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Subject: The Noncommutative Game Product rss

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David Stoner

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This isn't a new game per se. Rather, it's a way to take two games A and B, each played on a square grid, and create a product game, which I'll call (A x B). The idea is that the game A is being played overall, with each cell on A replaced with a copy of the game B. The rules are as follows:

-Player one starts by making a single placement in any of the game B boards.
-Thereafter, each player makes up to two successive placements in any of the game B boards. These can both be in the same board, or in two different boards.
-Whenever a B board is won by some player, that player places a stone in the corresponding square in the overall game A board, if that placement is legal (with respect to game A). Otherwise, the player must place an opponent's stone in that square instead, if that placement is legal. If neither placement is legal, the corresponding A space remains empty.
-Each individual move must be legal as a move in a B game, and the two placements are considered sequentially.

The player which wins the overall A game wins. The images below shows a game in progress (Kopano x Kopano) with black=vertical, white-horizontal. The first image shows board A separate from the corresponding board Bs, while the second shows them together.



Note that even though white is winning on the incomplete center board, he shouldn't complete that board as the game currently stands. Doing so would create an illegal weak connection in the overall A board, so if white wins that board it would currently be awarded to black.

Some comments on the product:
-I've only tested this when A, B are connection games, but theoretically (A x B) is well-defined whenever A is a placement game. (B could potentially be a movement game if we replace "two placements" with "two moves")
-If A, B are drawless, then (A x B) is as well.
-For connection games, (A x B) is most interesting when A is somewhat cold. Then one can try to force their opponent to complete grids corresponding to unfortunately placed stones in the overall A game. In the Kopano example, board which are diagonally and orthogonally adjacent are closely tied to each other due to the local Kopano rules. In general coldness leads to more interesting interactions between the different B-boards.
-The 2 moves per turn makes this far more interesting than it would be otherwise, I've found, due to the options of spreading out or concentrating on a single board.
-When A, B are 4x4, (A x B) is a 16x16 game, and that's currently the size I would suggest.
-Let's take = to be equivalence of games up to turn order. While the product isn't commutative in general, it is associative! (A x B) x C = A x (B x C), though I don't recommend actually playing those It also has an identity element; namely, the 1x1 trivial game where the person who places a stone wins. Hence the set of drawless placement games on rectangle grids forms a monoid under x!

What do you think? I wouldn't be surprised if someone had tried this construction already, but I've liked the result based on playtesting so far. Probably the most difficult thing practically speaking is finding games A, B which are sufficiently interesting on small boards.




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Russ Williams
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nonzer0 wrote:
What do you think? I wouldn't be surprised if someone had tried this construction already,


Sure, it's an idea that has been around a while. The most obvious famous example which immediately occurs to me is doing it to Tic-Tac-Toe with 9 Tic-Tac-Toe subgames. E.g. Ultimate Tic-Tac-Toe, Tic Tac Toe Times 10.

But it was neat to see the idea described with such formalism and generality here!
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Stephen Tavener
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Fire and Ice is also a close relative.
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Christian K
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I like this. ConHex is sort of this (although the smaller games are overlapping. You play he. But each space is an area majority game.

I have not seen the concept abstracted in this way before.
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Luis Bolaños Mures
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Ingriguing!

I like that you chose Kopano of all games.
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Dieter Stein
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luigi87 wrote:
I like that you chose Kopano of all games.


The games could be a whole set of Luis' large selection of connection games. Luis, you've got 17 of them already?
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Frederic Heath-Renn
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nonzer0 wrote:
It also has an identity element; namely, the 1x1 trivial game where the person who places a stone wins.


Vile nitpicking but this isn't, I think, correct: while A = I' x A for all A, A = A x I' isn't always true. If A has illegal moves (for example, the drawless placement game Crossway), then a player in A x I' can attempt an illegal move (by completing the I' board corresponding to that cell) and place an opponent's stone (if that's legal) or no stone, which they can't do in A.
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David Stoner

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Ah, that's a good point! We can fix things by extending the mentioned equivalence classes to include games with arbitrary impartiality restrictions and the option of passing; essentially, two games with the same individual stone placement restrictions (regardless of who is doing the placing) and goal are equal.

