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RR36: The Most Boring Episode Ever (Rerun: Math Games)
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This is a rerun of one of our favorite episodes! We hope that you enjoy it if you haven't listened to it yet. We'll be back next week with new content! Thank you so much for listening to Breaking Math!
Math is a gravely serious topic which has been traditionally been done by stodgy people behind closed doors, and it cannot ever be taken lightly. Those who have fun with mathematics mock science, medicine, and the foundation of engineering. That is why on today's podcast, we're going to have absolutely no fun with mathematics. There will not be a single point at which you consider yourself charmed, there will not be a single thing you will want to tell anyone for the sake of enjoyment, and there will be no tolerance for your specific brand of foolishness, and that means you too, Kevin.
Theme by Elliot Smith.
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Transcript
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This episode is distributed under a Creative Commons Attribution Share Like 4.0 international license. For more information, visit creativecommons.org.
Redefining the Planck Constant
00:01:14
Speaker
Before we start this episode, we have a little bit of news about the Planck Constant. It's been redefined, it will be in like 2019, but the decision was made this November, November 2018. Do you guys have any thoughts on that?
00:01:28
Speaker
That's pretty significant. That's going to change a lot of things, isn't it? Well, I mean, yeah, it's going to make things more accurate so that everybody can agree on a value. But basically now the kilogram, the second, all SI units are in terms of numbers and things that you can find anywhere in the universe. Is that why they decided to redefine it? Just so that it would be like standard?
00:01:50
Speaker
Yeah. They've always been trying to redefine it. Like in the beginning, uh, a meter was defined in terms of the world. It was one 10 millionth of the distance between the poles. He may have fuzzed it a little bit, but don't, you know, and, uh, and then, um, yeah, everything was based on unit on things that you could find in like a specific places and one by one they've been going away. And now we have all the units in a way that you can find anywhere.
00:02:16
Speaker
So real quick, what did they use?
Impact of New Measurement System
00:02:19
Speaker
Because actually, full disclosure, no, I did not read the article. I knew about it, but I didn't read it. What did they use to come up with the new plank constant? From what I could tell, they used something like they made a sphere with the same weight as it. I'm not sure exactly. I haven't read all the things about it.
00:02:40
Speaker
We'll learn more about it from the general press as time goes on, but they essentially counted the number of atoms. They're also redefining the Avogadro constants, the number of atoms in a mole. Wow. Interesting to me that we're updating our measurement system for a changing view of the universe. So it used to be like totally based on just the earth and now we're basing our measurement system on the whole universe. Yeah, it's pretty cool. I mean, it's still based on the original artifacts, but
00:03:06
Speaker
Yeah. Yeah. I wonder how things will change then. Like once this is done, like how much more precise can, can you go, you know? Yeah. Well, I mean, with this, you could literally be as precise as you possibly can be. Like there's no more precision that you can get from units, but I know that there's this, uh, there's this, um, scale that they're developing. Uh, it's so accurate that they measured the following, they measure the local gravitational constant in the laboratory by dropping something and looking at the interference pattern from a laser.
00:03:34
Speaker
Oh wow. And then they oscillate the matter back and forth and measure the amount of energy it takes to keep it oscillating. You know, as you say, this is a bit of a side topic. You know, when you talk about precision, I think that a good conversation on precision would be a great way to combat fake news, especially when it comes to topics like climate change, when you actually talk about
SI Units Explained
00:03:53
Speaker
the precision. You know what I mean?
00:03:54
Speaker
Oh, definitely. Precision is everything in science. I mean, there's reasons why metrologists have been getting better and better at this kind of stuff. They have really insane kind of stuff that they're doing. Yeah, things weren't always precise, you know, but obviously we've improved it. So, good topic.
00:04:10
Speaker
Yeah, and so just real quick, the second is defi- it will do all the SI units right here. You can skip forward probably about 45 seconds if you want to. But a second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfund levels of the ground state of the Season 1-33 atom. I don't know what a lot of that means. Okay, yeah. Wow, that was- That's intense. That was a lot.
