@app-2.diku.dk (Torben Ægidius Mogensen)
wrote:

>
> Atoll is only one of many generalisations of Hex.
>

This should be interesting.

>
> . . . a toroid variant of Hex: . . . The board is a torus with a
> hexagonal tiling. . . .


>
> In order to win, Black must make a closed loop that encircles the rim
> of the torus and White must make a closed loop that encircles the
> center of the torus.
>

> We can call this toroidal Hex variant "Torex".
>

You can call it Revenge of the Donkey Tail if you like, but it doesn't
work. Imagine a red stripe and a white stripe spiraling side by side
along your torus, like a candy cane bracelet, each stripe forming a
closed loop and encircling the torus in both orthogonal directions.

Atoll is the only one. I strongly suspect that it will remain the
only one, but it will take someone with your logical acumen to prove
it.

marks...@gmail.com

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Jan 30, 2008, 3:17:55 PM1/30/08
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On Jan 30, 2:25 am, Harald Korneliussen <vinterm...@gmail.com> wrote:
>
> It so happens that for the "Hex" pattern (and for the various n-Atoll
> patterns, all of which are expressible within Mind Ninja constraints),

No. Not even close. You can't say "In Mind Ninja you can choose to
play Hex, Atoll, or Havannah, etc., plus there's a bartering
protocol. Therefore Mind Ninja is a generalization of all of these
games." That's absurd.

> a failure of the builder to complete his goal means the blocker must
> have completed the converse goal...

Hex isn't just a game in which one player attempts to form some J-
random pattern like a happy face, and the other player tries to
prevent that pattern from forming. Hex is a game in which a randomly
filled board produces exactly one of: a connection of black stones
between the two black sides, or a connection of white stones between
the two white sides.

Yes, by thwarting your opponent you simultaneously achieve your own
objective. But that little coincidence does not make Mind Ninja a
generalization of Hex.

>
> It's just wrong that Atoll is the only possible generalisation of Hex.

Currently it's the only generalization of Hex. Is it the only
possible generalization? I suspect that to be true but I wouldn't know
where to begin in a proof of that.

> Depending on what aspects of Hex one sees as crucial, there are surely
> many other ways of generalizing it.

I think it's fair to say that by far the most crucial aspects of Hex
are that it's a connection game and that in a randomly filled board
exactly one player forms a stated connection goal. Not to mention
that it's a game of equal goals.

Now you're going to tell me that this totally unequal goals game in
which you barter to decide what type of shape one player is going to
form, among happy faces, yin yangs, and what not, and which the other
player is going to try to prevent from forming is a generalization of
Hex? All because it shares this one little property of Hex that
achieving your objective is equivalent to preventing your opponent
from achieving his?

And now that Atoll has been discovered, let's throw that into the mix
too. Atoll is now proclaimed to be "encompassed" by Mind Ninja.

Do you understand what a generalization is? I think some of us here
missed out on high school geometry.

Brian Tivol

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Jan 30, 2008, 5:56:13 PM1/30/08
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"marks...@gmail.com" <marks...@gmail.com> writes:
> There is only one generalization of Hex: Atoll.

Are these the rules of Hex?

The board is rhomboidal, made up of hexagonally tiled spaces, N
hexagons to a side. Two opposite edges of the board are colored red,
and the other pair are colored blue. The first player puts a red
token in a space; the second player then decides whether he or his
opponent will be the red player. Thereafter, starting with the blue
player, the players take turns placing a single token of their color
in an unoccupied hexagonal space. When all the hexagonal spaces are
filled with tokens, exactly one of these statements is true: a chain
of red tokens links the two red sides, or a chain of blue tokens links
the two blue sides. Whichever player finds his colors' sides
connected is the winner. (Note that Hex is a family of games, that
varies based on the value of N in the first sentence.)

So far so good?

I'd like to introduce three families of games that I'm making up on
the spot, some of which I'd consider to be generalizations of Hex that
are not encompassed by Atoll. Please let the group know if you'd
consider them to be generalizations of Hex, if you'd consider them
encompassed by Atoll, and why or why not. (These examples are very
simplistic, so you may be inclined to dismiss me as a smart-ass, but I
hope you'll give thoughtful answers.)

The first, "Colorful Hex", is very much like Hex as described above.
However, the first player first chooses his favorite color in the
world, and it replaces "red" in the rules above. The second player
then chooses a distinct color, and it replaces "blue". The players
use those colors instead of the standard colors for the rest of the
game. When players pick "red" and "blue", this family is exactly like
Hex.

The second, "Super Pie-Rule Hex", is very much like Hex as described
above. However, we replace the first clause of the third sentence
above with this one: "The first player places K red tokens and J blue
tokens in distinct spaces; ..." When K=1 and J=0, this family is
exactly like Hex.

The third, "Horrible Luck Hex", is also very much like Hex as
described above. However, the fourth sentence is a little different.
Instead of placing a token where he likes on his turn, a player must
declare where he wants to place his token, then roll an X-sided die Y
times and sum the results. If the total is at least Z, then he is
allowed to place his token. Otherwise, he must leave the space empty.
When X=1, Y=6, and Z=1, this family is exactly like Hex.

--brian

marks...@gmail.com

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Jan 30, 2008, 8:47:21 PM1/30/08
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On Jan 30, 2:56 pm, Brian Tivol <t1i...@m2it.e3du> wrote:
> I'd like to introduce three families of games that I'm making up on
> the spot, some of which I'd consider to be generalizations of Hex that
> are not encompassed by Atoll. Please let the group know if you'd
> consider them to be generalizations of Hex, if you'd consider them
> encompassed by Atoll, and why or why not. (These examples are very
> simplistic, so you may be inclined to dismiss me as a smart-ass, but I
> hope you'll give thoughtful answers.)
>

These are all variations on Hex. None of them is a generalization of
Hex. There's some serious confusion here about what a generalization
is. The word is being used to describe variations and families of
variations. If you list a family of games, all of which have one
little item in common with Hex you haven't generalized Hex.

You want to see a real nice example of a generalization? Look no
further than Atoll. You have a very succinctly, precisely defined,
infinite set of games. Every game in the set has all of the central,
defining characteristics of Hex. Both players are trying to form a
connection between two of their opposite segments. In a randomly
filled board, exactly one player achieves his cross board connection.
There are a multiple of four of perimeter segments, alternating
between black and white. It's a two player, equal goals game. Etc.,
etc., etc. The only difference between the games in the set is the
number of perimeter segments. One *special case* is when there are
four perimeter segments. That member is equivalent to Hex. Hex is a
*special case* of the more general Atoll. Is any of this making
sense?

Let's approach this from another angle. There's this amazing new game
that everybody must have heard of by now called Whopper 100. You have
a numbered list of 100 games starting with Chess, Checkers, and
Backgammon. Every game on the list is a two player, turn based game
in which each player has his own uniquely colored set of playing
pieces, and in which captures can occur. The pieces are moved on or
added to a square grid, one manipulation per turn, possibly resulting
in a capture. So all of these games have a fair amount in common.

To begin, one of the players, selected at random, rolls a 100 sided
die. Whatever number comes up, that's the game they play.

Oh, and the really magnificent thing about Whopper 100 is that it's
actually a generalization of Chess.

