S2E03 - Turing Machine (Computation)

#Computation #TuringMachine #AlanTuring #Logic #DeductionGames #BoardGames #Science #Math #STEM

Summary

Today we cover Turing Machine, a pure logic and deduction game where you use punchcards to identify the hidden code. We're joined by the inestimable Stephen Granade, grand high guru of the DragonCon Science Track, to help us understand who Alan Turing was, what a computer is, and how its logic works, plus cool facts about lasers and stuff.

Timestamps

0:00 - Introduction
2:11 - Laser cooling, plate tectonics, and DNA data storage
9:39 - Turing Machine game overview
18:37 - The magic behind the punchcards
23:15 - Who was Alan Turing and his machine?
32:54 - Data storage and punchcards
40:59 - Boolean math & quantum computing
46:06 - Nitpicks and final grades
54:38 - Final thoughts

Other stuff

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This episode of Gaming with Science™ was produced with the help of the University of Georgia and is distributed under a Creative Commons Attribution-Noncommercial (CC BY-NC 4.0) license.

Full Transcript

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Stephen 0:00
Music.

Brian 0:06
Hello and welcome to the gaming with science podcast where we talk about the science behind some of your favorite games.

Jason 0:11
Today, we'll be talking about Turing Machine by Scorpion Masque. All right. Welcome back. Everyone to gaming with science. This is Jason. This is Brian. And today we have very special guest Stephen Granade. Stephen we know from science track at Dragon Con, which I think people have heard us talk about before. He is the lead guru and ring master of the science track, and manages to keep all the things running and fight for our space and make sure we have the resources we need. So we are very grateful to him.

Brian 0:41
The Grand Poobah.

Stephen 0:42
You make me sound so organized.

Jason 0:45
All you need is the illusion of organization, and you're fine.

Stephen 0:49
That's right,

Jason 0:49
anyway. But if you can kind of introduce yourself to the guest, what's your background? What's your what's your story?

Stephen 0:54
So my background is that I spent my undergraduate years at a small liberal arts college as a member of the major of the Month Club. But as my chemistry professor said, I never dropped any of the majors. So after cramming four years into five, I had a Bachelor of Science, dual major, physics, chemistry, and then a bachelor of arts, theater arts, with a math minor. And I looked around at the options there and decided, you know, where the real business is. That's physics. Why I went to graduate school in physics. I studied atomic cooling and trapping, where we would use lasers to cool down atoms to ultra cold temperatures to the point where they started to act in concert. And you would get a basically a quantum super fluid, if you've ever heard of like liquid helium, where you cool it down enough that it doesn't have friction or things like that. We were doing that, but with dilute gasses of atoms. So also, again, just a great career decision. Lots of people wanting to cool atoms down a lot, but fortunately, it also involved lasers and optics. So I moved into working for companies doing sensors and image processing, which of course, turned into machine learning. So I just have this mishmash of different experiences and background

Brian 2:11
that is remarkable. I have tried to explain, tried to explain laser cooling to my oldest son, to no avail.

Stephen 2:20
Oh, yeah, I bet.

Brian 2:21
But can you explain? How do you cool something with a laser beam?

Speaker 1 2:26
The most straightforward way is a method called evaporative cooling that works like it sounds. It's sort of like what tends to happen if you've got a hot cup of coffee where the liquid in there is really, really hot, and so the water molecules bounce around. And then occasionally, they bounce around in such a way that one of them gets more energy and the other gets less energy, like the overall total is conserved, but one of them gets enough energy that it can turn into vapor and escape, leaving behind cooler atoms. So we would start with just as many atoms as we could pile into a trap that was formed out of the electromagnetic potential that you could create with atomic laser beams. And then you would it. So if you could imagine the atoms are all like rolling around in this laser beam, like trap that is sort of like marbles in a bowl, and then you keep lowering the edges of the bowl, and what that ends up doing is it lets the hottest atoms, the ones with the most energy after a collision, escape, and the rest of them get colder. And so then you lower the bowl a little more, more atoms escape with more energy. The ones that are left get colder. And you just sort of keep doing that and reducing it in temperature until you're as close to absolute zero as you can get.

Brian 3:37
You're almost like filtering out the slow atoms.

Stephen 3:41
Yeah, you're you're making it easier for the hottest atoms after they collide to escape, carrying off more than their fair share of energy, so that the remaining atoms drop in energy and are colder and are moving less fast.

Brian 3:54
Wow, I study onions.

Jason 4:00
Hey, there is some fascinating stuff going on with those onions and all the chemistry there. You told me about this. Anyway, we usually start with some sort of fun science fact. And Steven, as you are our guest, we give you priority. Do you have something fun you want to share?

Speaker 1 4:12
Yes, I love collecting science facts. I think that's part of what has ended up happening with me being involved in the Dragon Con science track, because I get exposed to all of these other scientific disciplines that I am kind of a dumb, dumb about. My favorite current fact is that your fingernails grow at about the same rate that plate tectonics move.

Brian 4:34
Yeah, I've heard that before. That's great. It gives such a fantastic visual representation of what's happening, right?

Stephen 4:40
Well, it's such it feels like so very different scales, because you've got the, you know, inch, like fingernails on your fingers, and then you've got these giant plates, and they're both moving and growing at roughly the same kinds of speeds. And that's just wild to me to think about.

Jason 4:55
and yet, when I think of it like that, I think, wow, those plates are moving really fast. Because, like, I've seen this. In his book of world records, of those people that never cut their fingernails. And that's like, feet of fingernails over a lifespan. So that's, that's more than I would have expected.

Brian 5:09
Jason has fast growing fingernails confirmed?

Jason 5:12
I no, I have the nervous habit of picking em. I've never let them get that long. Like, if they start getting over, like, two millimeters, like, Oh no, they're bothering me. I must get rid of them.

Brian 5:23
Should I do my fact too? I mean, absolutely. So I have to turn everything I have a hammer of bacteria and genetics, so I have to turn everything into a nail. So what I found is it's a it's a pretty old story, but it's actually, it's not such an old idea at this point. A lot of this work was pioneered by a synthetic biologist and DNA guru, George Church, of using DNA to store digital information. So I thought that was appropriate for this. So DNA is incredibly dense in terms of the amount of information that you can store in a really small space, and it's also very stable, like you can under the proper conditions have DNA retain that information for 1000s of years, which we don't really have good storage media that could actually hold up to those long term archival states.