00:04:36
Speaker
The meter is defined by how much light travels in 1,299,790,458th of a second. So the meter is determined by the second. Interesting. The ampere is defined by taking the elementary charge to be 1.602176634 times 10 to the negative 19th coulombs. The kilogram is defined now in terms of the Planck constant, the meter and the second.
00:05:04
Speaker
The Boltzmann constant is defined as 1.380649 times 10 to the negative 23rd joules per Kelvin. The Kelvin is defined in terms of the other units. A mole is now exactly 6.02214076 times 10 to the 23rd things. And the candela is measured in a given direction of a source that emits monochromatic radiation of a frequency
00:05:30
Speaker
of 5.4 times 10 to the 14 Hertz and has a radian intensity in that direction of one 684th of a watt per steradian. So all the units are dependent on one another now and future space colonists just listen to our podcast. That's interesting. So like when you say they redefined like the constant and the measurements, like it's like they took the old measurements, but they define them with more precision like Gabriel was saying.
00:05:58
Speaker
Yeah, they're all plus or minus a few like 0.000s, but now they're perfectly the same. And what that means is that because the kilogram basically was getting old, it gained about 20 micrograms in 300 years or so, which is too much for metrologists to handle.
Math and Humor
00:06:18
Speaker
And now onto our grave episode.
00:06:24
Speaker
Math is a gravely serious topic which has been traditionally done by stodgy people behind closed doors and it cannot be ever taken lightly. Those who have fun with mathematics mock science, medicine, and the foundation of engineering.
00:06:41
Speaker
That is why on today's podcast, we are going to have absolutely no fun with mathematics. There will not be a single point at which you consider yourself charmed. There will not be a single thing you'll want to tell anyone for the sake of enjoyment. And there will be no tolerance for your specific brand of foolishness. And that means you, Kevin. Episode 36 of Breaking Math, the most boring episode ever.
00:07:09
Speaker
you
00:07:16
Speaker
I'm Baka. And I'm Gabriel. And today we have on Adam Korlova. He's a friend of the show. He's been on, I think episode three, was it? Yeah, I think so. Yeah. And welcome to the show. Hi, guys. How are you doing, Adam? Good to have you. Now, before we get started with our serious episode, we have an ad for a serious poster about tensors, which are used in the serious theory.
Podcast Updates and Ads
00:07:38
Speaker
We have to stop with the serious stuff of general relativity. Yeah, this is an update. This is something you have not heard before. We actually have a price change.
00:07:46
Speaker
Yeah, we got a better printer, a different printer. Well, actually it was the same printer, wasn't it? They just did a different process. Yeah, they were able to more bang for our buck so we can actually lower our prices. And now our posters are pie to the E dollars each. And for you, Plebes, that's $22.46.
00:08:06
Speaker
That's right. So that's not quite e to the i pi, but if you were to... It's not inverted. What do you say when you swap it out? You swap the two elements of a power?
00:08:18
Speaker
Um, wait, if they're just multiplying by one another, wouldn't it just be the same power? So no, no, cause I'm, I'm, I'm relating the price of the poster to, uh, oilers. Awesome proof. I guess if you took a negative one, raise it to the ith power and negated it, you would get our price of the poster. I don't know. That might be the case, but I think that.
00:08:37
Speaker
Yeah, so we'll say that price again in terms of dollars. It is exactly pie to the e dollars and that's $22.46 you get that through our patreon and it used to be $35 and that's patreon.com slash breaking math and All you have to do is donate one time and you'll get a poster. Yep. Yeah, I know it will still say do you want this to be a Recurring thing. No. No, we don't just one time is fine
00:09:02
Speaker
Yeah. And so Oh, in shipping, we now have a garage full of posters. So the shipping in the United States takes a couple of days. That's it. Yeah. And of course shipping around the world because of restrictions and all that kind of thing, we're going to have to still charge shipping for international orders, but it is still available internationally. A couple of announcements. We've moved from SoundCloud to Anchor now.
