 23 42 Blockchain bleibt Blockchain und Bitcoin bleibt Bitcoin und Bullshit bleibt Bullshit. Willkommen am 4. Tag. Vielen Dank an Atoll gerade für den Workshop und jetzt geht es weiter mit PENTA. Der wird das PENTA Game vorstellen. Kennt ihr vielleicht vom Camp. Da gab es einen eigenen Tisch oder vom letzten Kongress ein Taktikspiel, das mithilfe einer Kickstarter Kampagne finanziert wurde und PENTA erzählt ein bisschen über die Geschichte, über Spieltheorie, was euch sonst noch einfällt. Könnt ihr eben Chat fragen. Die Fragen geben wir nachher in der Question and Answer Session weiter. Also jetzt viel Spaß und strengt eure Grauen Zellen an für das PENTA Game. Danke, E-Punk. So, last, this is PENTA Game here and these are my slides. Perfect. Thank you very much for having me. Last time I held a similar presentation on Congress, which we had in Leipzig. And it was also scheduled in English. I actually asked, is there anybody here who doesn't speak English and there was nobody there. So I decided to stick with English even though most of the members of the audience actually were capable of understanding it in German. But my slides are in English, so beg your pardon. PENTA Game is actually, I'm going to talk about PENTA Game, which is, as E-Punk mentioned, a strategic board game, which has some really interesting properties. And it has been developed over a number of years. Last year at Congress we had prototypes. Since then we had a successful Kickstarter campaign. So we actually have PENTA Game copies now, which is really cool. We could not quite have a release this year, obviously, because of the pandemic. So game fairs and so fair flats. However, I'm going to talk a little bit about the genesis of the game. So this is how the game came into existence and the existence of the game. Actually also some words about a possible pre-existence, because there are some things that make me think that or believe, give us reason to believe that there was an ancient board game that must have worked pretty much along the same lines. So this is the fun part. And then comes the essence part, which is more the game theoretical point of view, because having developed and played PENTA Game many hundreds and hundreds of times, really, obviously we have a lot of questions. Like, does it always end? And how complex is it really? And does it play in the same league as other games? Things like that. So that's the more mathematical part. And I finished with some links at the end. So the genesis of PENTA Game, how did it come into being? That actually started a long time ago, some years ago, really. The first thing was about 1996. And I wondered, because I had these, as we all have as kids, these game collection things, and you see these classic board games. So this is just a picture that you find anywhere. And these games are kind of different to, yeah, author games or German style board games, which come out every year. In that, they don't really have a topic as such, because inventor games, typically author games, have topics like the caravan and the desert or the ansiatic league or stuff like that, while these classic games seem to be different in many respects. And the most obvious aspect about this is, obviously they are abstract and they also have some sort of geometric patterns. And you see this particular one, which Ludo, based on the ancient Indian game of Pachizi, is basically circular. You run in a circle. Then you have games like this one, Chinese Checkers or Halmar, which is basically triangular. And you obviously have quite a lot of games, which are played on square shapes. The most prominent in the West, at least, is chess, but Go is certainly a game of the same family. So these are basic fundamental geometric shapes. And if you look at it, basically, you find out that this is actually something like a family of mathematical structures, because they are, well, you know, they have this geometric patterns, regular N shaped patterns. And then you have games with them. So you have linear, circular, as mentioned. You have triangular games. And you have square games. And we even have hexagonal games, quite popular today, where we are playing on hexagonal shapes. So, when you look at this, so as I looked at that, I actually asked myself, what about this pentagonal shape? How about the number five? It's kind of suspiciously missing. And by that day, I didn't know of any game that was actually played on a pentagon, or pentagonal shape. So this was the quest. Is it possible that there could be a game, just as simple, just as fun, just as deep and entertaining as these other games, but played on a pentagram or pentagon shape? So, when you look at the pentagram, then, you know, what you do is, or whatever I have done, obviously, is draw a pentagram, a pentagram. And to draw a pentagram, as anyone knows who has ever endeavoured to try, is not that easy. So, as you see, this is a pentagram construction. And as you quickly find out, is the lengths in the pentagram. So, this length, and this length, and so on. So, the long and the short distances within the pentagram, they are what we call incommensurable. So, this is not a rational relationship. It's actually the golden section. You find the golden section between this short line and that long line, and also between this line and the entire line, and so on, all over this pentagram. And since this is an irrational relationship, that means it's not so easy to just make stops on this board that have the same