Ed Murphy

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Jan 30, 2008, 11:09:01 PM1/30/08
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marks...@gmail.com wrote:

> These are all variations on Hex. None of them is a generalization of
> Hex. There's some serious confusion here about what a generalization
> is. The word is being used to describe variations and families of
> variations. If you list a family of games, all of which have one
> little item in common with Hex you haven't generalized Hex.
>
> You want to see a real nice example of a generalization? Look no
> further than Atoll. You have a very succinctly, precisely defined,
> infinite set of games. Every game in the set has all of the central,
> defining characteristics of Hex. Both players are trying to form a
> connection between two of their opposite segments. In a randomly
> filled board, exactly one player achieves his cross board connection.
> There are a multiple of four of perimeter segments, alternating
> between black and white. It's a two player, equal goals game. Etc.,
> etc., etc. The only difference between the games in the set is the
> number of perimeter segments. One *special case* is when there are
> four perimeter segments. That member is equivalent to Hex. Hex is a
> *special case* of the more general Atoll. Is any of this making
> sense?

I think so. This seems to agree with most of the second paragraph of my
previous post (of which you only responded to the first paragraph):

> you may well be intending "generalization" and/or
> "superset" to indicate that all cases within the larger set involve
> similar strategies to those within the smaller set, in which case it
> would be much more reasonable to say that Mind Ninja is not a
> generalization of Atoll after all; "connect opposite segments" involves
> different strategies than, say, "create a solid 3x3 block" or "create
> the corners of three different squares of any size and orientation".

Now, examining Brian's examples in this light:

Colorful Hex is trivially isomorphic to Hex, so there isn't (in any
meaningful sense) a larger set at all.

Super Pie-Rule Hex can involve notably different strategies than Hex,
depending on how far you deviate from K=1 and J=0. In particular,
K=0 and J=1 is Misere Hex, while K=N and J=0 degenerates into a trivial
first-player win.

Horrible Luck Hex is pretty much equivalent to Super Pie-Rule Hex with
K=random(0 or 1) and J=0.

Hmm, what about generalizing Hex to higher dimensions?

* 0-D tokens, 1-D goals, 2-D board = Hex
* 1-D tokens, 2-D goals, 3-D board = ?

Certainly not as elegant as Atoll, though, and moving beyond a 3-D
board would be awfully difficult for anything but a computer.

marks...@gmail.com

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Jan 31, 2008, 12:33:45 AM1/31/08
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On Jan 30, 8:09 pm, Ed Murphy <emurph...@socal.rr.com> wrote:
>
> Hmm, what about generalizing Hex to higher dimensions?
>
> * 0-D tokens, 1-D goals, 2-D board = Hex
> * 1-D tokens, 2-D goals, 3-D board = ?
>

I've spent a fair amount of time messing around with 3D shapes such as
the surface of a cylinder and a torus and so forth. I keep running
into this problem that you have to win a hex-like confrontation in two
different locations in order to win. So if you win one of the
confrontations and lose the other one, neither player makes it
through. There's also the problem that both players can make it
through by going around opposite sides of the same 3D shape.

I created a game, Box Hex, a couple years ago, which does work. It
can be argued that it's not really a 3D shape but rather a morphed 2D
shape. To me it seems like an essentially 3D shape because if you
want to have a regular hexagonal tessellation all over it (except for
the pentagonal cells in the corners) it has to be 3D. But you can see
the arguments on both sides. The fact that it is a morph of a 2D
board is probably what makes it succeed.

I no longer have Box Hex on my website because it's nothing real
special IMO, but here's a link:

http://www.marksteeregames.com/Box_Hex_rules.pdf

Harald Korneliussen

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Jan 31, 2008, 2:35:33 AM1/31/08
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marks...@gmail.com wrote:

> And now that Atoll has been discovered, let's throw that into the mix
> too. Atoll is now proclaimed to be "encompassed" by Mind Ninja.

If you find that offensive, perhaps you shouldn't have turned up on
Hexwiki like you did, apparently thinking your more general game was
Big News and should instantly be recognized as _the_ generalization of
Hex.

So you mean MN isn't a generalization because it has a mechanism (a
Hex mechanism, even) for combining the games it encompasses? I think
that is nitpicking, to use a moderate word, but you must at least
agree that all the subgames of MN beginning at move 2 constitute a
generalization of Hex.

Atoll may still be a good game, but your generalization isn't the only
one, or even the first one.

Torben Ægidius Mogensen

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Jan 31, 2008, 4:02:27 AM1/31/08
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"marks...@gmail.com" <marks...@gmail.com> writes:

> On Jan 30, 2:25 am, Harald Korneliussen <vinterm...@gmail.com> wrote:

> Hex isn't just a game in which one player attempts to form some J-
> random pattern like a happy face, and the other player tries to
> prevent that pattern from forming. Hex is a game in which a randomly
> filled board produces exactly one of: a connection of black stones
> between the two black sides, or a connection of white stones between
> the two white sides.
>
> Yes, by thwarting your opponent you simultaneously achieve your own
> objective. But that little coincidence does not make Mind Ninja a
> generalization of Hex.
>
>> It's just wrong that Atoll is the only possible generalisation of Hex.
>
> Currently it's the only generalization of Hex. Is it the only
> possible generalization? I suspect that to be true but I wouldn't know
> where to begin in a proof of that.

When you generalise something, you choose which properties you want to
retain and which you want to relax. There are no a priori right or
wrong choices for this, so you can't claim that a game that has Hex as
a special case isn't a generalisation of Hex, just because you don't
like the type of generalisations that are made.

At most you can say that Atoll is the only generalisation of Hex that
preserves (list of properties). But that is a vacuous statement if
the list of properties exactly define Atoll.

>> Depending on what aspects of Hex one sees as crucial, there are surely
>> many other ways of generalizing it.

Indeed.

> I think it's fair to say that by far the most crucial aspects of Hex
> are that it's a connection game and that in a randomly filled board
> exactly one player forms a stated connection goal. Not to mention
> that it's a game of equal goals.

These are indeed notable properties, but breaking some of them just
generalises the game even more.

And, BTW, Y fulfills the above properties but is not an instance of
Atoll.

> Now you're going to tell me that this totally unequal goals game in
> which you barter to decide what type of shape one player is going to
> form, among happy faces, yin yangs, and what not, and which the other
> player is going to try to prevent from forming is a generalization of
> Hex? All because it shares this one little property of Hex that
> achieving your objective is equivalent to preventing your opponent
> from achieving his?

You clearly haven't understood the concept of generalisation. You
might say that you think it is an overgeneralisation, because you lose
properties that you want to retain, but that doesn't make it any less
of a generalisation.

> Do you understand what a generalization is? I think some of us here
> missed out on high school geometry.

Heh.

Torben

Torben Ægidius Mogensen

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Jan 31, 2008, 4:15:18 AM1/31/08
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"marks...@gmail.com" <marks...@gmail.com> writes:

> On Jan 30, 2:53 am, torb...@app-2.diku.dk (Torben Ćgidius Mogensen)


> wrote:
>>
>> Atoll is only one of many generalisations of Hex.
>
> This should be interesting.
>
>> . . . a toroid variant of Hex: . . . The board is a torus with a
>> hexagonal tiling. . . .
>>
>> In order to win, Black must make a closed loop that encircles the rim
>> of the torus and White must make a closed loop that encircles the
>> center of the torus.
>

> it doesn't work. Imagine a red stripe and a white stripe spiraling
> side by side along your torus, like a candy cane bracelet, each
> stripe forming a closed loop and encircling the torus in both
> orthogonal directions.