Jason 6:07
Yes, although, as we learned in the last episode, a chunk of Amber is not an appropriate archival state for DNA.

Brian 6:13
No, it is not. No, it is not. In fact, they're even putting some archival information into living cells, so then you get the repair and replication mechanisms. So they're sort of like dropping little Rosetta libraries of human information into living organisms, kind of very like very sci fi. So it's great for archival. You can keep it for a long time. The problem is it's absolutely ridiculously expensive to write to DNA, because you are synthesizing new DNA molecules. You're storing digital information, you know, binary in the A, G, C's and T's, it's actually not too bad to get it back out again, because you can just sequence the information and turn it back into digital information. So that's not cheap, but it's a lot easier than synthesizing the DNA.

Stephen 6:54
How do they put that together? What does the synthesis step look like?

Brian 6:58
Oh, you can actually run through a process through just raw from raw chemical bases, synthesize a specific DNA sequence, like from scratch, stick it in a test tube. They were here. They're always working on, like, new ways to do this.

Jason 7:11
Yeah, I think currently the limits probably about 100 to 150 base pairs is all you can get in one but then there are tricks to stitch them together so you can, like, yeast is apparently very good. If the ends are are match up to a relatively small out, like 20, 30 base pairs, the yeast will take a look like, Oh, those belong together, and they will just glue them together at that spot. And I think that's how they made there's a lot of stuff about making a fully synthetic genome about 10 years ago, which they did through this process of like starting from these tiny, little building blocks and then slowly building it up through many, many, many, many cycles of this recombination. It's called to glue them together, and presumably many, many, many, many hours of technician and graduate student time actually making this work and figuring out how to work, and figuring out all the ways it could go wrong, because they had to do things like the other problem with DNA is it's fragile. It's literally at one atom thick, and so any shear forces can just rip it apart. And so the pipettes we normally use sheer DNA down to about 10,000 base pairs, which is way too small for anything actually living. So they had to do all sorts of crazy workarounds to be very gentle with the DNA as they were handling it, so it wouldn't just get ripped to shreds by them trying to move it around.

Stephen 8:25
Well, that's wild.

Brian 8:27
That artificial synthetic genome also has watermarks in it. They wrote their names and a poem and a couple things. They left an HTML link in the DNA of this organism. So it's really weird.

Stephen 8:39
That is amazing. Biology is weird and squishes.

Jason 8:46
That's why I love biology, is it's it's phenomenally complex, and you get all sorts of very interesting emergent properties out of it. I mean, when you get down to it, most of us are what hydrogen, carbon, oxygen and nitrogen, and they're just rearranging certain things, and you get phenomenally complex things like podcasters and plants and stuff. So

Brian 9:07
yes, biology is squishy and stinky and fun.

Stephen 9:11
Well, thinking back to my orgo days, I'm like, All right, so what's the synthesis steps that gets you from pile of chemicals to podcaster

Brian 9:19
a couple billion years, yeah,

Jason 9:21
so we had a previous episode in evolution, so we'll just link that in the show notes. Nice. Yeah, give it about 4 billion years and a whole bunch of trial and error, mostly error and right, what works will be around still, all right, but we're not talking about biology. Today. We are talking about computation. So let's dive into this game. So today we are talking about Turing Machine by Scorpion Masque, which was designed by Fabien Gridel and Yoann Levet. I hope I'm pronouncing those right. This is a very interesting game. Brian and I played this. This is outside our normal area. It is a deduction game. So the most famous deduction game that you're probably familiar with is Clue, the. Idea being that the point of the game is to uncover some particular piece of information and clue. It's like, okay, who did the murder, where and with what? And the designers of Turing Machine really liked the duction games, but they saw there was one major flaw, is that usually you are asking questions of the other players, so the players all have parts of the information, and you're trying to assemble the whole but usually some information is more valuable than other information, and so some players might get an advantage just by the cards that are dealt, or whatever include this would be, I'd say probably the rooms have the most value because they're the hardest to get to, and there's the most of them. So if you happen to get a handful of rooms, you have an advantage over everyone else. So I'll put in the show notes a link to the designer diary. It's actually really fascinating all the iterations they went through over this, but they eventually hit on the idea of, well, we won't have them ask other players. We'll have them ask the game itself. And that's how we have Turing machine. So the way this is set up is there's a central piece which is actually unnecessary. It's just a cool little visual piece.

Brian 10:58
We need a word for that, for things that are totally unnecessary for the function of the game, but just make the game look a little cooler. Oh, yeah. Like fun, clicky element to it or something. I still haven't come up with it yet. What did we come up with? Jason?

Jason 11:10
I think boondoggle was the closest we got.

Brian 11:13
Yeah,

Jason 11:15
goober, if we want to take from spider verse,

Stephen 11:18
yeah,

Jason 11:19
widget, that's probably too useful. It's too I mean, a spandrel is actually kind of useful. It does something. So it's not just that.

Brian 11:26
Well, listeners get in the discord. What term is this? We need to we are missing a term, and that's not allowed.

Stephen 11:33
You know, I would have originally said those kinds of things were absolutely optional. I spent, you know, a chunk of time playing cheap ass games where you got basically the rules and some cardboard, like, if there was going to be a board, you'd get sort of thin card stock with it on there. And then they're like, you have dice, you have tokens, you have, you know, chits to count up things. Just pull those from monopoly or whatever other sad game that you should put to one side to play cheap ass games. And I like that. I was in grad school. I had no money, but I have gotten to really like the ones where it's like, we spent a little extra money, and now the meeples have a little bit of heft. You're like, Oh, that's really nice. Actually, something nice and tactile about those things.

Brian 12:14
Yeah, I'm a sucker for some glossy card stock. I know that Jason is as well.