size. So, obviously, drawing a pentagram, putting some stops on it and trying to come up with a game brings you to the point where you find out this is actually mathematically a bit challenging. So, since the lengths are, this is this golden section relationship, and no one size of stop actually fits. So, what you would do is, and okay, you say, maybe we have different size stops in the middle, on the corners and here. But with two sizes of stops, you still don't really get it, since, and that's what I tried out. So, basically, this is the prototype gallery. This is just what I found at home. I actually have quite a lot of prototypes. So, my first prototype, the first very first pentagram prototype is this one here, and you see basically what I've done is I just got colorful stickers and I stuck them on, and I figured out that the sizes here kind of, that doesn't really quite fit. So, then this one here actually looks already much better, and then all the, at the end, it looks like this. So, you see, there's been quite a lot of development in the middle. To solve this math, I don't really want to get too much into it. Just, this is the corner condition, because what you find out, you have an 18 degrees angle, obviously, in the pentagram, and you have these stops along the line. So, if these stops are supposed to just meet here in point C, that means that this corner circle has to have the ratio of a square root of five, if this one has the radius of one. So, this is the corner condition. You can fit that into the entire pentagram, golden section equation, together with the number of stops you want to have, and then you actually come to these values. So, the terms are actually quite complicated, and I must confess that I had some sleepless nights until I finally found them, and it was a bit of a Heureka moment, that also explains why probably there haven't been many pentagonal pentagram-shaped balled games before. So, let's talk about the existence, because this is what we've got now. We have this, you've seen some of the prototypes actually contained colored stops. This is because when we started the development, trying out different rule sets, we found out actually to rule dice and count numbers is really tiring. So, we basically had color dice, and you just roll dice a color, and you move to the stop of that color. And then later we found out that this is not even necessary, and the whole game also works perfectly fine, just without these colors, and that's the final design. So, what is pentagame now? How is it played? I mean, this is the geometry, and obviously the journey towards a rule set that would also work, was a very long one. So, it took really some decades, and let me just quickly explain, for this I've got this table here set up for you, and I wanna just, so if I can have that different picture, yeah, thank you. Here, you see the pentagram, the pentagame, pretty much, this is a large table that we use at Seabase, it's filtering out, this is actually green here, but yeah, thank you, it's a transparency. Now, what it really is, is it's quite simple to explain, and I'm just going into the basics. So, basically you play one shape, so I could be this ball here, and I have a blue ball that starts on blue, I have a red ball that starts on red, I have a white one on white, a yellow one and a green one. And what they really wanna do, what my aim is now, to get this yellow one here, the red one here, green one wants to travel to green, and the blue one could travel any which way it wants, to blue and the white, to white. Which per se is simple, but since we have, obviously you are not alone on the board, so you have another player, that typically has another shape, so this is important, you're not one color, you're one shape, and has the exact same objective. So, now the thing is, you can't just move to where you want to move, but you have to get rid of the blocks. And what you can do now, when it's your turn, and say I'm this glass ball and I want to start, so I can choose any of my pieces, like say this one, and I'm not allowed to jump over anything, so I can either just go here, or I can just go here, or something like that, but I can also just beat these ones, and then place them somewhere else, where they might block my opponent more than myself. Like say here, or maybe here. And the other move that's possible is, I can always swap to a jack in pieces. So here, this one cannot jump over that red one, so either I stop somewhere on the way, or if I beat myself here, then I swap position. So these are the two basic possible moves, and when it's free, when the corner is free, then I can also turn corners, like this blue piece that now reached the blue stop, replace that, move out. For moving out, you then get another block, a gray block, that is to have the number of pieces on the board vaguely the same. So now, positioning this is a major strategic aspect of the game, because you can see this blue one could also now go to blue, this yellow one could now go here to yellow, this red one could go to red. So I put this here, and hence making it possible for my opponent to get anyone out. So a fairly simple game, really good fun to play. And yeah, thank you. So that's basically all the rules in a nutshell. Can I have back to the slides? Yes, thanks. And where have I put my pointer? Yeah. Okay, so this is the game. And since its inception, it has really been played hundreds and hundreds and hundreds of times. We had tournaments, you know, lots