Hmm. You are right. I'll have to think a bit more and see if I can
come up with different winning criteria that makes it work.


> Atoll is the only one. I strongly suspect that it will remain the
> only one,

What is wrong with Y?

> but it will take someone with your logical acumen to prove it.

Or maybe disprove it?

Torben

marks...@gmail.com

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Jan 31, 2008, 9:41:57 AM1/31/08
to
On Jan 31, 1:15 am, torb...@app-3.diku.dk (Torben Ægidius Mogensen)
wrote:

> >>
> >> . . . a toroid variant of Hex: . . . The board is a torus with a
> >> hexagonal tiling. . . .
> >
> > it doesn't work.

>
> Hmm. You are right. I'll have to think a bit more and see if I can
> come up with different winning criteria that makes it work.

I await with breathless anticipation.

>
> > Atoll is the only one. I strongly suspect that it will remain the
> > only one,
>
> What is wrong with Y?

There's nothing wrong with Y. It's just not a generalization of Hex
though as is sometimes claimed. However, don't despair. I have
discovered the generalization of Y too which I call Begird.

Begird rule sheet:
http://www.marksteeregames.com/Begird_rules.pdf

>
> > but it will take someone with your logical acumen to prove it.
>
> Or maybe disprove it?
>

I'd be satisfied either way.

marks...@gmail.com

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Jan 31, 2008, 10:28:29 AM1/31/08
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On Jan 30, 11:35 pm, Harald Korneliussen <vinterm...@gmail.com> wrote:
>
> ... but you must at least

> agree that all the subgames of MN beginning at move 2 constitute a
> generalization of Hex.
>

Sure no problem . . . if you'll agree that Whopper 100 (see a couple
of posts back) is a generalization of Chess. Surprisingly, it's an
even weaker argument than the one you're making.

marks...@gmail.com

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Jan 31, 2008, 10:44:36 AM1/31/08
to
On Jan 31, 1:02 am, torb...@app-3.diku.dk (Torben Ægidius Mogensen)
wrote:

> "markste...@gmail.com" <markste...@gmail.com> writes:
>
> >> Depending on what aspects of Hex one sees as crucial, there are surely
> >> many other ways of generalizing it.
>
> Indeed.
>

Ok, there is a subjective element. But you're actually claiming that
an unequal goals game in which one player tries to form a happy face
shape and the other player tries to prevent it captures the essence of
Hex? You have to be kidding me. You must agree then that Whopper 100
is a generalization of Chess, right? Somehow everyone overlooked that
argument a couple of posts back.

>
> And, BTW, Y fulfills the above properties.
>

No it doesn't. There are an odd number of perimeter segments, just
for starters.


marks...@gmail.com

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Jan 31, 2008, 10:48:50 AM1/31/08
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On Jan 31, 1:15 am, torb...@app-3.diku.dk (Torben Ægidius Mogensen)
wrote:
> ...

Somehow nobody noticed the argument that Whopper 100 should be
considered a generalization of Chess. Conveniently. I'd hate to see
rec.games.abstract become home of the Whopper.

Zachary Catlin

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Jan 31, 2008, 11:43:03 AM1/31/08
to
marks...@gmail.com wrote:
> Oh, and the really magnificent thing about Whopper 100 is that it's
> actually a generalization of Chess.

Not really. Chess is not a special case of Whopper 100--Chess is the
outcome of a certain roll of the die, yes, but Whopper 100 is a *certain
game,* not a class of games.

I'll agree with you that some generalizations make more sense than
others, but if you want to make that point, please try to come up with
an actual generalization.

marks...@gmail.com

unread,
Jan 31, 2008, 12:27:54 PM1/31/08
to
On Jan 31, 8:43 am, Zachary Catlin <zcat...@purdue.edu> wrote:

> markste...@gmail.com wrote:
> > Oh, and the really magnificent thing about Whopper 100 is that it's
> > actually a generalization of Chess.
>
> Not really. Chess is not a special case of Whopper 100--Chess is the
> outcome of a certain roll of the die, yes, but Whopper 100 is a *certain
> game,* not a class of games.

Ok, so let's call it Whopper Infinity. It's the class of all possible
turn based, two player games, played on a square grid, where each
player has his own set of uniquely colored pieces and in which there
is capturing, and in which each player makes one manipulation per
turn, possibly resulting in a capture?

Now Whopper is a generalization of Chess, right?

>
> I'll agree with you that some generalizations make more sense than

> others, . . .

Amazing.

marks...@gmail.com

unread,
Jan 31, 2008, 1:54:33 PM1/31/08
to
On Jan 31, 8:43 am, Zachary Catlin <zcat...@purdue.edu> wrote:
>
> ...

I keep seeing these claims that there are many possible
generalizations of Hex and many existing generalizations of Hex other
than Atoll, yet by a cruel twist of fate I have yet to actually see
one.

At least Torben *tried* to prove me wrong with his game, Torex.
Unfortunately Torex was fatally flawed, but you have to give him
credit for trying.

Brian Tivol

unread,
Jan 31, 2008, 2:40:44 PM1/31/08
to

I wrote:
>> Please let the group know if you'd consider them to be
>> generalizations of Hex, if you'd consider them encompassed by
>> Atoll, and why or why not.

Mark Steere wrote:
> These are all variations on Hex. None of them is a generalization

> of Hex. [...] If you list a family of games, all of which have one


> little item in common with Hex you haven't generalized Hex.

I'm disappointed not to see any further explanation why you think the
three examples I gave are not generalizations of Hex. I agree with
the final sentence, but I believe that each of my examples has far
more than "one little item in common with Hex".

> You want to see a real nice example of a generalization? Look no
> further than Atoll.

Great. I hear it's the gold standard of Hex generalizations, so I'll
see how my examples stand up to it.

> You have a very succinctly, precisely defined, infinite set of
> games.

Seems true for Atoll and true for my examples.

> Every game in the set has all of the central, defining
> characteristics of Hex.

Which you go on to define as...

> Both players are trying to form a connection between two of their
> opposite segments.

Seems true for Atoll and true for my examples.

> In a randomly filled board, exactly one player achieves his cross
> board connection.

Seems true for Atoll and true for my examples.

> There are a multiple of four of perimeter segments, alternating
> between black and white.

Seems true for Atoll and true for my examples (except that one of
examples parameterizes "black" and "white"-- do you find that crucial
to Hex and Hex generalizations?)

> It's a two player, equal goals game. Etc., etc., etc.

Seems true for Atoll and true for my examples.

> The only difference between the games in the set is the number of
> perimeter segments. One *special case* is when there are four
> perimeter segments. That member is equivalent to Hex. Hex is a
> *special case* of the more general Atoll.

What is the only difference between Colorful Hex and Hex? Is there a
member of Colorful Hex that is equivalent to Hex? Is Hex a special
case of the more general Colorful Hex?