Jason 12:19
Yeah, no, I like the tactile sensation. And so what this central thing? It's just a hexagon with some little like eight bit computer faces on it. It's just to organize things, because the real meat of it has there are six faces and around that you put anywhere from four to six cards, which are your condition cards. The goal of this game is to deduce a number that is hidden. And the thing is, the rulebook comes with about 20 of these. There's a website where they have, like, 7 million of them that they've programmatically generated. So like, you're never going to run out of puzzles to solve, but you have to get it because it is this very specific setup. So you set it up, you put out your four to six condition cards that have the conditions you're testing. I'll talk more about that in a minute. And then now each one of those is tied to a verification card, which tells you information. And the way this game works is you have a set of number cards, and so you put together a three digit number. The digits only go one to five. So there's five times five times five equals 125, possible numbers it can be and you take the three and there are tabs, whether it's the first digit, the second or the third. You put them together, and they have these punches in each of them, and it looks like an old fashioned computer punch card, and they designed it so that when you put your three digit code together, there is always exactly one hole open out of all of them, and every 125 different ones has a different hole open. And you hold this up to the verification card, and you get a check or an X. And that basically tells you that with its condition card, whether your number passed the condition or not, and you use that information to try to deduce what the actual real number is, because the real number passes all of the conditions out on the table. The one further wrinkle of this is that the condition cards do not have a single condition on them. They actually have multiple conditions, anywhere from two up to nine, and only one of those is actually being tested. So your job is first to figure out which of the many conditions on the board are actually in use, and then to determine from those which single number satisfies all of the conditions. And the authors are very clear in the rule book that in every puzzle, every single condition is needed. You can never just skip one and come to the answer, except by luck, like you need all of them in order to get it, which leads to some weird things where there are some conditions that can never be used in the game, because, like, one of the cards says the number of threes. There can be one three or there can be two threes, or there can be three threes. If there are three threes, that one card tells you the solution, therefore that condition will never be used. At least weird stuff like that. And the best way of thinking about this, this can be played one player or multiplayer. It's one to four players. This is one where I think maybe the one player version might actually be the most popular, because it's it's competitive puzzle solving. Think of it as if you were doing competitive Sudoku with a bunch of your friends. It's very much a solitaire. There's no way to interfere with each other whatsoever. Your little clues and stuff are hidden. You can play it cooperatively if you want. That's definitely an option. Or you can play it solo. And I did that a few times, and I could definitely see we say this a lot, actually, several of these games would be good apps. This would be a great app to just pull open and do with, like Wordle or connections or turning machine in the morning, just to kind of stretch your brain cells a little bit. I can see a lot of people liking it for that reason.

Brian 15:24
But wouldn't you lose something without the little cards getting to pick them up and make the grid and hold them up to the thing and the sort of weird mathematical magic of, how does this work?

Jason 15:34
Well yes, you would lose the tactile sensation of it. Yes. On the other hand, you wouldn't have to have Turing machine set up for you to play every single morning, which is difficult with cats and children

Stephen 15:44
So just the dedicated Turing Machine table off on one corner.

Jason 15:49
Yes, and there's an interesting history. So right now, the final version of the game is pure numbers. It's just you have the three digit number you're trying to solve. They actually tried skinning it a few different ways during development. So first, and for a long time, you were trying to find out, like the number of different animals on the farm, and each of the farm hands only knew one thing, like, there are more pigs than chickens, or if you add up all the animals, and they are greater than six, which seemed like some pretty uninformed farm hands, but it was the metaphor of the game. Then to try to tie it to Turing Alan Turing, which we'll talk about more in a bit, they themed it as being trying to discover components of the German army, like, how many tanks did they have? How many infantry did they have, et cetera. And at the end, they decided that, you know, we're just gonna stick to pure numbers. We're just gonna make it a pure number deduction game, which I think is, I mean, I could see the skin being nice, but I really think what they were going for, they really just like the pure logical deduction. And I think keeping it pure like that really reinforces the feel and the vibe of the game they're going after.

Stephen 16:46
Yeah, I feel like, if this, where games where you have that skin, where it really you see sort of the abstract working of the mechanisms underneath the different engines, or things like that, then I want the skin to still be thematically resonant with the game. You know, if I think about something like wingspan, which at its heart is just, you're producing more resources that you put out in the form of, you know, eggs and things like that, and you're laying out the places for the birds and all of that. So you're getting your birds. The theme informs what those things are in a way that feels like actual birds and their habitats. The skinned versions you're talking about sounds more like, well, this is the thing that you're going to end up mostly ignoring and treating as abstract anyway.

Jason 17:28
I agree. I think that's why it works better as just pure numbers. Because ultimately, even though it bills itself as like the punch card computer game, it's really a game about logical deduction, and it's not actually a computer. The rule book actually says it's a proto computer, but I guess whoever was in charge of the box wanted to say it was a punch card computer game,

Stephen 17:47
right

Jason 17:47
And it does feel a little magic to put all the punch cards together. There's always that one hole that's open and you get to look through and get your answer. And Brian, I've known this for several weeks, and I've hold off playing like I figured out the magic or how that happens, the mathemagic, the mathematic, they describe it very clearly in their design diary, and it turns out to be really, really simple.

Brian 18:08
OK! I'm excited to hear about it. But you, you made a point when we started this endeavor. We can't just do biology games. We have to mix it up. You know, in the whole idea of STEM science, technology, engineering and math. This is our first this is a Math game. That is what this is, right? I don't think there's a lot of those out there. I don't think so well, where they just completely go for it. It's like, it's fully abstracted, just numbers. We don't need any theming,

Stephen 18:34
right

Brian 18:34
Okay, now explain to me how this works, like, because I want to know.

Jason 18:37
So when I first saw it, I was like, Oh, I thought this would be like, Spot it, or Dobble which I saw a video on a year ago. I'll link it in the notes. In spot it, you have this collection of cards, and every pair of cards shares exactly one symbol. And there's this whole mathematical thing to ensure that happens. I thought it'd be something like this. So every trio of cards would share exactly one hole. And then it explains like, oh, you take all 125 numbers, you put them in a grid, and then you randomize them, and then you just take whichever card. So this is the first digit, and it's a one. So all the ones that have a one in the first digit have a hole in them, and all the ones that have a two in the second digit, when you go into that one, you put a hole in them there.

Stephen 18:38
It's decimalencoded. It's a decimal place encoded basically, oh, that's super cool.

Jason 19:16
The first versions, they actually had them in long strips because they were all adjacent to each other. They later realized it was actually much more fun when they randomized them. And so that's what we have, the final one. So there are basically 125 spots on the card where there could be a hole, anything that has that matches the digit of that card has this hole punched out. And so by definition, when you put all three of them together, there's only one hole left open. So

Stephen 19:39
you have, like, all of the 500 numbers punched on the 500 card and all of the, like, 30 numbers that have number three number on the three card in the middle. Yeah, that's really cool.

Jason 19:50
And it's not completely random. They did put in some rules, because, I mean, you could do this, like, a bajillion different ways. So I think what they did is they just generated a bajillion different ones, and then they screened. Them because they wanted to make sure, okay, nothing has more than two in a row. You don't form any weird angles that might catch and tear. And the end result actually looks a lot like old fashioned computer punch card, because you all have these, like one and two slot holes all next to each other, and they're all spaced out and in different directions and stuff. And it looks really cool. And I think you're about to ask Brian about the numbers. So there's 125 possible codes.

