christian freeling
Netherlands

"One sticking, one free" is a generic opening protocol for a certain class of games, as well as a combinatorial game in itself. It works for any grid and any shape or size of the board. I put it here as an optional tool for other inventors. Here's how it goes:
Quote: White starts by placing one stone on the empty board. From that point on players take turns to:
* Place a stone on a cell adjacent to the last stone placed by the opponent, and ... * ... place a stone on a on a cell that has only vacant cells as neighbors.
Both placements are compulsory. When the player to move can no longer make the second placement, then his turn ends and his opponent may start the second phase. 'Adjacent' on a square board may or may not include diagonal adjacency. In the latter (and also the usual) case it will lead to a more densely packed final position. Unless considered as a stand alone combinatorial game, this final position is the initial position of the second phase, which may be any game. I'll include three that emerged these past three months.
If you think about the implications, you'll find that when the protocol has reached its end, then
* the number of black and white stones will be equal, and ... * ... the total number of placed stones is set between a (as yet to be proven) minimum and a (as yet to be proven) maximum, and ... * ... there's no telling who's turn it may be.
Regarding the second, I'm not a puzzler, but it can be established for any grid in any form or size. Regarding the third: trying to be the first (or second) to move in the next phase is a combinatorial game in itself, and not a trivial one. In terms of strategy you'd have to know what are the best initial conditions for starting the second phase, and that again would depend on the game.
The protocol shines a new light on the turnorder balance issue, serving as such for the second phase. The question might be whether the protocol itself would need a turnorder balancing mechanism. If the answer is affirmative, then the pierule would be the most obvious one. For the time (and the games) being we've decides not to. Here are the three games I mentioned that employ it:
* Triccs * Argon * Inertia
I found Triccs when I found the protocol but couldn't immediately link it to the object I had in mind for the second phase. That object was "dynamic goal connection". But momentarily switching to "territory" I saw this simple game that could not derail in terms of mechanics and would always be decisive. I used it more than anything else to provisionally 'store' the protocol.
The other two games have "dynamic goal connection" as their object. Argon happened accidentally because I couldn't leave LOA out of the equation, but Inertia was the one I was after. It is inspired by Luis Bolaños Mures extraordinary game Ayu, a game I highly recommend.
christian freeling mindsports



Thanks for the post. I think general move protocols is an area that could be discussed more here, so I am glad to see some focus on it.

Nathan James
United States Covington Ohio

This protocol is very intriguing. Thanks for posting it.

Benedikt Rosenau
Germany Göttingen

Muemmelmann wrote: Thanks for the post. I think general move protocols is an area that could be discussed more here, so I am glad to see some focus on it. My thoughts exactly. I think the last decade has been fruitful.

Chris Huntoon
United States Florida

Interesting. This reminds me of a game mechanic that I fiddled around with a little bit several years ago, but never fully developed, as other subjects took my interest.
The basic idea was to combine in one game two completely contradictory piece placing rules. Black would begin by placing a single stone anywhere on the board. Thereafter, a player's turn consited of two phases. In phase 1, a stone can be placed next to an enemy stone but not next to a friendly stone. Phase 2: a stone can be placed next to a friendly stone but not next to an enemy stone.
But I never developed it beyond that basic concept. The intent was to turn it into a combinatorial game. But I remember thinking  what happens when a player is able to complete phase 1 but not phase 2? Does that count as a win, a lose, or a draw?

christian freeling
Netherlands

hartunga wrote: Interesting. This reminds me of a game mechanic that I fiddled around with a little bit several years ago, but never fully developed, as other subjects took my interest.
The basic idea was to combine in one game two completely contradictory piece placing rules. Black would begin by placing a single stone anywhere on the board. Thereafter, a player's turn consited of two phases. In phase 1, a stone can be placed next to an enemy stone but not next to a friendly stone. Phase 2: a stone can be placed next to a friendly stone but not next to an enemy stone.
But I never developed it beyond that basic concept. The intent was to turn it into a combinatorial game. But I remember thinking  what happens when a player is able to complete phase 1 but not phase 2? Does that count as a win, a lose, or a draw? In my experience simplicity has a habit of becoming obvious in retrospect: how could I have missed that?
In this particular case, the search for an opening protocol was fueled by Luis' Ayu. This essential "dynamic goal connection" game, with rules that appear necessary and sufficient, had made me realize that I had grossly neglected (if not underestimated) it as a stand alone goal, though I had employed it as a means to an end in Symple's goal.
Of course I had to consider other means to the same object, and one thought was: make a protocol that leads to a variable initial position. It had to divide as well as spread forces equally, and leave about half the board empty. The game it would serve was a later worry.
When I found it I managed to get it the wrong way around, placing the free stone first. I'm not sure about the role of stupidity in game inventing, but I can't seem to do without it. It actually took a week or so before I realized that reversing the order would always lead to an even division, and always leave the last move a sticking one.