I've liked games like Kopano for A due to their inherent coldness and downwards scalability. I still don't know what type of game B makes for the most intersting games, though.
 
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Nick Bentley
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gameception!
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Luis Bolaños Mures
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spielstein wrote:
luigi87 wrote:
I like that you chose Kopano of all games.


The games could be a whole set of Luis' large selection of connection games. Luis, you've got 17 of them already?

Let's see. You could play Brique on a1, Cation on a2, Flicker on a3, Konobi on a4, Morpheus on b1, Quaxtep on b2, Quentin on b3, Restitch on b4, Rhode on c1, Squer on c2, Squex on c3, Tex on c4, Trisq on d1, Vimbre on d2, Weft on d3, Whirlwind on d4 and Kopano as the meta-game. It adds up!

I'm not sure, but I think there might be a draw if a player's only legal moves on the mini-boards correspond to illegal moves on the big board and the other player is in the same situation with a different set of mini-boards.

If that's the case, though, I got you covered: draws are resolved by playing Stitch!
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David Stoner

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I'd be willing to try that crazy thing if anyone else is! I guess you'd resize Trisq, Squex, Tex and a few others to keep a full board at about 16 pieces.

About the drawing concern: if each player's only legal moves on all of the unfinished B boards are illegal in the large board (for both players), then the board A is necessarily in a state with no legal moves (each of the unfinished B boards are drawless, so they all have legal moves for some player). This is impossible if A is drawless.
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Luis Bolaños Mures
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nonzer0 wrote:
About the drawing concern: if each player's only legal moves on all of the unfinished B boards are illegal in the large board (for both players), then the board A is necessarily in a state with no legal moves (each of the unfinished B boards are drawless, so they all have legal moves for some player). This is impossible if A is drawless.

I can't quite wrap my head around this, but, just in case, I meant something like the following position in the large board:

sugarcoffeecoffeecoffee
sugarcoffeecoffeesugar
sugarcoffeecoffeesugar
blankblanksugarsugar

By Kopano rules, Black can only play on b1 and White can only play on a1.
What happens if

- Black has no legal moves in the b1 sub-board,
- all legal White moves in the b1 sub-board are immediate White wins,
- White has no legal moves in the a1 sub-board, and
- all legal Black moves in the a1 sub-board are immediate Black wins?

Not quite sure if that makes sense...
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David Stoner

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If I understand your example right, whoever is to move loses! To clarify, when white wins a board, then the following happens:
-If white can be legally placed in that square in A, that happens.
-Otherwise, if black can be legally placed in that square in A, that happens.
-Otherwise, nothing happens (for the moment). On every turn thereafter, the above algorithm is repeated, in that order. But the corresponding B board is finished, and neither player can move on it.

In particular, since unforced passing isn't allowed, your example means the player to move loses.

See the Kopano x Kopano example for another instance of this; if white takes the fourth center board in the near future, the corresponding square would go to black.
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Luis Bolaños Mures
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nonzer0 wrote:
-Otherwise, if black can be legally placed in that square in A, that happens.

Oh, nevermind. I had overlooked this rule. The game is obviously drawless with it.
 
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Dan Nunuyerbiznez
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nonzer0 wrote:
I'd be willing to try that crazy thing if anyone else is! I guess you'd resize Trisq, Squex, Tex and a few others to keep a full board at about 16 pieces.


Is there a need to have all sub-boards on A the same size? Not sure why there should be if you are playing different games on each 'tile'...
 
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David Stoner

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There's no need to have boards all similar size, though I feel that would be best to maintain some sense of balance across the games. I feel like smaller boards would probably inevitably be played out first otherwise, since those are easier to capture.