of fun. And yeah, I know that some of you who may be now listening might already have the experience that this is actually a fun game. So because the ultimate test for a game, obviously is whether it actually is fun. So whether it's entertaining, whether it gives you the opportunity to talk to people, have interesting conversation and also whether it's, well, complex enough to entertain you, but not so hard that you think I have so much to think about it. Okay, so this actually really does work. That's the empirical test. So we have this Penta game, obviously the idea. We had lots of prototypes and some of you who have been on last year's Congress might have these backed prototypes. So I basically made a hundred of these and sold them and give them away and so on and made tournaments and all that. And then finally we said we ought to have a crowdfunding campaign to get this whole thing off the ground. And at this point, I would like to say thank you, particularly to everyone who has supported all these developments and particularly the crowdfunding campaign. The first five in particular, the first five were the five Donators who donated the most and they have this golden hats. Andreas Grübel, Christian Jans, Jean-Martin Neu-Gerhard-Suchenek, Nathan Tubes. Thank you very much. So this is what we got now. This is the Penta game that I showed earlier. This is the, well, these are three of them. We actually have it really made by LudoFuck. This is the company in Germany that also makes games like Settlers. And we also have these translated rules in 16, 15 different languages in Latin in Low German in Indonesian in Arabic and stuff like that, all typeset nicely with latex. So it's good fun and pretty much self-explanatory and lovely, lovely little pieces also. We got grease and hedgehogs and stuff. So this is what we have. And I'm really happy about this. The only thing is obviously we were meant to have a party, we were meant to have a release, we were meant to go to fairs and all this fell flat because it's damn 2020. So thank you for being here and we will actually have this some more later. This is pretty much what I wanted to say about the existence of this game. But now I'm talking about the pre-existence because obviously in the whole process of developing this, I had a look at the pentagram as such and the cultural history of the pentagram, the colors associated with the corners within the pentagram and so on. And there was quite a lot of stuff to develop, to discover. And I find it's quite fascinating because basically the pentagram in antique times was the symbol of the Pythagoreans. It's something that we know from Luke here on. We also know that these corners were associated with far or even five elements and who's into that, I really, really recommend this book by Fasaks from 1917, fantastic. So taking this together, there are actually some chances that there could have been something similar but there isn't, I mean at least in the archeological record, we do not find anything that has anything that looks like anything like it. But we have this quote from Julius Pollux, again, who himself then quotes Sophocles. And I save us to try to read this out but it's pretty much intelligible. And basically what it means is that there was a five line board game in antiquity where each player had five pieces and this was also not exactly the same as dice games. So, and this existed. There has been a lot of discussion about what this was throughout the centuries. You can go back really in the earliest game theoretical literature. People say, was it pentagonal? Was it five parallel lines and so on? Well, I mean, given the fact that the golden proportion and also geometry was really important for the ancient Greeks, Plato and Platonic solids springed to mind. And also the importance of playing, gameplay for this, for culture and for the origins of culture. Thank you, Anheusinger. And the fact that the symbol of the Perthagoreans was the pentagram makes you actually believe that this could well fit into what we see here. Now, we don't have any pentagram shaped objects from antiquity at all, which, well, probably has something to do with the fact that, as we know from Lucian, it was a Pythagorean symbol. And the Pythagoreans were later in history considered to be heretics and so on. Again, I don't want to dive into too much. You might know already, associations about pentagrams, pentagrons. People are pretty much sentimental about it still. I remember I was playing this game in a pub once and there were some people looking at us and they were like, what are you doing? And we were like, I will play this game, it's a pentagram. And they were like, oh, we don't like it because we are Neo Druids. And we were desacrating the holy pentagram by a game. So, people are a little bit sentimental and superstitious even about these things. And at this point, I just want to tell you, I'm really not superstitious. For me, this is a beautiful geometric shape and so far no demons have been coming at us for playing this instead. It's actually from its history of a Pythagorean system, symbol, more a symbol of advanced science and mathematics and stuff like that. For me and probably also for you. So, why was no such, and I must say, I had an interesting discussion with Ulrich Schädler about this, who wrote about the antique game Pasea, this pentagramma, this five line games. And now, we don't have an archaeological record. But this could be because it's never existed, but