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christianF
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Re: Essay by Christian Freeling on inventing games
« Reply #1125 on: Jan 15th, 2013, 7:03am »

Abdessamad Elkasimi, who's bot Dyfficult9 won the CodeCup 2013, has made some modifications to the program and challenged me to a double game base-15, penalty-6:Phewww! Two fairly tight and quick games in between household distractions. In both games I ended up with less raw territory but also with far less groups, to make up for it. More than make up for it Smiley .
Since there are at least three players probably better than I am, we can say that Symple does well in terms of a certain resistance to being programmed. I started out last summer, losing against a bot that now, based on its ranking, must be considered mediocre. Programs also may evolve slower than humans. Then again, maybe they're not. And in any case they're nice sparring partners.
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Re: Essay by Christian Freeling on inventing games
« Reply #1126 on: Jan 18th, 2013, 5:56am »

Early this week we hit minus 18 C here, during the night, minus 8 during the day. The raccoondogs don't mind, they've been sleeping 22/7 for weeks already. I don't mind either, the shoppingmal is a 5 minutes walk. I'll survive Smiley
 
Miraculously the northern part of the Netherlands remained snowfree, in particular the province of Friesland. When the frost came, they immediately shut down the pumping stations to protect the ice, resulting in a nice dark icefloor, not nearly thick enough at places, but the predictions are that the frost will stay for another ten days. That might just be enough to start a uniquely dutch event called
 
De Elfstedentocht

 
For some reason this event unites the Dutch in a warmhearted crazyness like no other event can. Even non skaters like yours truly can't escape: it's totally captivating and all things unrelated seem to come to a complete standstill. It makes great television too. I hope it will happen. For me its the only excuse for having freezing temperatures in the first place Tongue .
 
Edit Jan. 21:
Snow, and a lot of it over Friesland. Higher temperatures as of next weekend. Thaw will turn the snow on the lakes and canals to "fondant ice". End of story (for the time being).
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Re: Essay by Christian Freeling on inventing games
« Reply #1127 on: Jan 19th, 2013, 11:16am »


Jos Dekker - Christian Freeling (0-1)

 
This is the final position of a game between Jos Dekker, a dutch Go player living in Germany, who has 33 games under his belt, and me. It was also the first game in which I employed a tactics that Hyperpape had used against me: free internal placements.
 
Reasoning starts from the premiss that having more internal vacancies in one's largest group than the opponent has in his largest group, is good. If both must fill in in the endgame, the opponent will be full first, and so be the first who must invade, creating new groups and accumulating penalty points for it.
 
Then, still very generally speaking, and always within the context of a 'fairly close' game, it became apparent that not only the size of a vacant internal area matters, but also its shape.
 

 
This is a couple of moves earlier. The 3x5 area bottom left is easier to invade (white just did it) than the 16-points white area top left. Apart from this 16-points area, white has a 9-point area, a 3-point area and a 1-point one. There are 6 dame points on the G- and N-line.
Black has 14, 12, 12, 8, so each filling his own territory results in a black win. White must invade and the largest area is most fit. In fact it was left that open to invite the invasion (black had been taking the dame-points on the outside instead).
 
After C2 black was counting on his two 12-vacancies groups and on his free internal placements. There are two in the top group: O19 and Q19. Then there's A10 and A12, and either S1 or S2. Why are they crucial here? Because there are almost no dame points left, so if a player must grow, this growth will be internal. At some point White must either grow internally or accept penalty, because he has only A19 and I19 as free placements.
 
Black does not want to grow internally, so he decides to grab two points by cutting at S14, and leave the invasion for what it is, for the moment.
 
In the final position white has an 11-vacancies group against two 9-vacancies groups, and he managed a solid invasion bottom left, but he has no free internal placements left, while black still has 3 and so black can slow down his internal growth and avoid having to invade, despite white's higher number of vacancies.
 