How about the other two games? The answers there are more
interesting.

> Is any of this making sense?

No. The points you listed to support your claim that Atoll is the
only generalization of Hex also appear to apply to the three examples
I defined.

> Let's approach this from another angle. [...] Oh, and the really


> magnificent thing about Whopper 100 is that it's actually a
> generalization of Chess.

So... this is sarcasm? Because Whopper 100 (or Whopper 101, I
suppose) would just as validly be a generalization of Hex as a
generalization of Chess, and you state that the only valid
generalization of Hex is Atoll.

I'd like to focus on the questions I posed. I thought you gave me a
sincere yet uncompelling argument, then followed it with an equally
uncompelling sarcastic one. It makes me wonder if I misread your
sincerity.

--brian

Zachary Catlin

unread,
Jan 31, 2008, 3:57:31 PM1/31/08
to
marks...@gmail.com wrote:
> On Jan 31, 8:43 am, Zachary Catlin <zcat...@purdue.edu> wrote:
>> markste...@gmail.com wrote:
>>> Oh, and the really magnificent thing about Whopper 100 is that it's
>>> actually a generalization of Chess.
>> Not really. Chess is not a special case of Whopper 100--Chess is the
>> outcome of a certain roll of the die, yes, but Whopper 100 is a *certain
>> game,* not a class of games.
>
> Ok, so let's call it Whopper Infinity. It's the class of all possible
> turn based, two player games, played on a square grid, where each
> player has his own set of uniquely colored pieces and in which there
> is capturing, and in which each player makes one manipulation per
> turn, possibly resulting in a capture?
>
> Now Whopper is a generalization of Chess, right?

Not a very interesting one, but yes.

>> I'll agree with you that some generalizations make more sense than
>> others, . . .
>
> Amazing.

Okay, I apparently misread you. Sorry.

Zachary Catlin

unread,
Jan 31, 2008, 4:13:39 PM1/31/08
to
marks...@gmail.com wrote:
> I keep seeing these claims that there are many possible
> generalizations of Hex and many existing generalizations of Hex other
> than Atoll, yet by a cruel twist of fate I have yet to actually see
> one.

Could you please lay out, in detail and in a single, coherent post, why
you think Atoll is the only possible generalization of Hex? I've seen
bits and pieces of your reasoning in various posts, but I don't think
I'm getting all of it. I can see the difficulties involved in attempts
to make other generalizations, but just because it's difficult doesn't
mean it's impossible, and I'd like a good argument as to why it might be
impossible.

marks...@gmail.com

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Jan 31, 2008, 4:33:08 PM1/31/08
to
On Jan 31, 1:13 pm, Zachary Catlin <zcat...@purdue.edu> wrote:

I never said Atoll was the only possible generalization of Hex,
although I do happen to strongly suspect that to be the case. Atoll
is currently the only generalization of Hex. I'm certain that I would
have heard about any others by now if they did in fact exist.

marks...@gmail.com

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Jan 31, 2008, 4:52:29 PM1/31/08
to
On Jan 31, 11:40 am, Brian Tivol <t1i...@m2it.e3du> wrote:
>
> > Let's approach this from another angle. [...] Oh, and the really
> > magnificent thing about Whopper 100 is that it's actually a
> > generalization of Chess.
>
> So... this is sarcasm? Because Whopper 100 (or Whopper 101, I
> suppose) would just as validly be a generalization of Hex as a
> generalization of Chess, and you state that the only valid
> generalization of Hex is Atoll.
>

Yes, that last bit about Whopper was intended to be facetious.
Whopper is not a generalization of Hex or Chess or any other game.
But the reasoning as to why Whopper (Whopper Infinity defines a class
of games, as covered in a previous post) should be a generalization of
Chess is the same reasoning used to claim that Mind Ninja is a
generalization of Hex.

Your three examples of Hex variants don't define a class of games.

It's a subjective matter as to whether or not a class of games
qualifies as a generalization of Hex. But there seems to be some
agreement here that the general case should capture the essence of, or
have most or all of the "crucial aspects" of the special case being
generalized. It's amazing to me that Mind Ninja is considered by
anyone to be a generalization of Hex. So it shares one little
property of Hex that in attaining your goal you prevent your opponent
from attaining his. And it doesn't even fully capture *that*
property. In Hex it's a two way street. Each player is trying to
form his own given construct and simultaneously trying to prevent the
other from attaining his. In the unequal goals game of Mind Ninja
only *one* player is trying to form a given construct and the other is
only trying to prevent it, not form one of his own. Just as Whopper
is a long way from being a generalization of Chess, Mind Ninja is
about that far from being a generalization of Hex.

Ed Murphy

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Jan 31, 2008, 6:04:24 PM1/31/08
to
marks...@gmail.com wrote:

> Your [Brian Tivol] three examples of Hex variants don't define a
> class of games.

Super Pie-Rule Hex most certainly does, if you consider each ordered
pair (K,J) as an individual game; (1,0) is canonical Hex.

> So it [Mind Ninja] shares one little


> property of Hex that in attaining your goal you prevent your opponent
> from attaining his. And it doesn't even fully capture *that*
> property. In Hex it's a two way street. Each player is trying to
> form his own given construct and simultaneously trying to prevent the
> other from attaining his.

This is trivially true, if "the player's construct" is defined as
"anything that blocks the other", i.e. G2 = inverse(G1). What Mind
Ninja fails to capture is the similarity of the goals in a positive
sense, i.e. G2 = rotation(G1).

marks...@gmail.com

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Jan 31, 2008, 8:20:44 PM1/31/08
to
On Jan 31, 3:04 pm, Ed Murphy <emurph...@socal.rr.com> wrote:
>
> > So it [Mind Ninja] shares one little
> > property of Hex that in attaining your goal you prevent your opponent
> > from attaining his. And it doesn't even fully capture *that*
> > property. In Hex it's a two way street. Each player is trying to
> > form his own given construct and simultaneously trying to prevent the
> > other from attaining his.
>
> This is trivially true, if "the player's construct" is defined as
> "anything that blocks the other", i.e. G2 = inverse(G1). What Mind
> Ninja fails to capture is the similarity of the goals in a positive
> sense, i.e. G2 = rotation(G1).

Exactly, but that's a huge disparity.

Hex to me is and probably always will be the ultimate game. Not as
something to play but as something to behold. It's not just any
member of the Atoll set. It's the simplest member. If you're going
to come along and proclaim that you've discovered a generalization of
Hex, you better have something serious - not some unequal goals game
where one player is trying to form a smiley face and the other is
trying to prevent it. (Again, not to impugn Mind Ninja. It's a fine
game and its creator, Nick Bentley, is a fine designer, with whom I've
agreed to disagree about this generalization thing.)

I've spent a lot of time trying to develop 3D connection games played
on a hexagonally tessellated surface and had only marginal success.
For example, if you have a tetrahedron sitting on a flat surface
missing the bottom face you could define the following game: the goal
is to encircle the peak (which could mean just claiming the peak
cell), and extend a line from that circle down to the bottom,
triangular perimeter. Two problems: It morphs to a 2D game, and
there's a huge first move advantage. I can't release a game that
requires an nth degree pie rule. I've designed a number of games
which were geometrically interesting but which never got released.