it could also be that they all just vanished, either because it's just been by material, they were drawn on chalk or by chance or on purpose because of the meaning of it. So, this is explainable and henceforth, it is possible that this is actually a recreation of a classic board game, which then was called Patea. And if you read Plato and there are many references to this game, I know it kind of fits. So, this would fit like a puzzle. I'm not saying it is this, but it's nevertheless a pretty beautiful thought where this is a kind of platonic game. So, that being said, how much time do I have? How long is this now? 30 minutes, okay, thank you. So, I'm trying not to rush too much through this because I know some of that is new for some of you. So now, we have this game and it works. It's fun and it's interesting, and we've been playing this and quite a lot of very bright minds have also been playing this excessively, really. And we figure out that we get better, but also we know that nobody actually knows how to play this perfectly. And this is quite surprising because normally you would think, you have games like NIM, they analyze it, then you know how it's actually to be played and it's solved and game over in a way. This has so far not happened with Pentagame. We're still learning, there are different openings and stuff like that, that happen, it's quite fun. So, my question was really, is this game actually really worth producing it or is it just a bad idea? So, when you think about that, again brings us back to the initial thought. Different classic board games have certain properties. You know, remember chess and Chinese checkers and all this. So, this is really a seminar article from Mr. Thompson, defining the abstract, talking about this and he basically was the first one to come up with these qualities. So, basically you say every proper board game has depth, which means it's actually interesting enough, it produces riddles, like you're posing a riddle and the other player has to solve it, find the best move, which then poses a riddle to you and it has enough depth to not be trivial. It also has to have clarity, clarity, easy rules. So, this is something we definitely have. Then we have to have drama, which means, well, I mean, a counter example would be monopoly. Who wins at the beginning wins at the end and that's clear pretty soon. So, it has to have, someone who's not an elite must basically have a chance to get into the lead so there must be surprises and action. And then obviously there's another necessary condition, which is decisiveness. And as game inventors, we know this and if you have ever tried to come up with a game, you find out it's pretty easy to not have decisiveness, so basically that means it always ends in a draw or something like that. So, the question is, it doesn't end at all. So, this is actually quite interesting because can we really prove that it always ends? So, this has also been discussed a little bit. Thompson himself also later added fourth one, which is elegance. Cameron Brown now talks about Shibui, which is like an Asian idea of aesthetics. Personally, I believe rather sublimeness, as opposed to rather just be beautiful. And also Narazio, which is something like drama, for me would be more important, that actually stories evolve on the board game, that pieces suddenly become important. So, you feel almost like, oh, this is my most important power and I love it so much. Things like that actually make games really good. Some of this we can prove and I just want to talk a little bit about depth because depth, this is obviously what you do when you start talking about it and looking at it. What are the different opening moves and how many options are there? Can I maybe learn these and find out which is the best one? So, I want to talk a little bit about the depth of the game. And there are different ways of measuring this depth. Obviously, you have state space complexity, average branching factor, game tree complexity, or even the complexity glass. So, here I've got a sketched out. So, basically, you can either ask yourself, how many possible positions are there? How many options do I have when it's my turn? What's the size of the game tree? So, how many different games are there actually possible in this hole? Is it just like always five different ways it can take? Or is it that we have billions? And finally, the hardness for an algorithm to solve this. Is it that we just don't know the best way? Or is it that we cannot know it? And those who have been observing AlphaGo and though they know what I'm talking about. So, I had to look at all these different things. Obviously, it's a little bit difficult to actually analyze this because you have to have some assumptions about what actually constitutes a possible move. Or a possible position. So, you have the legal positions, the legal moves. It's like anything you can do. But then some of them is plainly idiotic. So, you might not take that into account. So, you have reasonable moves, which is like, okay, that makes sense, and maybe that makes sense, and maybe, you know. But where do you draw the line is probably a matter of how well you play. And a subset of that, again, is the perfect move. In our game theory, we told us, there must be one perfect move, one perfect way of playing chess, and you would always win. The thing is, in chess and in Go, we know that this exists a priori. But we don't know it, so we can prove that we don't know this. So, what