Brian 19:51
Yeah,

Jason 19:52
they're laid out in a 12 by 12 grid, with some of the corners left off for some symbols you use to match it up. But there's a, I think there's 133 available slots, which means there must be eight squares that are just never used,

Brian 20:35
and that must be things like the 333, right?

Jason 20:37
No, 333, is a valid number. You'll have a hole going through it there. It could be a valid code, just not with that one card. It could be a valid code for a different set of cards.

Brian 20:46
Oh, man, yeah, I'm over my head. But that's okay. That's okay.

Stephen 20:50
I wonder if, if they added some blank ones so that they could ease the constraints of making sure that you wouldn't have cases where there were weird corners you could catch on, or they were too close, or things like that.

Jason 21:03
Maybe. I suspect it's just that a 12 by 12 grid is the smallest square grid you can fit 125 different numbers on. And so they just worked with that in the early versions of the game, it was actually a four digit code you were going after, and they dropped it down to three because it let them have smaller number of holes and bigger holes. So it worked out better, like bigger symbols, bigger holes, fewer of them. So that worked out well. They also used to have a lot more esoteric conditions. So now it's like, oh, the sum of these is equal to four or greater than four, whatever. Or this is odd. This is even. It used to be things like all of these things multiplied together, are this or this is a prime number, or other much more mathematical things that the the publisher said, No, these are too esoteric.

Stephen 21:45
There's one non prime digit

Jason 21:48
Exactly.

Brian 21:50
So they wanted to make sure that the conditions were things that didn't have too many assumptions about mathematical knowledge other than odds and evens bigger or lesser. Yeah, it's like that.

Jason 21:59
The thing I want to know still, though, is that there are 48 condition cards. Each has at least two conditions on it, and many have more than that. But there's only 95 verification cards, which means some of them have to be pulling double duty. And I don't know how that works. That's the part they actually intentionally did not explain in the designer diary, so they left that one a mystery. Anyway. That's the game. If you like Sudoku, you will probably like Turing machine. That seems to be the pattern from what I've gotten online. If you want to check it out, definitely watch some videos. It is not intuitive to most people the first time around, but once you get it, it's like, once I figured, I was like, Oh, I get it, then I could do pretty well about 90% of the time. And then I'd forget something and make some stupid mistake and completely mess it up.

Brian 22:40
I mean, when we played, we played once on normal difficulty. We both did great. We got it on the same round of guesses. I think you used one fewer piece of information, no issues. We clicked out the difficulty one notch. We could not solve this code. We both sccrewed up multiple different times, right?

Jason 22:58
Well, I know where I messed up.

Brian 23:00
Okay.

Jason 23:01
We had one of the ones where there were like, six different conditions, and I accidentally, I misinterpreted my result and thought I had eliminated five of them, when I'd actually only eliminated four of them, and so I was going off of wrong information. All right, so that's enough about the game. Let's move on to the science, and Stephen, I'm hoping you can help fill in the information here, because although I work in bioinformatics, like, I'm not a computer scientist. I'm not heavy into that. I want to start with the namesake. So this is named after Alan Turing. It's a Turing machine. So who was Alan Turing, and what was his machine?

Stephen 23:30
So Alan Turing was a British mathematician and ended up founding a lot of like fundamental thought about how digital computers would work. He was doing a lot of this work in the 30s, and then, of course, in the 40s, got swept up in the race to crack the German's Enigma machine that they were using to encode secret messages, where a lot more sort of approaches to and tactics about doing digital computation occurred at a point where there were no general purpose digital computers. Everyone was having to do analog circuitry and wind things together. But what Turing did was reason mathematically about how these computers that didn't quite exist yet might work. And He came up with a mathematical model that, I believe he called it like the A machine, and then it later his advisor renamed it to be the Turing machine, but it was, rather than an actual machine, sort of a mathematical model, a thought experiment about how digital computers could work and how you could describe a system that, then you could do mathematical reasoning about because Turing was really interested to dive into the question of like, Are there problems that a digital computer can't solve? He was playing in some of the same kinds of areas. If you've ever heard of Gödel and Gödel's conjecture that there are axioms in any mathematical system that cannot be proven, you can ever have a perfectly self proving mathematical system. There's always going to be some axioms that you just sort of have to assume as first principles.

Jason 25:06
And bit of vocabulary for listeners, axiom is basically the foundation, bedrock of math. They're the things you start it's like, okay, we know X, Y and Z from that we can derive other stuff. Your axioms are those that first bedrock stuff you lay down. It's

Brian 25:20
It's also the ship in Wall-E.

Stephen 25:21
It's also the ship in Wall-E. So Alan Turing was trying to look for some of those same kinds of principles in these digital computation machines that you know don't exist yet. So he came up with a system. The idea is that you've got this infinite roll of tape that can be slid forwards and backwards underneath this little device that is looking at each little cell on the tape. So the tape is divided into regions, and you've got this device called a head that can look at one of the regions, and it can read the symbol that is written there. It can write a new symbol on there, and it has this set of rules called states that say what it does when it encounters a given symbol. So you could imagine, for example, a four symbol tape. You can have A, B, C, or D on one of these cells, and so when it looks at say an A, it says, Okay, move the tape one cell to the right. If it sees a B, move the tape one cell to the left. If you see a C, write a D, if you see a D, don't do anything. So now you've got a simple set of rules for moving the tape around and manipulating these symbols on the tape. The next level up is you say, Okay, those are the first set of rules for A, B and C, but if you see a D, now you're going to swap in a second set of rules. You're going to go to another state in the state machine. So it shifts a new set of rules about what it does with A, B, C or D, and maybe there's a third set of rules and a fourth set of rules. So it can swap in and out these rules depending on what symbols that it sees. So it could have a, you know, a case where it says, All right, an A says I move left, and then a B says I move right. But if I see two B's in a row, now I'm going to change my rule, so now that B means I shift it to the left. So it's simple sounding but abstract. But it turns out that given those operations, you can simulate any computer algorithm like top to bottom, they can get really complex. You can build up a whole set of symbols and a whole set of these rules, these states that the machine swaps in and out. But you can use that to have any digital algorithm. You can describe it using that machine and the proper set of symbols and rules.