00:09:29
Speaker
That's right, Anchor is a newer platform for podcasters and our new address is anchor.fm slash breaking math podcast. But of course you'll still be able to pretty much seamlessly get our, I mean, you're listening to us now. You'll be able to get our stuff from anywhere unless you were listening directly on SoundCloud, but very few people were.
00:09:51
Speaker
So, any source that you previously heard us, we are still available on Apple Podcasts, Google Podcasts, which is actually a newer one, Stitcher, all of them. And Gabriel, you're launching a new podcast?
New Podcast on Speech and Debate
00:10:02
Speaker
That's right. Actually, it's not just me, it's Santa Fe Trail Media, which is the name that we use for all of our podcasts. It's the whole
00:10:09
Speaker
Umbrella Santa Fe Trail Media will include breaking the breaking math podcast and every other venture our brand new podcast I'm so excited about once again we have reached out to educators we have found a teacher with which to launch a new podcast this is a speech and debate podcast and we are so excited the host is a gentleman by the name of Trey Smith who has been involved in speech and debate for over 20 years he started a speech and debate club at a very small charter school
00:10:37
Speaker
And in very short order, I believe three years, they dethroned the 24 year reigning champions or the 23 year. They dethroned the local champions and they're very successful. The podcast name is The Soapbox with Trey Smith. You can find it on Facebook. You can also find it on anchor at anchor.fm slash the soapbox pod.
00:11:00
Speaker
And it's real exciting. It's all about stories from speech and debate. It's about how you build an argument, how you communicate. There are stories of debilitating defeats and really amazing victories. So I hope you find it on anchor.fm and enjoy it. Let us know what you think.
00:11:19
Speaker
And of course, our episode today, we're going to drop the pretense.
Rediscovering Math
00:11:23
Speaker
It's about recreational math. Yeah. The original title full disclosure was, um, fun with math. We just sort of went the other way with our theme there. Yeah. And, uh, yeah, as you heard from our new theme, I mean, we had to use that. Uh, so, uh, Adam, what, uh, what experience have you had with recreational math and how has it affected you?
00:11:46
Speaker
That's an interesting question like I was I went through math through like, you know like 12th grade and then into college and then I left college for a few years and Like before I left I really did not enjoy math I was like I had been through you know year by year like through high school and I didn't have the best teachers and it was just like I
00:12:07
Speaker
Like doing my math homework was not something that I wanted to do ever was like my least favorite homework of all but now since I've returned to college I like went back and started in like earlier algebra and worked my way back through and now I'm on trigonometry and Like so I went back through like a lot of math just to like just to make sure that I had a solid foundation and
00:12:30
Speaker
And I'm really enjoying it now. I'm finding that actually I look forward to doing my math homework now. I look forward to doing a test. It feels good to get a problem right. You get a little bit of a rush from that, especially a hard problem.
00:12:45
Speaker
Yeah, and I think that's kind of the driving force behind recreational math, is that humans are like crows, we love new stimulus. Definitely. And as soon as we can get new stimulus, we get this irrational, it is irrational because there's no rational end to anything in a certain sense, but just having this reaction to discovering new things has seemed to help us very much over the centuries in millennia.
00:13:13
Speaker
Yeah, absolutely. Yeah, goodness. I may take a stab at the same question, if I may, on recreational math.
Interview Puzzles
00:13:20
Speaker
So I think, you know, as a young student, you know, in elementary school, I had a lot of teachers who were very good at it, you know, a lot of fun math games, they made a game out of everything, including things like how quickly can you do your timetables, but it's obviously it's not just that. I think in our nation's history, there are several examples of very stimulating mathematical columns in newspapers that include puzzles, I think,
00:13:40
Speaker
Steven Strogatz is one of the authors who does a lot of great puzzles. I'm not sure but one thing that you did I just have to bring up because you mentioned our nation's history is that President Garfield has a proof of the Pythagorean theorem. President Garfield? Yeah.