we are dealing here with this. So, trying to get these different numbers and figures for Penta Game obviously requires some assumptions about what is reasonable. And now, what is reasonable for Penta Game? Typically, you have two ways of opening moves. Either you go into the center and you eat one of those blocks that you put somewhere else, which you put on one of the 20 lines, actually. Or you do the swap move. So, you basically just swap two pieces here on the outer rim. This is the two reasonable opening moves. So, the guy who starts actually doesn't really have much of an advantage here, because he can either do this or that. It's a little bit like in Chessia, open games, closed games, E2, E4, D2, E4, like this. So, das is interesting, because at the beginning you think, oh, it's quite trivial. It's just symmetric, it doesn't really matter. But, this is the replace branch. So, I play a replace move first. I go into the center, I take a block, I put in one of the 20 lines. Now, my opponent can then maybe do a swap. He's got five different ways of doing this. And then maybe I've got four different ways of doing a swap, because I'm not going to do the same. So, multiplying this out, you already get like 400 different positions. And this applies also for the other branches here. So, you see that even in this branch of replace games, I play, you play, I play, we can have up to 800,000 different positions on the board. And these are actually different to one another. They are not the same, trust me. So, this is the same for the swap branch. Pretty similar, big figures. So, you can say, okay, how many options do I have? Because here I've got three, there I've got 100, there I've got 200. So, how big is it? So, we get to different estimations of the size of the game tree and the complexity of the game. So, the game tree is just like this. You basically get from one position to the next and to the other ones. Or you could have the state space, where it's actually possible, where each possible position is only just once. So, you can have game tree complexity or state space complexity, you know. This basically means that here there's no history and it doesn't matter how you got to this point. Well, here you see, you is twice. So, basically, that means you have some sort of a memory. So, the game tree complexity asks, how many different games are there? And this basically means, how many different reasonable positions exist. So, how does it go? So, basically state space, I basically tip this graphic here on its side and then it comes to something like that. So, basically seeing that we start from a very ordered position, then we do the swap or replace and swap or replace and different moves. And it branches out massively, really, really, really large. And then it boils down again. And at the end, you funnel into the actual winning or losing position. So, you got one winning and the other one. So, we don't really know how, what shape this has. But what we can know is, we can actually think about how many positions do we have here in the middle? So, that is this one. So, here you got the number of total positions, which is basically the idea. I just scatter pieces randomly on the board. How many positions do I get? Well, I mean, you know, if you can say, okay, for two players and for 100 stops, you get this number. So, that's quite an upper limit because there cannot be more than that. And that means basically, that's a number larger than the number of atoms in this my body or the stars in the entire visible universe. So, that's quite a lot of positions. And also, if you say, maybe half of this make no sense, but then you're still in the same order of magnitude. Or if you say, only a tenth of that makes sense, then you got 10 to the power of 25, which is, you know, a little bit less. So, this alone shows that it's quite complex. And these are basically all the nodes that are in here, from which you can then deduce other things like the average branching factor and so on. So, how does it actually come out? So, state-space complexity of games, logarithm 10, so you got different state-space complexity measures. And most of these are basically, just take them from the internet and it's not quite comparable, not that easy to compare, since you don't really know what method these people have been using. So, here I see basically, we are kind of in the area of Othello or something, but definitely more complex than nine-man's morals. But also, this is just the state-space complexity. Now, pentagame is a really short game. When you play a two-player game, it's about 20 moves. So, all of this is basically crammed into a very short time. So, in this one here, basically, what you got here on the bottom, this is really short. So, the average branching factor then will be much higher. So, you get basically doing this, you can again have different assumptions, I come to these values and you got lower bound, upper bound and something I call the best estimate. And that basically means, this basically is somehow the proof that this game plays along in the same area as do other classic board games that we all love and cherish. So, that was that. That was quite satisfying to see, because it fits into the observation that this is a fun game. And then, let me just allow one more minute about the decisiveness. So, does it always end? In practice, yes, it does. You basically say here, the thing is, when you design a game, as soon as you start losing and you see you