The tentative conclusion is that the groups with the largest number of internal vacancies should shape these areas so as to make invasions difficult, that is a bit 'tentacled', because large groups are targets for invasions. Groups with less internal vacancies however, should shape themselves so as to leave a maximum (or at least 'enough') of possible free internal placements.
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Re: Essay by Christian Freeling on inventing games
« Reply #1128 on: Jan 24th, 2013, 1:13am »

on Jan 19th, 2013, 11:16am, christianF wrote:
The tentative conclusion is that the groups with the largest number of internal vacancies should shape these areas so as to make invasions difficult, that is a bit 'tentacled', because large groups are targets for invasions. Groups with less internal vacancies however, should shape themselves so as to leave a maximum (or at least 'enough') of possible free internal placements.
Note that this is, yes, largest groups, plural. Not only the largest group of each player, but also any groups that are likely to become largest after the largest has suffered some invasions. But to throw another wrench into the metaphorical machine, not only is it important to know which groups qualify for this, but it is also important to know the relative number of these groups between the two players. A player with two equally large groups has a clear advantage over a player with only one group of the same size: The opponent has to invade the two groups separately in order to reduce the number of eyes that the larger of the two groups will have. This makes invasions against the player with more, smaller groups fundamentally less effective than invasions against the player with one large group, all else being equal, and suggests a mutual damage strategy very different from the one I have seen from you when playing the side with one large group.
 
(I made a reference to this idea in March during the analysis of our second game- this lack of understanding that because you had the more numerous groups you had the obligation to invade me cost you four very clear points in the final stages of that endgame, and probably several more in the earlier phases as well. In my opinion you had winning positions in both our second and third endgames but failed to press your advantage correctly using this principle. Not that I can fault you for not understanding the principles of the game before you have been made aware of them. Smiley)
 


With all of the above said, I now make a bold claim that you may not agree with: The group penalty parameter is no more than another bug posing as a feature and not an essential component of the game. Occam's razor begs to be applied to Symple once more. Having played a handful of games with already medium to high penalties, I now feel confident in saying that deciding games entirely by the number of groups each player has would not break the balance between connection and territory, but it would simplify the rules and remove the need for counting so many stones at the end of the game if it is ever played on a physical board, though that is admittedly not the biggest of concerns nowadays. A bigger concern in my mind is what to do about the possibility for a draw under these rules, but if it must remain drawless, giving the win to the player who moved last when the two players have an equal number of groups seems natural. That player can be thought of as having an extra spare half tempo in the endgame where one's primary goal is to have as many extra tempi relative to one's opponent as possible when the position inevitably leads to zugzwang.
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Re: Essay by Christian Freeling on inventing games
« Reply #1129 on: Jan 24th, 2013, 5:27am »

on Jan 24th, 2013, 1:13am, clyring wrote:
Note that this is, yes, largest groups, plural. Not only the largest group of each player, but also any groups that are likely to become largest after the largest has suffered some invasions. But to throw another wrench into the metaphorical machine, not only is it important to know which groups qualify for this, but it is also important to know the relative number of these groups between the two players. A player with two equally large groups has a clear advantage over a player with only one group of the same size: The opponent has to invade the two groups separately in order to reduce the number of eyes that the larger of the two groups will have. This makes invasions against the player with more, smaller groups fundamentally less effective than invasions against the player with one large group, all else being equal, and suggests a mutual damage strategy very different from the one I have seen from you when playing the side with one large group.

Agreed, but I feel acquited by the fact that I'm a beginner, and not different from anyone else in this respect. Insights regarding strategy necessarily start out in a very low resolution, so refinements are only to be expected and I thank you for them.
 
on Jan 24th, 2013, 1:13am, clyring wrote:
(I made a reference to this idea in March during the analysis of our second game- this lack of understanding that because you had the more numerous groups you had the obligation to invade me cost you four very clear points in the final stages of that endgame, and probably several more in the earlier phases as well. In my opinion you had winning positions in both our second and third endgames but failed to press your advantage correctly using this principle. Not that I can fault you for not understanding the principles of the game before you have been made aware of them. Smiley)

My point exactly, and I'm improving, but not to the point of being the leading strategist or anything. Actually, as a player I lack some of the properties to excell at any game, including my own. My max rating in Havannah at LG was 2000+, but not that much "+", and there are several players with significantly higher rankings. Inventing abstracts and playing them are two very different diciplines.
 
on Jan 24th, 2013, 1:13am, clyring wrote:
With all of the above said, I now make a bold claim that you may not agree with: The group penalty parameter is no more than another bug posing as a feature and not an essential component of the game. Occam's razor begs to be applied to Symple once more. Having played a handful of games with already medium to high penalties, I now feel confident in saying that deciding games entirely by the number of groups each player has would not break the balance between connection and territory, but it would simplify the rules and remove the need for counting so many stones at the end of the game if it is ever played on a physical board, though that is admittedly not the biggest of concerns nowadays. A bigger concern in my mind is what to do about the possibility for a draw under these rules, but if it must remain drawless, giving the win to the player who moved last when the two players have an equal number of groups seems natural. That player can be thought of as having an extra spare half tempo in the endgame where one's primary goal is to have as many extra tempi relative to one's opponent as possible when the position inevitably leads to zugzwang.