Torben, if you had succeeded with Torex, I'm not sure what that would
have proved. You'd have had one very interesting connection game with
an obvious, strong relation to Hex.

Harald Korneliussen

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Feb 1, 2008, 2:15:14 AM2/1/08
to
On Jan 31, 4:28 pm, "markste...@gmail.com" <markste...@gmail.com>
wrote:

The "subgames" of "Whopper 100" have some common traits, but they
aren't a generalization, because they aren't a set of ALL games that
share certain traits.

Harald Korneliussen

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Feb 1, 2008, 2:18:50 AM2/1/08
to
On Jan 31, 4:48 pm, "markste...@gmail.com" <markste...@gmail.com>
wrote:

Give me a day to reply, will you? Generalizing a game is taking it,
and relaxing some properties so that you get a category of related
games. Chess and Checkers have some common properties, but a
generalization of both games would have to contain ALL games that
shared the chosen properties.

Harald Korneliussen

unread,
Feb 1, 2008, 2:28:44 AM2/1/08
to

Oh, and I see that you talk about a "Whopper infinity" below. If you
DO find a rigorous way to enumerate all possible games that share (one
or more of) the properties of chess and checkers, I will certainly
recognize it as a generalisation of both, and a pretty impressive one,
too.

Torben Ægidius Mogensen

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Feb 1, 2008, 4:31:41 AM2/1/08
to
"marks...@gmail.com" <marks...@gmail.com> writes:

> On Jan 31, 1:15 am, torb...@app-3.diku.dk (Torben Ćgidius Mogensen)


> wrote:
>> What is wrong with Y?
>
> There's nothing wrong with Y. It's just not a generalization of Hex
> though as is sometimes claimed.

Why not? It has the unique-winner proerty, it has symmetric (in fact,
identical) winning conditions and it is about making connections on a
hex-grid.

Torben

Torben Ægidius Mogensen

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Feb 1, 2008, 4:35:42 AM2/1/08
to
"marks...@gmail.com" <marks...@gmail.com> writes:

> On Jan 31, 1:02 am, torb...@app-3.diku.dk (Torben Ćgidius Mogensen)


> wrote:
>> "markste...@gmail.com" <markste...@gmail.com> writes:
>>
>> >> Depending on what aspects of Hex one sees as crucial, there are surely
>> >> many other ways of generalizing it.
>>
>> Indeed.
>>
>
> Ok, there is a subjective element. But you're actually claiming that
> an unequal goals game in which one player tries to form a happy face
> shape and the other player tries to prevent it captures the essence of
> Hex?

Perhaps not, but that doesn't disqualify it as a generalisation.

>> And, BTW, Y fulfills the above properties.
>
> No it doesn't. There are an odd number of perimeter segments, just
> for starters.

You didn't list that in the properties. It may be a property of
Atoll, but you only mentioned symmetric winning conditions, making
connections and having a unique winner on any filled board.

In any case, why is having an even number of perimeter segments an
essential property of Hex that you can't relax when making
generalisations?

Torben

Torben Ægidius Mogensen

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Feb 1, 2008, 6:08:30 AM2/1/08
to

Taking a step back, I'll start by summarising Hex:

1. The game is played by two players on a board which is an NxN
rhombic hex grid.

2. The object of one player (Black) is to create a connection between
the top and bottom edges using black pieces. The object of the
other player (White) is to connect the left and right edges using
white pieces. The edges are usually precoloured to show which
edges each player need to connect.

3. Moves are taken in turn by the two players until one wins. A move
consists of placing a pice of your own colour in any empty hex.

It can be shown that there is always a unique winner because any
filled board will obey the winning criteria of exactly one of the two
players. We will call this the unique winner property.

Atoll changes Hex by the following:

1. Changing the shape of the board.

2. Introducing 4N precoloured segments on the edge.

3. Changing the winning criteria so a player must connect any pair of
opposite segments of his colour by pieces of his colour.

Y changes Hex by the following:

1. Changing the shape of the board to a triangle.

2. Changing the winning criteria so a player must connect all three
sides of the triangle with pieces of his colour.

Both games have the unique-winner property.

You can generalise both by the following class of games:

1. The board is a finite hex grid of any shape.

2. Certain hexes can be numbered.

3. Players take turn placing a piece of their colour in any empty,
unnumbered hex.

4. Each player has a finite pre-specified set of sets of numbers.

5. Winning is done by having a connected group of pieces of the
player's colour such that the group touches hexes of all the
numbers in any of the player's set of numbers.

6. The game has the unique-winner property.

You can instantiate this to Hex by choosing a rhombic grid and
numbering the edge hexes by numbers that identify the edges, i.e.,
1,2,3,4 clockwise. Black's sets are {{1,3}} and White's sets are
{{2,4}}.

You can instantiate this to Atoll by choosing an allowed Atoll board
shape, numbering 2N segments of hexes by the numbers 1,2,...,4N
clockwise. Black's sets are {{1,2N+1},{3,2N+3},...{2N-1,4N-1}} and
White's sets are {{2,2N+2},{4,2N+4},...{2N,4N}}.

You can instantiate this to Y by numbering the edge hexes (excluding
the corners) with the numbers 1, 2 and 3. Both players have the sets
{{1,2,3}}.

So this class of games generalises both Hex, Atoll and Y.

Note that I have put the unique-winner property as a requirement for
each game in the class, as you can easily define games with properties
1-5 that don't have this property.

To show that there exists games other than Atoll and Y in this set,
here is an example:

1. The board is a hexagon.

2. The center hex is numbered 1, and the edge hexes are all numbered
2.

3. All other hexes have their own unique number.

4. Black has the sets {{1,2}}. White has all sets that describe a
closed loop around the center hex. This is a large, but finite
set of sets.

This has the unique-winner property. Proof:

At most one: A loop around the center and a path from the center to
the edge will, obviously, cross. (And, this time, there is no
mistake in this reasoning).

At least one: Assume you have a black path from center to edge.
Replacing part of this path by white pieces such that these do not
touch white pieces already on the board will not cut the path in
two. Assume, conversely, that you have a closed white loop.
Replacing parts of this by black pieces that do not tough black
pieces already on the board will not break the loop. Hence,
according to the proof I made in an earlier posting, there will be
at least one winner.

You might not like that I changed the unique-winner property from a
consequence of the other rules to an explicit requirement, but that
doesn't change that the above defines a class of games that generalise
Hex, Atoll, Y and other similar games.

Torben

Zachary Catlin

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Feb 1, 2008, 10:25:55 AM2/1/08
to

Because a generalization includes the object being generalized.

Zachary Catlin

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Feb 1, 2008, 10:56:35 AM2/1/08
to
marks...@gmail.com wrote:
> It's a subjective matter as to whether or not a class of games
> qualifies as a generalization of Hex.