Jason 27:38
Do our computers work like that? This is one thing I've never been able to figure out. Like, is the Turing Machine an abstract concept, or is this essentially the foundation on which all actual hardware is built?

Stephen 27:48
It is an abstract concept because it turns out, number one, if it's going to be an actual Turing machine, the tape has to be infinite, if you know,

Jason 27:58
okay, only minor, minor issue there.

Stephen 28:00
It's a small, small wrinkle in building one of these things, yeah, because it turns out, if the tape is not infinite, let's say it's just merely super, super, super big, like you've got a Googol's worth of cells, there are some algorithms you're not going to be able to describe. So it's more that it was simple enough that you could do mathematical reasoning about it and create mathematical proofs to answer some of the questions that Alan Turing had about how digital computers would work.

Jason 28:28
This almost seems like in biology, we have our model systems which are like our little like the lab mice and the E coli bacteria, which have been like they're very simple, they're very reduced, but we can do a lot with them to try to understand how things work out in the real world is that kind of this for computation? It's a very simplified model system that you can find out some rules and then apply that out into real computers.

Stephen 28:48
Yes, you you have stripped the idea of a digital computer down to like the fewest things that you can get away with and still be able to describe all of these different algorithms.

Jason 28:59
Okay, And in looking up research for this episode, I ran across a phrase called Turing complete, which apparently describes something about computers. But I couldn't figure out exactly what, what is Turing complete?

Stephen 29:10
So if a device is Turing complete, it means that it can do all of the digital operations necessary to express these computer algorithms. So if you have a device that is not Turing complete, you are limited about the kind of algorithms that you can use, that you can describe with it, that you can, you know, create on this device. If it is Turing complete, then it matches the same characteristics as this hypothetical Turing machine, and should be able to express any digital algorithm that you can come up with,

Brian 29:42
Even though it doesn't have an infinite stretch of tape

Stephen 29:45
Even though it doesn't have an infinite tape. Yes, yeah, thankfully, they managed to figure out how to do it without infinite tape.

Brian 29:51
Think anytime you need infinity, maybe it's only it because very esoteric and abstract.

Stephen 29:57
Oh listen, I come from physics. We ran into infinities. So we're like, you know how you get rid of those infinities? You divide them by other infinities, and we will call it renormalization.

Jason 30:06
So it sounds like most of our digital computers nowadays are Turing complete, at least, like for me as a user, it seems like they're capable of running basically anything we want on them. Is that right? Like, is my cell phone Turing complete?

Stephen 30:18
Yes, it has all of the operations necessary to create these digital algorithms, you end up being limited by the amount of memory that you have available, whether that's memory where you can shove all of the numbers that it's operating on in working memory, or to store it onto a disk or other medium, but it's able to do all of the operations that make it equivalent to that Turing device

Jason 30:41
got it so I said it Turing complete in Theory and Practice, like storage becomes an issue, but in theory, right? What's what's not Turing complete then?

Brian 30:49
yeah, like, what's an example from modern life of something that doesn't qualify?

Stephen 30:54
I really should have looked up this, because at this point, I'm not sure we've got a lot that isn't Turing complete, the fact that you can, you know, run Doom, literally on your toaster. Computer chips have become so cheap and so readily manufacturable that it is, I think, harder to get a device that is not Turing complete, at least on the computer side. A lot of this came out of the point where, as I mentioned, they were building computers using analog circuits, and there were a limit to the numbers and kinds of operations that you could create by winding resistors and getting capacitors and soldering them all together. So I think that was a much bigger barrier when you're talking about the 40s with the early steps of digital computing, with the UK's bombes that were working to crack the Enigma code, than it is at this point where computer chips are super cheap and super easy to get and you just don't do much in the way of analog computing anymore.

Jason 31:58
So I just did a quick search on this and pulled up the Wikipedia page, and apparently, I think you're right, because there's, there's a list of things that are accidentally Turing complete, which includes stuff like Microsoft Excel, Minecraft and Magic, The Gathering. Yes,

Brian 32:12
wait, I'm Sorry. What does that mean?

Jason 32:16
Run computer programs in magic. And I have seen the YouTube video that explains how to do so I'll link it in the show notes.

Brian 32:22
Yes, please put that in the show notes.

Stephen 32:24
You can do it in Excel with its formulas. You can create basically computer systems in Minecraft using the redstone bridges and the, you know, linking them all together and the switches that they've got available. You can create the different kind of logic gates that are required to build up to more complex mathematics.

Brian 32:43
but Magic, the Gathering too. So yes, you can design loops in magic.

Jason 32:47
Yes, you can is the world's most boring game of Magic, but you can do it.

Brian 32:54
So when you were talking about the Turing machine, the analogies that you made, you had a read head and a writehead, you know, a long line of tape, symbols. I mean, all this sounds like magnetic tape. Is that? That's not a coincidence? Is it?

Stephen 33:07
No, and it gets to the thing you were talking about earlier, about like, how do you store information? Turing's machine, his mathematical model, used a symbol table, and you could have four symbols or five symbols, or things like that. But it turned out that it was a lot easier to build up these systems. If you just have two symbols, you have true and false, you have one and zero, and then you start building those up from there. And it turned out that you could store this in the direction of a magnetic moment. You know, if you think about the little old bar magnets, where you've got a North End and a south end, you can say, all right, if the North End is pointing up, that is one or true. If it is pointing down, then that is zero or false. And you can encode that on a magnetic tape by having the ferromagnetic particles where you hit them with a magnetic field to flip the direction that they're in. And then it persists, at least for a while. Eventually, the magnetism sort of wears off, depending on, you know, if you leave a tape in your hot car when you're, you know, 10, and then come back a month later and discover that it's all kind of gone.

Jason 34:16
Yeah, but so tapes weren't actually the first one, though, and this plays right into the game, because the game has the whole punch card thing, and punch cards were among the first actual ways of storing information. They're not digital, like magnetic tape, but what I was reading like the Code Cracking for World War Two, they were generating like, 2 million punch cards a week to try to run all their computational stuff. Have you ever run a punch card computer?

Stephen 34:39
I have not but I have used punch cards and the ways that they were never meant to be done, because I was in a laser lab at a university where the professor had gotten what was probably multiple graduate students, entire PhD researches on punch cards, and then taken them away because they have little holes in them if you're trying to align a laser. Beam. It's really nice to have a place where you can have a little hole and pull it away, and a little hole and pull it away. So that's what we used our punch cards for.