don't have any more chance to win, you will try to force a draw. So, this is a big pitfall for anyone who designs a game. This draw thing, how do you avoid a draw? You want to have a cool game that is kind of open in the result until the end and then still allows somehow drama. But not a complete stalemate or the stalemate should be the absolute exception. So, looking at this, it's like, well, is it actually possible to block someone, to hinder someone from winning? I cannot play the strategy, I don't want to win, I just want the other guy not to win. So, and oops, oh, missing a slide here. Okay, so, sorry, I'm missing a slide, but I can just briefly verbally tell you that there is a proof, actually, because to be able, if you are able to block the other one and make it impossible for the other one to win, you yourself are in a winning position. So, this involves a little bit of geometry and I can possibly put that up on the table, but that is basically the proof. So, again, a result that I'm quite happy with, because it means, don't worry, it'll always end. You will not get stuck in an endless loop. It takes like 20 or maybe 30 moves, but then the game will end. So, that's good fun. Then one more thing, while we're at this kind of abstract stuff. Remember the starting point of this whole idea was that we are dealing with two-dimensional structures or these different board games. Now, as Go-Players and Chess-Players know, it makes a difference, whether you put your horse, your knight at the rim of the board or in the center of the board, because there is a rim and there is a center. Now, how is that in Pentagame? If you really look at it the way it looks like. Let me see, we've got it again, picture of Pentagame. This one. So, you see here in Pentagame, there isn't really, I mean, we start at the outside and we move to the inside. We could also start on the inside and move to the outside. There isn't really a rim, since if I want to get from this red point to that red point, I have to cross a long line, a short line and another short line. Or a short line here on the outside, which are also just three stops and a long line and a short line. So, it's always like two short and one long line, whichever path I choose. So, this means that there isn't really a rim and what you could possibly do is like, fold this up, you could take this middle bit here, if you want, and lift it up so that that ring kind of collapses like a cloth. And then what you then would get is a geometric shape that is called an antiprism. And the antiprism looks like this. So, here basically you see it, I would move from the small stop A to the big stop A, long path, short path, short path and I'm here. Or I could go any of the other ways. So, this is quite funny and interesting and obviously sitting together and discovering this, we thought, well, we must actually come up with something like a three-dimensional pentagram pentagame where we build this out of wire and can play it like three-dimensional. However, we have similar problems to what we had before, like what sizes of balls or spheres would then be the stops so that again in these points here, they are kind of just a jacket. So, this is a geometrical riddle for those of you who like to solve mathematical riddles. It's on the table, who is the first to solve this, be welcome, be my guest. Brings us to some other aspects. We got the games, we got the tournaments, we got a trophy that we compete for, if we are allowed to. All these things really got fun. We haven't yet got a proper game engine that people can log in. And I'm very thankful to Nikki and to Kobayd, who are working very hard to make this possible. However, it seems also there that there are surprisingly, surprisingly, surprising obstacles, really. Like it starts with, if you go to, got the original pentagram shape, how do you call the different stops here? So, it's not that easy. In chess, it's like E2, E4 in here. So, because this can rotate. So, you have to find somehow something that really, well, good news is we got that solved. But then also, if you have a user interface and you want to click somewhere, you got x and y coordinates. But how does that x and y coordinate transform? So, what's the actual coordinates of any of these stops in a Cartesian coordinates? So, obviously, you know, you think, oh, it's circular. So, you got to have kind of work a little bit with complex numbers. Again, it's solvable, but it's not that trivial. So, it's a bit of a price. And since we are here, I'm talking to a lot of people who are into this kind of stuff. We would welcome very much if you want to be involved with this and maybe help us come to a solution so that we can finally play this. So, I've got a number of links here. And maybe that's worth showing this slide a little bit longer. This is where you can get the Kickstarter copy. But there are also these GitHub repositories and stuff like that. And plus, there's also my email address. If you have questions regarding this, if you want to be involved, then please be my guest. So, and with this long story, really, I can only just say thank you very much for this. And I hope I see you all very soon back in the flesh when the pandemic is over for the next Pentagon Cup. And that was won by Anna Redlich, by the way. Happy birthday, Anna, also in 2020. And we will have an actual get-together, an actual tournament and the whole social aspect of playing this beautiful game again together in the coming year. Thank you.