Regarding this claim I admittedly do see the draw problem. As for its context, you're right that I may not agree, but I tend to disagree more cautiously nowadays Wink .
 
The thematic angle
I've always used an arbitrary and incomplete division of object themes. Two of them, checkmate and elimination, are existential in that the object is to kill/annihilate/eliminate head&heart and body&limb respectively. I'm no longer sure why these object should be considered 'higher' than those of coexistential games. It may simply have to do with Chess being considered 'the royal game' as opposed to anything not Chess, like the proletarian 'Draughts'. A cultural thing, most likely. I'm not sure if Go- and Shogi players in the East have cultivated a similar hierarchy in which one is considered 'superior' to the other.
 
A game's object has little to do with its depth, its clarity, or its appeal, so from a purely abstract point of view, stripped from any reference to a 'metaphor for life', the object is irrelevant. But humans are not purely abstract in their abstract thinking, and I for one have preferences. I've done my share of chess variants, but seldom play one. If one sucks at Chess, as I do, then trying Grand Chess is silly. In the last decade I've played an occasional game of Shakti. I do love Draughts, but I'm not even mediocre, and knowing how much I don't see in games like Dameo and Bushka is annoying. So I've been drifting away from the killing fields.  
 
In the Arena we feature both Mark's Oust, an elimination game, and Luis' Ayu, an approach game. In terms of simplicity, clarity and depth I consider these two of the highest category. Yet I prefer the latter, precisely because there's no capture and the object is constructive rather than destructive. It's friendlier and age has made me milder.
 
Territory versus connection
As a 'metaphor for life' one conquers territory and uses connections in the process, not vice versa. That's why I've always considered the relationship similar to the one between primary and secondaty colors. It makes 'territory' the slightly higher theme.
From a purely abstract point of view it is a dual relationship between equals. Despite this equality I'm more territory inclined, while Luis leans more towards connection. I don't think anyone has done more (or is doing more) in the 'square connection' department than he. I once added an ad hoc list in the Query rules and it keeps growing. Six by Luis so far. His insights have also been very helpful in that he initiated Symple's switch to compulsory placement, and the generic restriction rule in Scware.
 
Dynamic and static goal connection
This is an important distinction and rather than trying my hand at a definition, I'll give examples:
  • Hex, Havannah and Slither use static goal connection
  • LOA, Ayu, Xodd & Yodd, Ketchup and Symple use dynamic goal connection
Static goal connection usually makes for the sole object of a game, but Star was the first to quantify it using group penalty.
In LOA and Ayu 'dynamic connection' is the sole object, but it also loves thematic company:
  • Ketchup merges it with the 'largest group'
  • Xodd & Yodd merge it with the 'minimum number of groups'
  • Symple quantifies it and merges it with 'territory'
Simple Symple
As to your suggestion to alter Symple's goal, to make it 'simple Symple', I have no objections. Everyone is invited to alter my games (and I don't exclude the possibility of improvements) as long as due reference is made to the original game. But I consider the merger of 'dynamic goal connection' with 'territory' in its simplest form (the number of stones) to be preferred above a merger with a 'minimum number of groups' object. The latter lowers the resolution of the game and introduces a less than rare possibility of a draw. It might end 10-10 and something would have to be done about it. Your suggestion seems too much of a means to an end, and I also have some doubts regarding the "I now feel confident in saying that deciding games entirely by the number of groups each player has would not break the balance between connection and territory ...". Where's the 'territory' if only the existence of a group counts, regardless of size? What would keep a player from growing only one group? What am I missing here?
 