Which would seem to be a difference between you and everyone else in the
thread. I personally consider a generalization of a game to be a class
of games defined by some rules (for example, the constraints on possible
boards in Atoll) that keeps some properties of that game. Now, I *would*
consider some generalizations to be more interesting than others (i.e.,
they keep more of the interesting properties of the original game
intact), and in that sense, Atoll looks to be the most interesting
generalization of Hex that currently exists (with the caveat that I
don't know much of what is out there).

marks...@gmail.com

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Feb 1, 2008, 11:16:59 AM2/1/08
to

Ohhh, ok. So Whopper isn't "rigourous" enough for you. But Mind
Ninja is rigorous, why? Because it *almost* shares one property of
Hex? (Half of the property that both players have equal, positive
goals, the attainment of each of which is simultaneously a prevention
of the opponent's goal.)

That totally explains it. Thanks for clearing that up.

marks...@gmail.com

unread,
Feb 1, 2008, 11:32:23 AM2/1/08
to
On Feb 1, 7:56 am, Zachary Catlin <zcat...@purdue.edu> wrote:

> markste...@gmail.com wrote:
> > It's a subjective matter as to whether or not a class of games
> > qualifies as a generalization of Hex.
>
> Which would seem to be a difference between you and everyone else in the
> thread. I personally consider a generalization of a game to be a class
> of games defined by some rules (for example, the constraints on possible
> boards in Atoll) that keeps some properties of that game.

I've had to backpedal a tad here. Instead of insisting that Mind
Ninja can't possibly be considered a generalization of Hex I've taken
the softer stance that to me it seems absurd to call a game that
shares half of one property of Hex a generalization of Hex. And if MN
is a generalization of Hex then Whopper is a generalization of Chess.
The word "generalized" can itself be so generalized as to have
virtually no meaning, which has been the whole thrust of the counter-
argument here.

> Now, I *would*
> consider some generalizations to be more interesting than others (i.e.,
> they keep more of the interesting properties of the original game
> intact), and in that sense, Atoll looks to be the most interesting
> generalization of Hex that currently exists (with the caveat that I
> don't know much of what is out there).

Thank you. There doesn't seem to be a whole heck of a lot out there
in this specific class of games. Not that I have an encyclopedic
knowledge of connection games - far from it. But I'm relying on those
who do.

marks...@gmail.com

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Feb 1, 2008, 11:34:32 AM2/1/08
to
On Feb 1, 7:25 am, Zachary Catlin <zcat...@purdue.edu> wrote:
>
> >> On Jan 31, 1:15 am, torb...@app-3.diku.dk (Torben Ægidius Mogensen)

> >> wrote:
> >>> What is wrong with Y?
> >> There's nothing wrong with Y. It's just not a generalization of Hex
> >> though as is sometimes claimed.
>
> > Why not? It has the unique-winner proerty, it has symmetric (in fact,
> > identical) winning conditions and it is about making connections on a
> > hex-grid.
>
> Because a generalization includes the object being generalized.

Amen.

marks...@gmail.com

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Feb 1, 2008, 12:07:44 PM2/1/08
to
On Feb 1, 3:08 am, torb...@app-5.diku.dk (Torben Ægidius Mogensen)
wrote:

>
> A loop around the center and a path from the center to
> the edge will, obviously, cross. (And, this time, there is no
> mistake in this reasoning).
>
> You might not like that I changed the unique-winner property...

What I don't like is that you've now trashed the equal goals
property. This creates a huge disparity between your "generalization"
of Hex and Hex itself. It opens the floodgates on dime-a-dozen,
unequal goals games with overwhelming move order advantage, such as
the typical, uninteresting one you provided as "proof". You've
basically just given Mind Ninja a new face.

Torex was more to the point, except that unfortunately it didn't
work.

nicko...@gmail.com

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Feb 1, 2008, 12:12:08 PM2/1/08
to
On Feb 1, 11:34 am, "markste...@gmail.com" <markste...@gmail.com>
wrote:

Hi! I'm the inventor of Mind Ninja. Mark and I have discussed Mind
Ninja privately, so I'm not going to talk about it here, but I do want
to say something about Y. There is a sense in which Y includes Hex.
After the first few moves in a game of Y, you can end up with a
position such that the remainder of the game is exactly a game of
Hex. So, the Hex decision tree is embedded in the Y decision tree, as
a subgame under certain circumstances. I wish I had a link to picture
of this situation, but I don't. According to the set-theoretic
definition of "generalization", Y can be considered a generalization
of Hex. Whether it's an interesting generalization is up for
debate.

That seems to be what this debate is really about: whether or not
certain generalizations are more interesting than others. One thing
Mark is right about: the definition of generalization that I use is
very, uh, "general". Whopper 100 is a generalization by the set-
theoretic definition, and an extremely uninteresting one at that.
There is no requirement that a generalization be interesting. But of
course, we game designers who claim that our games are generalizations
are really trying to imply that there is something interesting about
our games! That's publicity!

Here's the set-theoretic definition of "generalization":

B is a generalization of A if all the instances of A are included in
B, and B contains some instances not in A.

Note that there is no reference to "shared properties" in this
definition. Of course the only generalizations that will matter to us
are those that are defined by shared properties that we care about.

nicko...@gmail.com

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Feb 1, 2008, 3:05:50 PM2/1/08
to
OK, I said I wasn't gonna talk about Mind Ninja, but I thought someone
might like this little thing I wrote about how Mind Ninja came to be:

http://www.mindninja.com/history.html

marks...@gmail.com

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Feb 1, 2008, 9:22:52 PM2/1/08
to
On Feb 1, 9:12 am, nickobe...@gmail.com wrote:
>
> Hi! I'm the inventor of Mind Ninja. Mark and I have discussed Mind
> Ninja privately, so I'm not going to talk about it here, but I do want
> to say something about Y. There is a sense in which Y includes Hex.
> After the first few moves in a game of Y, you can end up with a
> position such that the remainder of the game is exactly a game of
> Hex. So, the Hex decision tree is embedded in the Y decision tree, as
> a subgame under certain circumstances. I wish I had a link to picture
> of this situation, but I don't. According to the set-theoretic
> definition of "generalization", Y can be considered a generalization
> of Hex. Whether it's an interesting generalization is up for
> debate.

It's a fine point but I'm going to go into it anyway. It is
interesting that the Hex game tree is embedded in the Y game tree.
But Y is a distinct game with a distinct set of rules. Y is one game,
not a set of games or a class of games. Just grammatically and
logically it doesn't make sense to say "Y is a generalization of Hex,"
or "Hex is a special case of Y". Either way you're implying that Y is
a set of games, which it isn't. The fact that two players can
cooperate and play Y down to a position such that the remainder of the
game is equivalent to Hex (a position that's unlikely to occur without
cooperation) doesn't make Y a generalization of Hex.

>
> That seems to be what this debate is really about: whether or not
> certain generalizations are more interesting than others. One thing
> Mark is right about: the definition of generalization that I use is
> very, uh, "general". Whopper 100 is a generalization by the set-
> theoretic definition, and an extremely uninteresting one at that.
> There is no requirement that a generalization be interesting. But of
> course, we game designers who claim that our games are generalizations
> are really trying to imply that there is something interesting about
> our games! That's publicity!
>
> Here's the set-theoretic definition of "generalization":
>
> B is a generalization of A if all the instances of A are included in
> B, and B contains some instances not in A.
>
> Note that there is no reference to "shared properties" in this
> definition. Of course the only generalizations that will matter to us
> are those that are defined by shared properties that we care about.