Nick Bentley
United States Madison Wisconsin

I like this concept a lot. It feels simple and fundamental. In fact, my initial impression is considerably more positive than my present feeling for your Symple mechanism (Placing more than 34 stones on a turn has alwasy felt unwieldy to me).
I've been trying to think of a simple lastmover wins game to use your "one sticking, one free" mechanism in, such that the first player who can't place "one free" loses. I'm still missing an ingredient or two though.
[edit] I explored a related line of thinking some time ago, which lead to another mechanic to deal with initial stone spread: I call it the Shifty mechanic after a game by the same name (although I found what may be a better use for it in another game, called Shello). The rule is:
Place a stone adjacent to a friendly stone, or move a stone (usually by a queen's move, although it can vary depending on the needs of the game) to a space with fewer adjacent stones than the space it started from.
Not as simple as yours but creates a nice dynamic where movement constrains placement and placement constrains movement in interesting ways.

christian freeling
Netherlands

milomilo122 wrote: I like this concept a lot. It feels simple and fundamental. In fact, my initial impression is considerably more positive than my present feeling for your Symple mechanism (Placing more than 34 stones on a turn has alwasy felt unwieldy to me). I know, and anyone feeling the same way would probably agree. I've played so many games of Symple that I can enjoy the options to combine moves without any unwieldy feelings.
That being said, the Symple protocol is a move protocol that has an inherent strategic dilemma with specific opening consequences, but without any clear division between an opening phase and the 'rest of the game'. It also has an embedded turnorder balancing mechanism. It's a bit more than a simple opening protocol.
It's generic character is shown all the more in Scware and Multiplicity, two games that complete this season's batch. I hope to discuss these two later and in a separate thread. Multiplicity is a new generation abstract that is dependent on features provided by the applet, and there is no applet yet.

christian freeling
Netherlands

In this first base7 Inertia game you can see the protocol in action.
The object of the game is, loosely speaking, unification of one's groups. Red's final comment:
"Strategically speaking I'm in a lost position here because you're connected through the center and I can't isolate any black group, at least not without a so obvious preparation that you can defuse the attempt. Very well played, congrats ."

christian freeling
Netherlands

Here's an update after a number of increasingly interesting games we played in the exploration of Inertia. In this game you'll find some hopefully useful comments on strategy.
Jos Dekker (ger)  Christian Freeling (nl) (running)



I just thought of a protocol like remove an opponent stone and place two of your own. It could be used in Hex or Havannah for example. Does anyone have expirience with something like this?

christian freeling
Netherlands

Muemmelmann wrote: I just thought of a protocol like remove an opponent stone and place two of your own. It could be used in Hex or Havannah for example. Does anyone have expirience with something like this? Matteo Perlini's Ecalper features something like this. But the notion seems to ring some other bells too, rather vaguely. I have no experience with Ecalper, but from what I've see at iGGC it looks like an interesting game.

Carlos Luna
Spain Rubí Catalunya

christianF wrote: (...) the total number of placed stones is set between a (as yet to be proven) minimum and a (as yet to be proven) maximum, (...) I'm not a puzzler, but it can be established for any grid in any form or size.
Has anyone worked out these bounds for rectangular MxN grids with either the Von Neumann or the Moore neighborhoods? Any (rough) estimation?

christian freeling
Netherlands

CarlosLuna wrote: christianF wrote: (...) the total number of placed stones is set between a (as yet to be proven) minimum and a (as yet to be proven) maximum, (...) I'm not a puzzler, but it can be established for any grid in any form or size. Has anyone worked out these bounds for rectangular MxN grids with either the Von Neumann or the Moore neighborhoods? Any (rough) estimation? Estimates are easy but quite useless. Formulas for nxm bounds would be great, but that may not be easy. Hexboards are an issue too.
For any particular board you can make it a challenge: find the least number and find the highest, look who wins.