In Symple the equivalent of 10-10 would more likely be in the 100-100 region, the tenfold resolution, The closest one can get to a draw is something like this, an annoyingly funny and slightly ironic ending to a recent game against Abdessamad Elkasimi's bot:
 

 
The last move would bring my score to 100. Hurray! Wink
 
Changing the object the way you suggest still isn't entirely clear to me, but you've obviously put some serious thought into it so I hope you will elaborate. In any case it would take the penalty out of the equation, and therewith the, for lack of a better word, 'tension parameter'. Luis has argued towards an ideal value with less than ideal reasoning. Hyperpape has shown that things were slightly more complicated and came to a slightly higher value. In any case there's a concensus that shifting the value alters the character of the game. That is a rare property for a game, and off the top of my head I know only one other, Nick Bentley's Odd. So you're right, I don't agree where Symple is concerned. At the same time I'd be interested in more precise rules regarding your alternative.
 
Occam's Razor
on Jan 24th, 2013, 1:13am, clyring wrote:
The group penalty parameter is no more than another bug posing as a feature and not an essential component of the game. Occam's razor begs to be applied to Symple once more.

I'll try to explain why I think the last suggestion is wrong. I've argued that Symple is "quintessential" and there's no precise definition of what that is. Precise definitions are required in mathematics, but in everyday life we couldn't do without fuzzy concepts. Try defining a "chair". We use fuzzyness to communicate without making a fuzz over everything. But I can say plausible things about any particular chair, and I can try to explain why I consider Symple quintessential (like say Ayu, Oust, Hex and Emergo).
 
There have been four attempts that I know of, to quantify static goal connection using penalty points, Star, *Star, Superstar and YvY. I know of no game besides Symple that quantifies dynamic goal connection. In the static goal games, a group amounts to the number of 'special cells' it touches or occupies. The first thought that led to Symple was to take the groups size as its value, instead of how much cells it would touch. The immediacy of it seemed far more logical than the roundabout way of counting in the other games. Unhooked from the 'special cells' it implied both a simple territorial object and a shift to dynamic goal connection. So the territorial object was there, right from the start. The merger with dynamic goal connection sparked the move protocol, ensuring the dilemma between the speed of growth and the accumulation of penalty points.
 
So regarding Occam's Razor, it depends on what you aim for. You can design a giraffe, aiming at a high food source. Of course you can then use Occam's Razor to cut out the neck and argue that there's enough grass around to do without it. And you'd be right, but at the same time it's no longer a giraffe, is it? Wink
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Re: Essay by Christian Freeling on inventing games
« Reply #1130 on: Jan 25th, 2013, 7:15am »

on Jan 24th, 2013, 1:13am, clyring wrote:
... but it would simplify the rules and remove the need for counting so many stones at the end of the game if it is ever played on a physical board, though that is admittedly not the biggest of concerns nowadays.

It seems not too far fetched to suggest that abstract board games will become abstract tablet games in the not too distant future. You can have animated tutorials and you can play anywhere with anybody, chat, store games, include a bot, play 3D (generally speaking, it would imply the possibility to go 3D with a whole load of concepts and mechanics) and, barring the tablet, never lose stuff. I think it opens a huge new realm for inventors. Symple is just one example. You don't play the game by counting all the time. As in Go, there are moments to consider the total count, but usually you go for the best move in local situations without counting. Unless you like things to be difficult, it's convenient to be able to check the score at a glance. Not just in Symple.
 
It doesn't stop there. Bashni is still manageable if you use Backgammon type pieces, but Stapeldammen isn't, unless you're a fanatic. Now imagine you'd want to make a game of towers that would raise far higher, for instance with the object to have the tallest tower at the end of the game. Imagine there some natural feature in the mechanism that, given a certain boardsize, would eventually terminate growth at some point, say between 40 and 60 pieces high. That could be an interesting game, and nothing would prevent it from being played on a tablet, while in the 'real world' it would be impossible. Come to think of it, any argument that requires abstract games to be playable in the real world is a bit strange, since abstract games aren't the real world in the first place. They're virtual, and the only handicap till now has been that this virtuality had to be implemented in wood.
 
To make an even more compelling argument: you could play Symple-3D in say a base-5 or base-7 cube. Same rules.  
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Re: Essay by Christian Freeling on inventing games
« Reply #1131 on: Jan 27th, 2013, 11:25am »

As a domestic point, we've upgraded Chad to the ArenA (the applet will be updated soon *) and downgraded Symple Hex and Mu_levis to the Pit. Mu_velox is now completely separated from its tamed relative. Rotary has also appeared in the ArenA, but Ed is still working on the applet. The rotational pieces have sort of kept it in the pipeline for a long time, but it had to be done sometime and this seems as good a moment as any.
 