Thank you. You did effectively summarize the debate and there are two
ways to interpret the word generalization as you said, one being the
"set theoretic" sense and the other being the every day sense.

My complaint is this: when I first heard that Mind Ninja was supposed
to be a generalization of Hex, to me that implied that Mind Ninja
captures the essence of Hex. That's a tall order. Hex is considered
by many to be the world's most elegant game. Now you've encompassed
it in a universe of unequal goals games, whose size in limited only by
the number of shapes one can dream up. The vast majority of games in
that set are effectively unplayable due to overwhelming move order
advantage. They outnumber the playable ones by a factor of infinity,
as do the games in the set that bear no resemblance to Hex.

Now if it were the set of games with equal goals, in which achieving
your goal is equivalent to preventing your opponent from achieving his
(if play were to continue beyond a win), and if you tossed in a few
original examples, that'd be totally different. But as it is, the
claim "Mind Ninja is a generalization of Hex" without the qualifier
"in the set theoretic sense" is objectionable, at least to me.

Ed Murphy

unread,
Feb 2, 2008, 12:36:34 AM2/2/08
to
marks...@gmail.com wrote:

Hmm, can Y be generalized in an Atoll-like fashion? If so, then would
you accept (Atoll + Atoll-like generalization of Y) as a generalization
of both Hex and Y?

marks...@gmail.com

unread,
Feb 2, 2008, 5:32:35 PM2/2/08
to
On Feb 1, 9:36 pm, Ed Murphy <emurph...@socal.rr.com> wrote:
>
> Hmm, can Y be generalized in an Atoll-like fashion?
>

Yes, Y can be generalized and has been generalized - by me. I call
the generalization of Y Begird:

Begird rule sheet:
http://www.marksteeregames.com/Begird_rules.pdf

>
> If so, then would
> you accept (Atoll + Atoll-like generalization of Y) as a generalization
> of both Hex and Y?
>

No. Atoll is the generalization of Hex and Begird is the
generalization of Y. That's the extent of it.

By the way, someone here said I was upset because I expected Atoll to
be met with accolades and it wasn't. Believe it or not, I'm not quite
that stupid. I knew Atoll would be a big, bitter pill. People in a
given field automatically reject major discoveries in that field, at
least initially. It'll take people at least a decade to accept Atoll
as the generalization of Hex.

I half expected Atoll to be deleted from Hexwiki. The so-called
"editors" (pronounced with a chuckle) surprised me though by not being
quite as dick headed as I thought. They put Atoll into the variants
category which I knew for a fact they'd do if they kept it. As I
mentioned earlier, Atoll is the generalization of Hex, not a variant
of Hex. A set is not a variant of one of its members.

Harald Korneliussen

unread,
Feb 4, 2008, 12:25:21 PM2/4/08
to
On Feb 2, 11:32 pm, "markste...@gmail.com" <markste...@gmail.com>
wrote:

> I knew Atoll would be a big, bitter pill. People in a
> given field automatically reject major discoveries in that field, at
> least initially. It'll take people at least a decade to accept Atoll
> as the generalization of Hex.

Certainly nothing wrong with the size of your ego, Copernicus. By the
way, no one has rejected Atoll as _a_ generalization of Hex.

marks...@gmail.com

unread,
Feb 4, 2008, 4:34:01 PM2/4/08
to
On Feb 4, 9:25 am, Harald Korneliussen <vinterm...@gmail.com> wrote:
> On Feb 2, 11:32 pm, "markste...@gmail.com" <markste...@gmail.com>
> wrote:
>
> > I knew Atoll would be a big, bitter pill. People in a
> > given field automatically reject major discoveries in that field, at
> > least initially. It'll take people at least a decade to accept Atoll
> > as the generalization of Hex.
>
> Certainly nothing wrong with the size of your ego, Copernicus.

I'm entitled to it.

> By the
> way, no one has rejected Atoll as _a_ generalization of Hex.

Well, until someone else comes up with one, it's _the_.

marks...@gmail.com

unread,
Feb 4, 2008, 5:23:28 PM2/4/08
to
You can just about prove that Atoll is _the_ generalization of Hex.
Hex can't be played on a closed three dimensional surface, i.e. no
holes. Box Hex, for example, (http://www.marksteeregames.com/
Box_Hex_rules.pdf) is played on the outer surface of a "box" but it
has an open top and so can be morphed into a 2D shape.

Call this an "inspiration" for a proof. If you have a globe with the
equator divided into four segments, two black and two white, Black can
make a connection across the Northern hemisphere and White can make a
connection across the Southern hemisphere. Now, assumptions here are
that there are at least two black "zones", for lack of a better word,
and two white zones. And that both players have equal goals.

Hex has to be played on a 2D surface, and the black and white zones
have to be on the perimeter. They must also be contiguous,
alternating black and white. There can't be any "blank" zones.
Again, to emphasize, you can only play on one side of a two
dimensional surface or you'd have something equivalent to the globe.

You could print a Hex board on a rubber sheet with no tessellation,
only the colored perimeter segments, morph it into some other 2D shape
and then add tessellation, but this wouldn't do anything to further
generalize it.

The only thing you can really do to vary this is to further divide the
perimeter segments, increasing the number to larger multiples of four
and continue to connect opposite, like-colored segments. You need to
add the little trick of allowing indirect connections via multiple,
separate bridges linking multiple, like-colored segments so that you
can maintain the "filled board produces exactly one winner" property.
And you need that property. Don't even talk to me about
"generalizations" of Hex that don't have that or don't have equal
goals.

I mean if you want to call Atoll _a_ generalization of Hex, fine. I'm
tired of arguing about it. But personally I believe it can be proved,
along the lines of the above "inspiration", that Atoll is _the_
generalization of Hex. I've got donuts to think about here. Let's
move on.


marks...@gmail.com

unread,
Feb 4, 2008, 6:24:46 PM2/4/08
to
There is actually Eight Sided Hex which uses a rhombus with eight
perimeter segments. It's only one game though, not a class of games,
and the goal isn't to connect opposite sides. It should be mentioned
in this discussion though. It appears in Cameron Browne's book
Connection Games.

David Bush

unread,
Dec 29, 2008, 4:19:28 PM12/29/08
to
> ... There is a sense in which Y includes Hex.

> After the first few moves in a game of Y, you can end up with a position
> such that the remainder of the game is exactly a game of Hex. So, the
> Hex decision tree is embedded in the Y decision tree, as a subgame under
> certain circumstances. I wish I had a link to picture of this
> situation, but I don't. According to the set-theoretic definition of
> "generalization", Y can be considered a generalization of Hex. Whether
> it's an interesting generalization is up for debate.

Here is a simple image which shows what I believe you are talking about.
A size 13 triangular Y board becomes a 7x7 Hex game after some horrible
moves by each side.

This text graphics image requires a fixed width font:


B B B B B B
* * * * * * *
W * * * * * * *
W * * * * * * *
W * * * * * * *
W * * * * * * *
W * * * * * * *
W * * * * * * *

Not all of the triangular board is shown. B indicates moves by black and
W indicates moves by white.

Technically, this is NOT a 7x7 Hex game, for two reasons.