00:13:55
Speaker
Oh wow, fascinating. Yeah, so there's certainly an enjoyable aspect to it. It's the same way as puzzles. I am going to include games of Tetris where you have to strategize how you fit things in. So it's really not difficult to talk about mathematical games as just a topic of pure enjoyment.
00:14:18
Speaker
Yeah, but as far as you're mentioning, you were helped by recreational math during a job interview. Yes, I really look forward to sharing this story. I actually landed a job thanks to recreational math. Now, there's more to the story than just that. Essentially, I was interviewing a while back for a job that was essentially an entry level engineering job right out of school.
00:14:39
Speaker
I was fairly excited about it, and I sat down with the interviewer, and he told me, full disclosure, during this interview, I have a few mathematical puzzles that I'm going to give to you to solve, and I'm very interested in how you go about solving them. That is very important information for me, and I think that it helps me to assess whether or not we're a good match. So I was like, oh, okay. I got a little bit nervous, and he gave me a couple of puzzles, and I'm going to share two of them with you all.
00:15:07
Speaker
The first puzzle he gave me was, um, what do you call them? One of the blacksmith puzzles. Um, yeah, I think that's what they're called. The one, the iron puzzles that are in squiggly shapes. And then the goal is to separate the pieces. The goal is to separate the pieces. And this particular puzzle was, uh, it had like a, uh, a long thick, thick screw with a bolt on, on one end and the bolt looked like it had been soldered. It was like a real sloppy solder job. That was, uh, just like, I don't know how to describe it. It looked like.
00:15:36
Speaker
is bumpy yeah bumpy and it was like melted ice cream like it was all over the place it was like they didn't do a very good job so on the other end of it um oh yeah so the other end of it was a hook and it was not a straight hook the end of it was slightly bent to the right so it looked like if our oh and i also didn't um mention there was a ring that was attached to the hook
00:15:59
Speaker
And the idea was to remove the ring from the end of the hook. And, you know, based on how it was bent, I was sure that there was some way that I could orient that ring in order to pull it off. So, you know, during the interview, I'm telling them all about my experience in college and all of my engineering projects and all this stuff. And I'm trying to bend this thing and undo it. He kept asking me if I would like a hint or if I'd like any help. And I thought, this is an interview. This is a test.
00:16:28
Speaker
My answer was no. No, thank you. I'd like to do it myself, which is what I thought he wanted to hear. I, you know, I kept going and during the entire interview, I didn't get it. He then finally stopped me and said, the reason why I asked you for, if you wanted some help is I wanted to see how, if you got frustrated, would you ask for help?
00:16:46
Speaker
And at the time I was like, Oh, I thought I, you know, did I blow it? And then how I recovered from that, as I told him, Oh, I certainly plan to ask for help. I just believe in exhausting all my options first before asking. I'm not opposed to it. I hoped that I could show that I was a hard worker or something.
00:17:03
Speaker
Now this kind of went on back and forth and then finally I said, in the interest of this interview, I'm going to stop myself and say that I will accept your help. And then he said that he took the puzzle from me and he said, when you have tried everything that you think it could be, and it doesn't work, it has to be something that is impossible.
00:17:28
Speaker
He then took the end where the screw was welded, shut, and he just turned it and unscrewed it. It was a fake weld. It wasn't welded at all. And then he just took the ring off that way. So yes, I was completely duped. I didn't even think to try that.
00:17:46
Speaker
It reminds me of that there's a puzzle in the show Star Trek, the Kabayashi Maru that's supposed to be impossible to solve task. And it just kind of reminds me of the same thing. But what is interesting is that you must have had an enjoyment for recreational puzzles to have had the muscles in your mind to keep going at this like very frustrating. I've done these tasks and they're frustrating. Yeah. Yeah.