Also a base-13 and a base-15 board will be added to the Ayu applet shortly. *
 
* Done
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Re: Essay by Christian Freeling on inventing games
« Reply #1132 on: Feb 4th, 2013, 10:14am »


 
For what it's worth, here's an idea for an initial position for HexAyu.
 
Here's Ayu itself.
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Re: Essay by Christian Freeling on inventing games
« Reply #1133 on: Feb 4th, 2013, 12:20pm »

on Feb 4th, 2013, 10:14am, christianF wrote:

 
For what it's worth, here's an idea for an initial position for HexAyu.
 
Here's Ayu itself.

 
Interesting. I had suggested this one instead:
 

 
Quite conveniently, both of them have (roughly) as many stones as empty points, just like the square version. Your suggestion is more symmetrical and maybe just better. I considered it as well, but I was worried that occupying the central point would be too powerful. Now I'm not so sure.
 
EDIT: Yours has the slight disadvantage that boards with an odd number of cells per side can't be used (the set-up is not symmetrical for both players on those boards), but that's not a problem.
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Re: Essay by Christian Freeling on inventing games
« Reply #1134 on: Feb 4th, 2013, 12:58pm »

on Feb 4th, 2013, 12:20pm, Luis Bolaños wrote:
Quite conveniently, both of them have (roughly) as many stones as empty points, just like the square version. Your suggestion is more symmetrical and maybe just better. I considered it as well, but I was worried that occupying the central point would be too powerful. Now I'm not so sure.

Neither am I, it's difficult enough to understand tactics and strategy on a square board. I can't say anything about the hexboard, other than that it is a similar game and that it will raise similar problems. On the face of it, considering object and mechanism, the centercell would not appear to have any special significance.
 
on Feb 4th, 2013, 12:20pm, Luis Bolaños wrote:
EDIT: Yours has the slight disadvantage that boards with an odd number of cells per side can't be used (the set-up is not symmetrical for both players on those boards), but that's not a problem.

Going from the premiss that 13x13 will become the 'main variant' I figured that a 169 cells hexboard would be completely in line. But yes, the adjacent alternatives are farther away on a hexboard: base-6 much smaller, base-10 much larger.
On the other hand, you need odd-based boards for the square game, in the suggested setup it's
  • square - 121-169-225
  • hex - 91-169-271
I'm working on the applet graphics. Appletwise, Rotary comes first, but some time in the not too distant future ... Smiley
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Re: Essay by Christian Freeling on inventing games
« Reply #1135 on: Feb 8th, 2013, 6:17am »

Multiplicity is a concept of a game, or maybe a game already. It follows a discussion at rga around "multiplication of sizes of groups" as an object.  
 
Nick Bentley's Produto and Luis Bolaños Mures' Alpha were the main actors, but I also include Néstor Romeral Andrés' Omega, because it has the same scoring criterion as Multiplicity: The product of the sizes of all of a player's groups.
These three games all have move protocols that allow placing stones of the opposing color. I can see the implications of that in this particular theme, as well as in Luis' Yodd and Xodd, but I don't feel much affinity. So I thought, why not introduce the Symple move protocol? Moreover, the product theme itself profits tremendously from having the score tracked and displayed by the applet. Nestor reassures us at BGG, regarding Omega:
 
Quote:
"You will soon realize that you don´t need to calculate your score during play. Use an intuitive strategy instead. How? You must figure it out by yourself."
Yeah, right. "We know it's there, now you go and find it!" Grin
 
But to be fair, I know what he's getting at. You don't count all that much in Symple either, or in Go for that matter. You often have a fair idea of the priorities in any given position. But believe me, it helps having the score displayed at all times. And in the evolution of board games to tablet games, it's a logical step.
 
A concept
It may be a game already, because it should work ... but is it fun? In some circles this criterion has fallen into disrepute, but I still embrace it. It also doen't have a tie-breaker, because I cannot as yet evaluate whether it needs one. There are large enough scores in the pipeline here, to make equal scores a less than frequent occurence, possibly, and for the record: I'm not a member of the Church, so I don't object to draws on principle Wink .
 