Jason 35:07
oh my, those poor graduate students, they're probably, I know, hopefully they weren't dead, but if they were, they probably were rolling over in their graves.

Stephen 35:15
Yeah, I used to be pretty blase about the fact that we were using ex grad students graduate life work basically as a cheap way to not have to use a hole punch on cards. I was like, you know, whatever. But then I sort of realized what that felt like, because we were doing atom cooling and trapping, and there was a lot of race to try to be the first group to create a Bose Einstein condensate that had been predicted by Bose and Einstein back in like the 30s, and it was this giant effort. People couldn't get it to work and couldn't get it to work, and eventually did in like 95 and it was impressive enough that they were getting Nobel Prizes for it in 98 which is just a ridiculously fast turnaround for that. And then, like, by the year 2000 it was a very common undergraduate lab, like everybody was making Bose Einstein condensates in the lab just, you know, over two days, it's like, okay,

Brian 36:10
Nobel Prize to undergraduate laboratory exercise in a couple of years. Amazing,

Jason 36:15
yeah. But no, when I was looking up this punch card things, I mean, I did not appreciate, I understood that punch cards were used to program computers back in the day. I didn't appreciate they were data storage. They were the floppy disks of their day, because there wasn't actual permanent computer storage. And so you'd have all these punch cards, and they had to be read in in order, because you flip two of them, you completely mess up the program, you mess up the data. And so they'd have these big stacks. They'd have like, Sharpie lines on them to help you figure it out. And multiple places, I found something to the effect of woe betide the graduate student who dropped their stack of punch cards, and they just scattered everywhere, and you had to try to figure out how to get them back in order.

Brian 36:52
Were they not numbered?

Stephen 36:53
Well, originally, they were just like long pieces of cardboard with a row of numbers, and then, you know, multiples of those rows, and they would be punched by hand, and then later, they would type in the thing that you wanted, and it would encode it into the space of the punch cards, because it's, it's spatial information storage in the same way that the game is storing information about the numbers. You know, each of those cards, for example, has the position of every number that starts with a three. Every three digit number that starts with a three is encoded by holes positionally punched on those cards. And I think because you would end up with just like stacks and stacks and stacks of them, because it was not a very information dense way of storing that information. You had bunches of them, and it took a while to get to numbering them. You would occasionally have ones that were a certain statement or operation that would be pretty common, like adding two numbers or, you know, moving information around. And so you could have punch cards that were commonly created that you just like, oh, I need one of these. Perhaps, instead of like, I'm going to hand create every single one each time. So yeah, I think there were some quality of life things that people had to come up with, like, numbering them,

Jason 38:10
probably via lots of pain, because that's generally what happens. Like someone messes up, like, oh, I should have done that. Let's do that from now on, yes.

Brian 38:19
Oh I'm sure I'll never drop it. It'll be fine. I'll be careful.

Jason 38:22
So the history of these things is fascinating. I didn't realize I think about them with the 50s. So apparently, the first wide scale use was in the 1890 census. Because they're like, it's going to take us 15 years to tally the census we have to do every 10 years. Like this is not going to work. And there's someone who came up with a punch card system that you could just count. It was always doing, was tabulated, it was counting stuff, but it made it so they got it in like, three or four years, so super fast relative to what they thought. And that guy formed a company that, like, three or four name changes later became IBM, International Business Machines, which is still around. That's a pretty good record for a company, I think. And if you go back further, like, the first one that everyone brings up is the first punch card, like proto computer thing was used for weaving.

Stephen 39:05
Yes, the Jacquard loom,

Jason 39:07
yeah, it would determine where which parts would get lifted up and which parts would stay down. So you could make patterns and can make them reproducible. Some version of this is what powers the player pianos you see in the old westerns, where they've got all the holes in the circular paper that's determining which keys get pushed. Okay, in the grand scheme of technology, 200 years is not a long time, but it's way older than I thought it was.

Stephen 39:27
Yeah, folks started to realize that you could encode information in holes in paper much earlier than you might imagine, especially if you're used to thinking of, yeah, computers are the things that kind of came out of World War Two and then really got a, you know, a jump start during the space race, and they're like, oh yeah. Jacquard loom,s were creating patterns for weaving that you would just bolt onto the side of your loom and have it do operations based on the holes in the cards.

Brian 39:53
So stupid question. Is a player piano Turing complete?

Stephen 39:58
I do not believe so. Because it is not doing any operations, it has no ability to swap rules in and out. It's just when this hold comes by, I press this key,

Jason 40:08
I think a player piano is basically read only,

Stephen 40:10
yeah.

Brian 40:11
Okay, so it's that, it's that is a key difference. You can program something that can just read. So a music box is programmable, but it has no ability to change what is on the recording, right? You can't write, you can only read, and that's the key difference?

Stephen 40:26
Yeah.

Brian 40:26
Okay, cool, yay.

Stephen 40:27
Now I'm, I'm curious if there are any Turing complete devices that would be read only, and I don't know, I guess if it's an infinite tape, but maybe you can get away with it, right?

Jason 40:42
Hey, y'all quick aside here. So after we recorded this episode, I looked this up, and it turns out that while you can have a read only Turing machine, it is not Turing complete. Some operations just can't be done when you can't write to the table. So just want to put that in here, so we had the answer. And now back to the show. We probably actually need to start wrapping this up. There's been a great conversation. There's a few things I want to touch on before we do one is the great mystery of computers to me, how do you do math with just ones and zeros? You talked about how it's much easier to do all these operations if you just use on and off with the magnetic fields. But when you get down to how those actually mesh inside the CPU and are doing math and are doing the calculations that show what's on my computer screen, all that I do not understand how that works, like what is going on there in the guts of the CPU.