00:18:11
Speaker
Yeah. Well, and also I've learned something from now on. Anytime there's a puzzle happening, I'm not going to rule out ridiculous things because it is possible. It is very possible that the person giving you the puzzle is, you know, trying to just trick you. And that's all there is to it, which was the case here. Game theory. Yeah. Now he gave me another puzzle as well. This one was, I was real excited. The other puzzle he gave me, I solved in 30 seconds because it was very similar to something that we had done on a previous
Logic Puzzle Cliffhanger
00:18:36
Speaker
episode.
00:18:36
Speaker
So my research that had gone into the Breaking Math podcast had really worked in my favor here. I'll go ahead and share this puzzle. It's a prisoner hat puzzle. So in this next puzzle, according to the story I was told, there are four prisoners who are arrested for a crime, but the jail is full and the jailer has nowhere to put them.
00:18:57
Speaker
So he eventually comes up with a solution of giving them a puzzle so that if they succeed, they can go free. But if they fail, they are executed. That sounds pretty intense. So essentially in this puzzle, you all may have heard it before. The Jailer sits three men in a line. Let's just call them A, B, C, and D. B faces a wall. C faces B. D faces C. And then there's B, of course. Now, the fourth man,
00:19:25
Speaker
Is put we hold up. So what regiment are they in b faces the which wall sure let's see here. Um, so okay, so so B faces the wall c faces b. Okay, so the b is facing the wall and c's looking at him So the reason why I mentioned that is is that b cannot see anything right now. Okay. Yeah Okay, so b doesn't b can't see anything c can cb. Yes, that's correct. And no one else. Yeah d faces c. Oh, yeah The other thing is d the last prisoner the fourth prisoner
00:19:52
Speaker
faces C and B, so he's got a little bit of a leg up. The fourth man, A, is put behind a screen or in a separate room perhaps. The jailer gives all four men party hats. He explains that there are two black hats and two white hats that each prisoner is wearing one of the hats and that each of the prisoners sees only the hats that are in front of them
00:20:20
Speaker
but neither on themselves or the one behind them. Now, the fourth man behind the screen can't see or be seen by any of the other prisoners. No communication among the prisoners is allowed. So if any prisoner can figure out what color hat he has on his own head with 100% certainty without guessing and tell the jailer all four prisoners go free. If any prisoner suggests an incorrect answer, all four prisoners are executed.
00:20:47
Speaker
So it's a double pronged problem where you have to figure out who knows and how they know. Yes, yes, absolutely. And they can't communicate with A. Yes. The puzzle is to find out how the prisoners escaped. Now, here's what I did and what I'm going to let you all do as listeners and for you, my co-hosts.
00:21:04
Speaker
What I did is I began strategizing and thinking aloud, and I just told him what I would do. I basically, I wrote down all the information he gave me, and then I categorized it. Now, without telling you how I did that, what do you guys think? What would you do? I would probably do something like...
00:21:23
Speaker
draw a tree of all the different possibilities, something like that. Yeah. This is kind of funny because my math teacher this semester actually started our class with a version of this problem and it completely stumped me. But the thing that I did do was draw a picture first because I tend to be really visual with stuff like that. Yes, exactly. Exactly.
00:21:39
Speaker
Now, what I did is in problems like this, I think before we did the Breaking Mouth podcast, I didn't think very carefully about it and I didn't look for hidden clues. There's some hidden clues here that are only apparent when you diagram every single bit of information. So just ask yourself, what is every single thing that we know and what do we know about what the prisoners know? Should we, I don't know, should I give the solution now?
00:22:06
Speaker
I think that we should give it on next episode. Okay. Absolutely. Absolutely. So in the next episode, now don't cheat. We will give the solution to the prisoner hat problem on the next episode.