The provisional board is a base-5 hexhex:

 
Say you conquer half the board, some 30 cells, how large a score is possible? To get an idea, lets divide 30 evenly, or as evenly as possible:
 
group divisionscore
30x11
15x232.768
10x359.049
7x4 + 232.768
6x515.625
5x67.776
4x7 + 24.802
3x101.000
2x15225
1x3030

Clearly having many 3-groups doesn't hurt. The extremes score extremely low. So here's a dilemma: you need growth, but connections are usually bad. You need singles to grow, but isolated singles are useless. Compulsory placement eventually may force players to make undesirable connections, or force them to place isolated singles, giving a cold ending to a basically hot game.
 
Turn order balance
Without the balancing rule, white could take the growth initiative at any moment, while black can not afford to start growing because he then would have a group less than white, so white could immidiately double the score. The balancing rule makes that either can trade the growth initiative for one or more "2-groups" already present, as a compensation for going second into the actual growing phase. Timing is important, but as to the moment to actually trade it. I'm not sure yet.
 
Timing of growth
Nor am I sure about the timing of growth after the trade: when do I start the actual growing phase? It's nice to have ten groups and grow them all twice, but say your opponent starts growing at eight groups while you have also eight. Can you afford to place another single? Compare eight groups of 3 [6561] to nine groups of 2 [512], two moves later. Taking a compensation into account, say two 2-groups for the second player, it would be seven groups of 2 and two groups of 3 [768]. So I'd say you can't, and a mutual obligation to keep growing may start early, like at five, six or seven groups each.  
 
As I said, it's a concept, maybe a game.
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Re: Essay by Christian Freeling on inventing games
« Reply #1136 on: Feb 9th, 2013, 12:03pm »

The graphics of the Ayu applet have been updated to match those of the diagrams, and there have been minor updates in the Yodd and Xodd applets (larger stones).
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Re: Essay by Christian Freeling on inventing games
« Reply #1137 on: Feb 10th, 2013, 6:11am »


 
Kobus had caught a severe cold after the power outage, breathing like a dentist drill, while shedding at the same time. He had not eaten for about two months, all in all, but now he seems fully recovered. It's a big rabbit too, because I had it for about the same period, and during that time it could eat as much fresh lettuce and carrots as it wanted. Kobus needs those vitamins, especially in winter.
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Re: Essay by Christian Freeling on inventing games
« Reply #1138 on: Feb 10th, 2013, 9:00am »

"One sticking, one free" - a generic opening protocol with an exceptional property
I assembled Triccs mainly to park the opening protocol somewhere, lest I should forget. Not that I forget easily. Smoking marihuana on a daily basis not only improves one's memory, but also ... Huh no, wait, it'll come to me ...
 
Anyway, the opening protocol was a near miss. Here's a very similar one called "one sticking, one free", which is shorter and has a very exceptional property. We've already adapted Triccs to it. Here it is:
 
White starts by placing one stone on the empty board. From that point on players take turns to:
  • Place a stone on a point 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.

 
Here's how the board might look at that point.

 
Note that the number of stones on the board will be equal, but note also that whoever started the first phase, has no bearing on whose turn it will be at the start of the second phase. Depending on the 'compactness' of the players' placements, either may end up being the one to start the second phase. That's quite an unusual property and it is also the reason that games using this protocol won't have much use for a pie.
 
The opening protocol is generic in that it results in an evenly spreaded position with an equal number of black and white stones covering about a third (hex) to half a board. Any game could start from here, including games using a normal move protocol (1-1-1 ...), and Triccs is certainly not the best game to serve it. Looking at the board I imagine you could start Ayu this way, but unfortunately for me that's not an option. So it's stuck with Triccs for the time being. Eventually something may turn up to that is more deserving of it.
« Last Edit: Feb 10th, 2013, 11:15am by christianF » IP Logged
Luis Bolaños
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Re: Essay by Christian Freeling on inventing games
« Reply #1139 on: Feb 10th, 2013, 12:49pm »

It's a very interesting protocol and definitely an improvement on the one you had originally used in Triccs. I agree that it would make for a nice Ayu variant as well.
 
The protocol even works as a combinatorial game all by itself, but obviously an opaque and boring one.  Smiley
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