Stephen 41:30
If you go all the way down to the base level, you're looking at Boolean algebra that was created by a mathematician named Boole back in, I think, the 1890s where he was looking at what kind of algebraic operations you could do if you had just ones and zeros. There's more to it, but for our purposes, let's just stick with ones and zeros. So every number is a one or a zero, and you can stack those together. So in decimal, where we would go 012345, instead, you go 01, 10, 11, 100, 101, 110, 111, 1000 so it's more properly base two numbering, where we're used to base 10 numbering, and then you introduce, three operations, you introduce, AND you introduce, OR you introduce NOT. So if I have two one digit numbers that can be either 0 or 1, if I AND them together, the result is a one if both of the numbers that you started with is one. So if I have a 1 and I have a 1 and I AND them together, I get a 1. If I have a one and a zero, if a zero and a one half a zero and a zero, I get a zero. So both the first number and the second number have to be one to give you a one. The OR operation is like that, but it's a little more permissive. If one of the numbers or the other number is one, then it results in a one. If both of the numbers are zero, it results in a zero, and then the NOT operation changes the number. If you have a zero, the not of the zero is one. If you have a 1, the NOT of the 1 is 0. So you can start to take those operations, and you can take longer numbers, so you can and two bit numbers together. So like an 11 and a 10 added together, gives you 10, because the first digit is one on both of them, but on the second digit, only one of them is if I or a 10 and 11, I get 11. If I take 10 and I not it, I get a 1. So you can build up longer algorithms using only those three operations and OR and NOT. There are algorithms that let you start to create, for example, addition. It lets you create multiplication. If you want a really deep dive into this, I'd say, go look up a YouTube video describing how the half adder h, a, l, f, half adder algorithm works for adding numbers together using just bits and ANDs, ORs, and NOTs.

Jason 43:58
It sounds like we're back in the Turing Machine territory, where you've got a very simple setup, and then through this like layers and layers of complexity, you manage to build up to something that we would recognize,

Stephen 44:08
Yep, yeah. So that, like, computers don't natively know how to do division, but we've got clever algorithms that let the computers do operations that result in a division that are stacks of these logic gates, they're called ANDs, ORs, and NOTs, and rules about how you apply them.

Jason 44:26
Okay, well, thank you, Brian, do you have any last questions or comments you want to put out?

Brian 44:30
I have my one joke about binary. Well, you've probably heard before there are 10 types of people in the world, people who understand binary and people who don't

Stephen 44:37
Now, the really fun thing as somebody who came out of laser cooling and trapping, one of the things that people end up doing once you can start to manipulate these atoms in their quantum states, as they start to build quantum computers, where you have qubits, Q, U, B, I, T, S, where they are particles that are in a quantum superposition. So maybe they're both one, maybe they're both zero, we're not going to know until we actually make the measurement. And all kinds of wacky algorithms are possible once you start to have enough qubits together that you can do in effect, kind of parallel, probabilistic computation.

Jason 45:18
I hear about those every now and then, although my understanding is that they're facing the infinite tape problem. In that last I checked like they can get like 15 or 20 qubits, maybe 50.

Stephen 45:28
There are systems where you can have more of them, but they're not in sort of the classical superposition. There's this thing called an Ising state, about where you can get atoms in a crystal aligned properly in a ways that let you do some aspects of quantum computing, but not the ones that people are really both interested and afraid in, like breaking cryptographic numbers really, really fast, so that all of your banking accounts can be sucked dry.

Jason 45:55
Yes, that's mostly what I hear. Is that if we manage to get quantum computing to work, all our current encryption goes to pot.

Stephen 46:02
Yes, that is the headliner application for it.

Jason 46:06
All right, so Brian, you have your nitpick corner. Do you have any nitpicks about this game you want to bring out?

Brian 46:11
I mean, it's not a computer. So that's, that's kind of a thing, like, you know, it says it's the punch card computer, but it's not really a computer. I don't know what it is, but it's not that.

Jason 46:21
It's a proto computer the rulebook clarifies.

Brian 46:23
What does that mean?

Jason 46:24
I don't know.

Stephen 46:27
Yeah, I guess it's using like computer, like storage or encoding of information, but there is no computation going on. It's sort of like the old school version of computers, which were people who did computations. You're the one doing the computation by putting them together.

Brian 46:43
That's true. We're, we are the punch card computers.

Stephen 46:47
You are your own Turing machine. Congratulations.

Jason 46:51
I have two nitpicks, and these are little ones, but one is, it's a four player game, but they only include three sets of the number cards. I assume that's a cost saving thing, because they seem like they're either very nice cardboard or maybe very thin plastic. That's probably just a cost thing, because I assume custom punching out all of these different numbers is kind of expensive. The other one is actually, recently, you talked about the hardest thing about a game is writing the rule book, because by the time you get to that, you know it so well. You're not you don't know how to explain it to a newbie. I think that happened here, because nowhere in the rule book do they actually say, by the way, you're the code you're looking for is the one that satisfies all the conditions on the table. It sort of alludes to that in one or two places. It kind of assumes you get that, but never actually says that outright.

Brian 47:33
I actually got that from watching a YouTube tutorial. It's like, by the way, if you want to play this game, you need all of the condition cards.

Jason 47:40
So that's my nit pick. Is, like, that's a fairly important part of the game that I think they left out of the rule book. It probably needs to be corrected.

Stephen 47:47
Gosh, yeah, that's a great point. So my dad was a historian, and his speciality was Civil War, and so he would play the Avalon Hill game Gettysburg with his classes. And so I grew up with this Avalon Hill Gettysburg game, which had all these pages of densely typed rules, and they were all subsections. So you'd like, oh, according to 3.2 point 7.1, point four, when it is muddy, my terrain is modifier is such and such.

Jason 48:16
Oh no this is steller Horizons

Brian 48:18
The game ran by an engineer.

Jason 48:20
All right, so let's get on to letter grades. Brian throw this to you. What do you think about gameplay?

Brian 48:25
I dont know this is a weird one, because this isn't the kind of game that we would typically play. But if you just, like, want to play a light game or something this, like, I kind of agree with you. This might be the one game that's worth having in solo mode, where you just set it up every once in a while. I guess I'll just give it a B. I think it's, I think it's kind of its own little unique niche. There's nothing really competing with it.

Jason 48:47
Yeah, apparently there's this whole genre of deduction games I was unaware of, so there might be some others competing with it, but like in our sphere, no, I I'm gonna give it higher, I'm going to give it an A-, and I'm gonna give it that because for what the game sets out to do, I think it does very well. The only reason I'm getting a little lower is because I think there's a barrier to entry that can put off a lot of people. In fact, when I was doing research, I ran across a Reddit thread by someone who's a professional game explainer who is basically asking for help, because no one ever understood Turing machine the first time he explained it to them, and so he's asking for help trying to figure out how to explain it better to people, and usually by the second or third time, they figured it out. But he just had trouble with that. And I wish they'd made that barrier to entry lower, but once you get past it, I think it can be fun. And if this is your jam, then I think it's a great game. It's not my jam, but I can definitely see what the appeal would be.