The most important reason is, all experienced Hex players and all the Hex
servers on the Net use the swap rule. After the very first stone is
placed on the board the second player has the option at that moment only
to swap sides. This makes the game much more balanced and interesting.
But here, after 6 moves by each side, swapping sides in Y is forbidden.

Also, in the Y game it would be perfectly legal, albeit useless, for
either side to play in the sections of the triangular grid which are not
shown here. This is effectively the same as passing, which is not legal
in Hex. This is a minor point, since it is never advantageous to pass in
Hex.

Another hit against this generalization is, both sides had to cooperate,
intentionally or unintentionally, to reach this position. This goes
against a basic tenet of two-player abstracts. It means that any result
from studying a 7x7 hex game teaches us nothing about opening theory on
an order 13 Y board.

I would also like to point out that in order for an NxN Hex game to be
"encompassed" by a triangular Y game, the Y board would have to be 2N-1
cells on each side. So any claim that this somehow shows the superiority
of Y over Hex would not be saying much. No one in these posts is making
such as claim AFAIK.

Phil Carmody

unread,
Dec 30, 2008, 3:56:08 AM12/30/08
to

A non-issue, surely? If you trust your opponent to build the borders
before starting the alternative game, then you should trust your
opponent to play the first move of the alternative game, and offer
you a swap, before the artificial borders are constructed.

Phil
--
I tried the Vista speech recognition by running the tutorial. I was
amazed, it was awesome, recognised every word I said. Then I said the
wrong word ... and it typed the right one. It was actually just
detecting a sound and printing the expected word! -- pbhj on /.

David Bush

unread,
Dec 30, 2008, 10:20:23 AM12/30/08
to
On Tue, 30 Dec 2008 10:56:08 +0200, Phil Carmody
<thefatphi...@yahoo.co.uk> wrote:

>> Technically, this is NOT a 7x7 Hex game, for two reasons.
>>
>> The most important reason is, all experienced Hex players and all the
>> Hex servers on the Net use the swap rule. After the very first stone is
>> placed on the board the second player has the option at that moment
>> only to swap sides. This makes the game much more balanced and
>> interesting. But here, after 6 moves by each side, swapping sides in Y
>> is forbidden.
>
> A non-issue, surely? If you trust your opponent to build the borders
> before starting the alternative game, then you should trust your
> opponent to play the first move of the alternative game, and offer you a
> swap, before the artificial borders are constructed.

The bad Y moves that led to the 7x7 Hex position were LEGAL.
Swapping sides in the middle of a Y game is ILLEGAL, regardless of what
the players might think. So now you are talking about some modified rule
set. The claim that Hex is some sort of "subset" of Y, now turns out to
be a claim that Hex is some sort of subset of a variant of Y.

Phil Carmody

unread,
Dec 30, 2008, 6:15:31 PM12/30/08
to
David Bush <tw...@cstone.net> writes:
> On Tue, 30 Dec 2008 10:56:08 +0200, Phil Carmody
> <thefatphi...@yahoo.co.uk> wrote:
>
>>> Technically, this is NOT a 7x7 Hex game, for two reasons.
>>>
>>> The most important reason is, all experienced Hex players and all the
>>> Hex servers on the Net use the swap rule. After the very first stone is
>>> placed on the board the second player has the option at that moment
>>> only to swap sides. This makes the game much more balanced and
>>> interesting. But here, after 6 moves by each side, swapping sides in Y
>>> is forbidden.
>>
>> A non-issue, surely? If you trust your opponent to build the borders
>> before starting the alternative game, then you should trust your
>> opponent to play the first move of the alternative game, and offer you a
>> swap, before the artificial borders are constructed.
>
> The bad Y moves that led to the 7x7 Hex position were LEGAL.
> Swapping sides in the middle of a Y game is ILLEGAL, regardless of what
> the players might think. So now you are talking about some modified rule
> set.

No I'm not. Read what I wrote again but this time please ensure
that you've engaged your brain.

David Bush

unread,
Dec 30, 2008, 11:31:33 PM12/30/08
to
>>>> Technically, this is NOT a 7x7 Hex game, for two reasons.
>>>>
>>>> The most important reason is, all experienced Hex players and all the
>>>> Hex servers on the Net use the swap rule. After the very first stone
>>>> is placed on the board the second player has the option at that
>>>> moment only to swap sides. This makes the game much more balanced and
>>>> interesting. But here, after 6 moves by each side, swapping sides in
>>>> Y is forbidden.
>>>
>>> A non-issue, surely? If you trust your opponent to build the borders
>>> before starting the alternative game, then you should trust your
>>> opponent to play the first move of the alternative game, and offer you
>>> a swap, before the artificial borders are constructed.
>>
>> The bad Y moves that led to the 7x7 Hex position were LEGAL. Swapping
>> sides in the middle of a Y game is ILLEGAL, regardless of what the
>> players might think. So now you are talking about some modified rule
>> set.
>
> No I'm not. Read what I wrote again but this time please ensure that
> you've engaged your brain.

Oh, perhaps you mean the first move of the Y game should be within the
Hex board region. For that matter, you could ignore all those border line
moves, and both players agree to play within the Hex region, and the
loser of the Hex game will agree to resign. But what does this have to do
with the claim that Hex is some sort of subset of Y? Isn't that what this
discussion was about?

Phil Carmody

unread,
Dec 31, 2008, 5:24:15 AM12/31/08
to
David Bush <tw...@cstone.net> writes:
>>>>> Technically, this is NOT a 7x7 Hex game, for two reasons.
>>>>>
>>>>> The most important reason is, all experienced Hex players and all the
>>>>> Hex servers on the Net use the swap rule. After the very first stone
>>>>> is placed on the board the second player has the option at that
>>>>> moment only to swap sides. This makes the game much more balanced and
>>>>> interesting. But here, after 6 moves by each side, swapping sides in
>>>>> Y is forbidden.
>>>>
>>>> A non-issue, surely? If you trust your opponent to build the borders
>>>> before starting the alternative game, then you should trust your
>>>> opponent to play the first move of the alternative game, and offer you
>>>> a swap, before the artificial borders are constructed.
>>>
>>> The bad Y moves that led to the 7x7 Hex position were LEGAL. Swapping
>>> sides in the middle of a Y game is ILLEGAL, regardless of what the
>>> players might think. So now you are talking about some modified rule
>>> set.
>>
>> No I'm not. Read what I wrote again but this time please ensure that
>> you've engaged your brain.
>
> Oh, perhaps you mean the first move of the Y game should be within the
> Hex board region.

Well, given that in this context the alternate game on a Y board
is hex, yes, saying that the first move should be a move from the
alternate game, yes it will be within the area reserved for that
hex game.

> For that matter, you could ignore all those border line
> moves, and both players agree to play within the Hex region, and the
> loser of the Hex game will agree to resign. But what does this have to do
> with the claim that Hex is some sort of subset of Y?

Nothing. It has something to do with your claim that because the
swap rule couldn't be implemented if you needed to first set up
the borders, it couldn't be implemented at all. Which isn't true.
However, you are right that the simplest contrivance is to simply
agree to resign.

> Isn't that what this
> discussion was about?

Nope, believe it or not it's possible to make more than one point
in a usenet post, and for those who respond to it to chose which
subset of those points they wish to address, and which they do not
wish to address.

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