Richard Walter
United States San Jose California

So I was bored this evening and threw together a quickndirty program to brute force all starting games on rectangular grid and keep track of the min & max number of pieces before the game ends.
So, for NxM rectangular grid with Von Neumann neighbors (ie: only the 4 orthogonally adjacent spaces are neighbors), the min/max values are:
\2345678 2  2,2 2,4 4,6 4,8 4,10 6,12 6,14 3   4,6 4,10 6,12 6,16 8,18 4    6,12 8,16
For NxM rectangular grid with Moore neighbors (ie: all 8 adjacent spaces are neighbors), the min/max values are:
\2345678 2  2,2 2,4 2,4 4,6 4,6 4,8 4,8 3   2,6 2,8 4,10 4,12 4,14 4,16 4    4,8 4,12 4,12 5     6,16
If the program ran for more than about 1 minute, I stopped it, which is why the tables are blank after size 4x7 or so.
Also, I did make sure to only try the upperleft 1/4 of the board for the first move, as any game whose first move is somewhere else on the board is symmetrically equivalent (via reflections) to a game that does start there.
If there is interest, I might try to make it more efficient and/or let it run longer to squeeze out an extra row or two.
It may also be interesting to print out the board state at these extremes to see what patterns emerge that may help with determination of larger board sizes.
It also appears that any value within the range is reachable. ie: on a Moore neighborhood board of 3x7, you can get every (even) value from 4 to 14. (I added this feature late and didn't retry every location, but the ones I did try all show this behavior.)
Turning these numbers into an equation of sorts is left as an exercise to interested readers...
Richard

christian freeling
Netherlands

Thanks, that's very enlightening and the range goes somewhat beyond my intuitive estimates. Which is a good thing because it is indicative of a corresponding range of strategies. And where, in this wide range, is the clue for getting (or avoiding, as the case may be) the first move of the second stage?

Mi Mimi
United States Fountain Valley California
Why is there no Word Games Forum or Subdomain?
There should be a Word Games Subdomain, or at least a Word Games Forum!

christianF wrote: Quote: .... Both placements are compulsory. When the player to move can no longer make the second placement, then his turn ends and his opponent may start the second phase. .... If you think about the implications, you'll find that when the protocol has reached its end, then * the number of black and white stones will be equal, and ... .... What if the player can't make the first move?

christian freeling
Netherlands

Phil Fleischmann wrote: christianF wrote: Quote: .... Both placements are compulsory. When the player to move can no longer make the second placement, then his turn ends and his opponent may start the second phase. .... If you think about the implications, you'll find that when the protocol has reached its end, then * the number of black and white stones will be equal, and ... .... What if the player can't make the first move? You mean "what if a player to move cannot make a move next to the last free placement of his opponent"?



You would need a board with a space with no neighbouring spaces

Mi Mimi
United States Fountain Valley California
Why is there no Word Games Forum or Subdomain?
There should be a Word Games Subdomain, or at least a Word Games Forum!

christianF wrote: Phil Fleischmann wrote: christianF wrote: Quote: .... Both placements are compulsory. When the player to move can no longer make the second placement, then his turn ends and his opponent may start the second phase. .... If you think about the implications, you'll find that when the protocol has reached its end, then * the number of black and white stones will be equal, and ... .... What if the player can't make the first move? You mean "what if a player to move cannot make a move next to the last free placement of his opponent"? D'oh! That'll teach me to reply before I've had any caffeine.



Phil Fleischmann wrote: D'oh! :blush: That'll teach me to reply before I've had any caffeine. So you think better without caffeine and take it so as not to think so well?

Richard Walter
United States San Jose California

I wrote: Turning these numbers into an equation of sorts is left as an exercise to interested readers... While thinking about how to make the program more efficient, I actually stumbled onto what I think the formula is for the maximum number of spots for Von Neumann neighbors.
For a NxM rectangle, assume N is smaller or equal to M (if not, turn the board 90degrees and N is now smaller or equal to M...)
if N is odd and M is even: Max = (N * M)  (N1) otherwise Max = (N * M)  N
These are based on a bricklike tiling pattern: +++++  B A  B A  B A  B A  ++++++++++  B A  B A  B A  B A  ++++++++++  B A  B A  B A  B A  +++++++++
That shows a 3x9 square board covered in dominoes that lie horizontally. Each A & B indicates one space of the board.
If the players use the algorithm: a) For a free placement, choose one of the leftmost A's that is available. b) For a sticky placement, choose the B space in the same domino whose A was just taken.
Then they can fill the entire board except for the holes in every other row along the left & right edges of the board. Those holes are either N or (N1) spaces depending on the even/odd sizes of the domino tiling, resulting in (N * M)  N or (N * M)  (N1) filled spaces.
You can also do the tiling vertically, but that will leave M or (M1) empty spaces, and since N is less than M, that will be more spaces.
Now, this doesn't *prove* that those are really the max values; it merely puts a lower bound on the actual max value. However, that formula works for all of the entries in my table for the Von Neumann neighbors, so I'm happy with it. I'm also pretty amazed that it's possible to fill up most of the board.
A rigorous proof is left as an exercise for interested readers...
Richard (Who's probably going to end up looking at the patterns for Von Neumann minimums next. *sigh* )

christian freeling
Netherlands

I really appreciate your efforts. I knew the protocol was interesting, but it's nice to see someone prove it!