Brian 49:33
Does it have Quick Start rules like, Hey, play this puzzle. These ones. Let us explain how this works. Does it do that, I don't think it does. And that, honestly, just like here, let us walk you through a simple puzzle. Yeah,

Jason 49:46
it gives an example round, but it does not walk through an entire game deduction. It gives examples of the components, but it never puts it all together until, like, Oh, here's your this is your first game of Turing machine, so you understand how it works.

Brian 49:58
This is the perpetual. Challenge. When you have experts trying to talk to amateurs, right? It's, it's very, very difficult to keep that beginner's mindset.

Jason 50:07
But like I said, overall, I think for what the game sets out to do, I think it actually does it very well, especially we talked about the lack of skinning it. It's like, it's a very pure This is a logical deduction game, and there's very few bells and whistles around it. It is trying to be a logical deduction game, and I think it does that great. Steven, do you have thoughts? I mean, you didn't have a chance to play it because we're unfortunately too far apart from each other.

Stephen 50:27
Right, I did get to watch some of the aforementioned YouTube videos. I really admire games like this that can pull off an experience without a lot of the sort of what I think of as traditional skinning and other elements of it to add to the experience. You know, if you think about like bluffing games, like Sheriff of Nottingham or something like that, like part of the fun is pretending to be the people smuggling the food in and out around the Nottingham wood. Here, it's just numbers and operations, like you mentioned Sudoku, like you mentioned Wordle, I think when that kind of game is done, well, I really admire it, because I think that feels to me like a much harder lift to come up with something that is that abstract and still interesting, that doesn't feel like I am playing Excel, the spreadsheet as my game tonight,

Brian 51:22
or magic the computer game.

Stephen 51:24
Yes, magic the computer game.

Brian 51:27
It still is a pretty game, but there's no fluff. It just it is what it is.

Jason 51:32
I mean, the punch cards, they did a little bit of fluff there, but not very much. I mean, the randomization is really the only fluff component there, and that's mathematically equivalent to having them not randomized. So you might as well do it.

Stephen 51:43
Yeah,

Brian 51:44
and a hexagon of unnecessary little digital faces

Jason 51:47
There we go. That is the one unnecessary aesthetic thing they put in there, the hexagon. Actually, many of the people who play a single player apparently don't use that. They just lay them out straight in a row. So onto science.

Brian 51:58
How do you grade this? It's math.

Jason 52:02
I'm going to give the mathematical answer and say that the science grade is undefined.

Stephen 52:06
Nice.

Jason 52:07
This game is not trying to represent any scientific concept, which I don't think we fully realized when we picked it up and put it on the show schedule. It's a game of logic and deduction. There's not a scientific process. It's trying to represent. And so I don't think it's fair to give it a science grade, because it's that's not what it's doing.

Brian 52:24
Yeah, it's like, again, it's in STEM science, technology, engineering, math, it's math. So we can give it a math grade, and it did math.

Stephen 52:32
Yeah, it feels like it is touching on some of the tools, the building blocks that you use as part of scientific inquiry. The idea of, like, if I've got a system, how do I query it? What questions should I be asking? How do I get information out of this system that I'm dealing with? And it is very much, in that case, a toy model, but it's an interesting exercise, I think, to go through to force yourself to do that in this constrained environment.

Brian 53:02
That'san interesting way to think about it.

Jason 53:04
Yeah, I hadn't thought about that because I often say that science is the world's biggest game of guess and check. That's that's what we do when we make a hypothesis and we test is we are making a guess and we're checking to see if we're correct, which makes it sound bad, except that the alternative is guess and not check, which is what a lot of other things do. So yeah, I hadn't thought about you're right, because, you know, there's some conditions you don't know which you make a hypothesis of your number. You check it to see if it actually fits or not, but you only get partial data. You have to figure out that's actually really cool. I hadn't thought about that.

Brian 53:35
There are some examples of logical deduction in biology. Again, we were talking about it with the grid, right? We figured out by pure principles, that to be able to code for 20 amino acids, that you'd need to have at least three digits to do it, because you can't do it with two. So the smallest number would have to be three, which means we've got more options. And like that was just sitting down and thinking about what made logical sense.

Jason 53:57
Well there's another tie back to Turing. So he came up with what's called a Turing pattern, which is basically you have these very simple rules about things that are like making some molecules, usually at least two different types, one of which has a different lifespan than the other, and they diffuse out. And from very simple rules, you can get super complex patterns. Everything from like cheetah spots to fingerprints to the folds of the human brain, are thought to arise from these Turing pattern processes. I'm actually studying one in corn right now, where I think that a Turing pattern is involved in how this certain feature comes out.

Stephen 54:29
Oh, that's neat.

Brian 54:30
Science is really fun, actually. Like, I was joking about onions, but like, they are interesting too. Corn is interested. Everything is connected,

Jason 54:38
all right. Well, that's where we should probably wrap it up, Stephen. Thank you so much. It's been wonderful having you on. Thank you for teaching us about laser cooling and computational operations and all sorts of stuff like that. And it didn't come up here, but I'm going to link in the show notes what I think is probably your greatest and most lasting contribution to the field, which is the science vs. movies panel at DragonCon, which for those of you who haven't seen this, there's no real science in this panel, except by accident. It's where Stephen makes poor scientists suffer through some of the worst science shown in Hollywood, and then explain why it's right, actually. And it's hilarious. I'll throw some links in the show notes. They are totally worth watching.

Brian 54:38
Have you done The Core? Did you subject people to The Core? You must have.

Jason 55:15
I have subjected people to The Core.

Brian 55:15
so Stephen, if people want to find you, where should they look you up?

Stephen 55:18
So you can search for my name, I have won the Google search for Stephen with a pH granade, G, R, A, N, A, D, E, my website is stephen.granades.com because one of the French branches of the family that are out in California got granade.com Before I could but I got my revenge. He ended up having to link to me early on, where he was like, Yeah, you're probably looking for this Stephen granade.

Jason 55:53
All right, well, then we'll call it there. Thank you everyone. Thanks for listening. Have a great month and happy gaming.

Brian 55:59
Have fun playing dice with the universe, see ya this has been the gaming with Science Podcast copyright 2025

Jason 56:03
listeners are free to reuse this recording for any non commercial purpose, as long as credit is given to game with science. This podcast is produced with support from the University of Georgia. All opinions are those of the hosts, and do not imply endorsement by the sponsors. If you wish to purchase any of the games we talked about, we encourage you to do so through your friendly local game store. Thank you and have fun playing dice with the universe.

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