That being said, it might be interesting to balance the A-board by making the central subboards larger than the others, kind of like Fractal.
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David Bush
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There is a Hex variant which is close to what you describe, but not quite. It it played on any rhombus grid of hexes of even size, for example 14x14. The grid is partitioned into four sub grids. In my example these would be 7x7 each. There are five sub-games going on at the same time, the four smaller grids plus the overall grid. Whoever wins three of those games wins the overall game. In Hex there are never any forbidden vacant hexes. There is a swap option to the second player after the very first (traditionally black) stone is placed. I suppose this might also work for an overall grid of odd size, if the sub grids share borders and the central cell is part of all five games.

So, maybe if game A is also game B, an alternative to what you describe might be to say the winner of the majority of games is the overall winner. My apologize for off topic drift.
 
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Maurizio De Leo
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Very interesting, although not new, as per the Tic-Tac-Toe examples brought up by Russ.

I think a big simplification would be to restrict to placement games which have no "illegal" moves if the cell is empty, like Tic-Tac-Toe, Hex, Connect6 etc.
For example the statement
Quote:
If A, B are drawless, then (A x B) is as well.
is not strictly true, as A could be not drawless anymore once we add "forced empty" cells due to illegal moves.

Speaking of Connect6, the 1222.. protocol is neat, but not really necessary for this games product.
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David Stoner

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megamau wrote:

For example the statement
Quote:
If A, B are drawless, then (A x B) is as well.
is not strictly true, as A could be not drawless anymore once we add "forced empty" cells due to illegal moves.


I could be mistaken, but I think this isn't an issue; spaces on A are never "forced empty" since legality conditions are checked for unfinished boards at each turn, in order of completion. The whole motivation behind this construction was to have A cold (so, lots of illegal moves) so that the different boards of B interact in an interesting manner.

As for the 122... it's probably not strictly necessary, but I've found it much, much favorable to the alternative.
 
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Maurizio De Leo
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nonzer0 wrote:
megamau wrote:

For example the statement
Quote:
If A, B are drawless, then (A x B) is as well.
is not strictly true, as A could be not drawless anymore once we add "forced empty" cells due to illegal moves.


I could be mistaken, but I think this isn't an issue; spaces on A are never "forced empty" since legality conditions are checked for unfinished boards at each turn, in order of completion. The whole motivation behind this construction was to have A cold (so, lots of illegal moves) so that the different boards of B interact in an interesting manner.


I think it is an issue, unless you somehow define moving in a B game after someone has won it.
What do you mean by "unfinished" boards ? Does this include the B games that have been won but not "assigned" to a player because they were illegal placements for both at that time ?

If so every time a stone is played anywhere, you have to recheck for all the moves in A that might have become legal for one or both players. It becomes quite tedious to play in real life.

EDIT: improved the syntax
 
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David Stoner

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Right, the rules are exactly as you describe in terms of games won but not assigned; they're checked at the end of every turn, in the order they arose. But this has never actually happened in my playtests, and I think that for most A, B, that rule basically never comes into play. That's because in most A boards (at least when A is a connection game) moves are very rarely illegal for both players. For example, in Kopano this only happens when both colors would create a crosscut, which requires a very specific and unlikely configuration of six stones to happen. In fact, in games like Kopano you can ignore this rule completely, because "illegal for both players" moves can never in the future become legal for either player. It's really only there as a technicality, to keep drawlessness in all circumstances.
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This can create a whole new feeling to the overall game when there are very limited legal placements, such as Othello. The small games can effectively force permanent "holes" in the overall game.

So who's up for a game of Go x Go?

And of course, there's no reason why you need to limit this to square grids. Or even that the grids of the small games be the same as the grid for the big game. Hex x Cairo Corridor, anyone?
 
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Maurizio De Leo
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Go is problematic due to passing and determining when one board is "won".
Also, it requires minimum 9x9 and 9x9 for a total of 6561 moves.

What about a Connect Four x Tic Tac Toe (7x6 by 3x3) ?
 
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