Carlos Luna
Spain Rubí Catalunya

rw2e wrote: So I was bored this evening and threw together a quickndirty program to brute force all starting games on rectangular grid and keep track of the min & max number of pieces before the game ends.
Great job!
rw2e wrote: If there is interest, I might try to make it more efficient and/or let it run longer to squeeze out an extra row or two.
It may also be interesting to print out the board state at these extremes to see what patterns emerge that may help with determination of larger board sizes.
This will be very interesting, indeed. In fact, it is even more interesting than the concrete numbers for small boards since once we find the most efficient patterns for the small boards we can use them to "tile" larger boards.
rw2e wrote: It also appears that any value within the range is reachable. ie: on a Moore neighborhood board of 3x7, you can get every (even) value from 4 to 14. (I added this feature late and didn't retry every location, but the ones I did try all show this behavior.)
This is secondary for me, but finding an unreachable number in the [Min, Max] range will be interesting so keep this feature in your program.
Anyway, if you compute these numbers it will also be very interesting to take a look at the "distributions" or, at least, at the "mean number" of pieces placed in each board.
Following your supertight Von Neumann configuration I've tried to obtain some rough bounds for all 4 cases. I will express them using Big O notation:
Von Neumann neighbors (4 orthogonally adjacent spaces) in a NxM grid (with N<M):
MIN <= O(NxM / 3)
Follow the pattern below:
+++++      ... +++++++++  1 2  3 2  4 3  5 4  ... +++++++++      ... +++++++++      ... +++++++++  6 5  7 6  8 7  9 8  ... +++++++++      ... +++++++++
MAX >= O(NxM)
Follow Richard's instructions above:
+++  2 1  7 6  ... +++++  5 4  ... +++++  3 2  8 7  ... +++++  6 5  ... +++++  4 3  9 8  ... +++++
Moore neighbors (8 orthogonally or diagonally adjacent spaces) in a NxM grid (with N<M):
MIN <= O(NxM /7)
Tile the grid using these regions that occupy 14 cells and contain 2 pieces:
++++ + + + +   4  ... + + + ++ + + +  1   ... + + + + ++++  2   ... ++ + + + + + ++   2  ... + ++++ + + + +   3  ... + + + + ++ + + +  3   ... ++ + + + ++++  4   ... + ++ + + + + + +
MAX >= O(NxM/2)
Fill the grid row by row:
+++++++++  2 1 3 2  4 3  5 4  ... +++++++++      ... +++++++++  6 5  7 6  8 7  9 8  ... +++++++++      ... +++++++++
These bounds seem (asymptotically) tight* so we can deduce that:
With Von Neumann neighbors the OSOF protocol will fill between 1/3 and (almost) the whole of the board.
Whereas with Moore neighbors the OSOF protocol will fill between 1/7 and 1/2 of the board.
Graphically:
< Von Neumann > ##################### < Moore >
Which means that, with the Von Neumann neighborhood we obtain a wider spread of initial position densities (approximately twice as wide as the one obtained with the Moore neighborhood).
But so far we have studied only the extremal cases (which inform us about "what it is possible"). Now it will be interesting to obtain some results about the "average case". This can be done by computing the exact density distribution for small boards and then extrapolating the results or (preferably) by performing a series of MonteCarlo random simulations.
Are you willing to continue investigating, Richard? What programming language are you using (perhaps I could help you)?
* I encourage everybody to try to provide exact formulas (as Richard did) or, at least, try to tighten the bounds given above!

Carlos Luna
Spain Rubí Catalunya

CarlosLuna wrote: Moore neighbors (8 orthogonally or diagonally adjacent spaces) in a NxM grid (with N<M):
MIN <= O(NxM /7)
Tile the grid using these regions that occupy 14 cells and contain 2 pieces:
++++ + + + +   4  ... + + + ++ + + +  1   ... + + + + ++++  2   ... ++ + + + + + ++   2  ... + ++++ + + + +   3  ... + + + + ++ + + +  3   ... ++ + + + ++++  4   ... + ++ + + + + + +
Note that the 1/7 bound is tight for very large boards. For small boards the lower bound will probably be closer to 1/6 since these alternative regions (occupying 12 cells with 2 pieces) fit better in a rectangle:
+++++++    ... + + + + + + +  1  2  ... + + + + + + +  2  3  ... + + + + + + +    ... +++++++    ... + + + + + + +  3  4  ... + + + + + + +  4  5  ... + + + + + + +    ... +++++++