00:22:15
Speaker
Oh, and yeah, speaking of episodes and things like that, a quick thing. I said that, I can't remember which episode it was. I said that Omar Khayyam helped to transcribe the Thousand One Nights. I was mistaken. It was actually a test I showed you for ninth grade. It was a mnemonic divide. I was taught that he had something to do with it. So I literally, I remember the song that I sang. I'm not gonna sing it for you guys, but
00:22:44
Speaker
Anyway, sorry, I give you guys bad information. Yeah, and thank you for our listeners for pointing out when we have inaccuracies so that we can correct them. We do our best, but sometimes things slip through the cracks and we appreciate your feedback.
00:23:02
Speaker
Hey, breaking math fans. First, I want to thank you for listening. I have an important message for everyone. You can start your own podcast right now with Anchor. Anchor lets you create and distribute your own podcast. Just get an idea, record, and upload. It's just that easy. Anyone can do it. I'm on my way to accomplishing my dream, and you can too. Just get on your device's app store and download Anchor. It contains everything you need to make a podcast. With Anchor, you can put your podcast on all the big platforms.
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Speaker
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00:23:51
Speaker
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Speaker
This week we want to feature a wonderful course on machine learning. It is one of many courses in data science available on Brilliant. So what are you waiting for? Sign up at brilliant.org slash breakingmath. The first 250 listeners get 20% off the annual subscription. That's brilliant.org slash breakingmath.
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Speaker
This episode is all about how mathematics has never been any fun for anyone, and we discuss serious topics like puzzles. To that end, our partner Brilliant.org has a full course all about boring mathematics and why you should always fear it.
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Speaker
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00:25:11
Speaker
Serious and boring math is important for your soul, and yay, it is required. To support your education in the grave arts, math, and physics, go to brilliant.org slash breakingmath and sign up for free. The first 200 breaking math listeners get 20% off the annual subscription, which we have been using. And now, back to the episode.
Ramanujan and the Number 1729
00:25:33
Speaker
So we're gonna talk real quick about Ramanujan and the taxicab numbers. And I think this is a good story just because it's just about pure recreational mathematics and a very famous mathematician.
00:25:44
Speaker
So this one is a very simple story. Romano John and some mathematician were in a taxicab, and Romano John is really well known for a lot of really incredible mathematics, a lot of things like nested radicals, like square root of one plus square root of two plus square root of three plus square root of four plus, but like not the square root of each number, like square root of, anyway, you keep really weird things like that. He's good at that kind of thing, or he was in the 1910s.
00:26:12
Speaker
But they were in a taxi cab, and the number of the taxi cab was 1729. And the mathematician with Romano John said it was a really boring number. And Romano John is like, no, it isn't. It's the smallest number expressible in more than one way as a sum of two cubes. And he's like, it's 12 cubed plus one cubed, or nine cubed plus 10 cubed. Interesting. Yeah, that's crazy. He just like came up with that. Yeah, like instantly. That's incredible.
00:26:40
Speaker
Yeah, and it just goes to show like these mathematicians, they're in a taxicab. They weren't doing other things, but like, I don't know. They had no goal in mind, but mathematics still was something that they decided to do and to the point where they called these numbers the taxicab numbers from then on. Numbers that are expressible in more than one way as the sum of two different cubes.
00:27:03
Speaker
Wow. That's interesting. Can you imagine if you're the kind of person who just like went around and every single number that showed up, whether, you know, it was the amount of P's on your plate, whatever you, you figured out the significance of that number. Man. I mean, you could do something kind of like this, um, like, uh, just, I mean, I'm sure everybody would have been lucky on every single number. Yes. But like if you, if you're driving down the road, uh, try to find patterns in the numbers and letters in the different things that when somebody rings you up at the check checkout counter,
00:27:32
Speaker
Try to think of the fastest way to remember that number forever. It's actually really good. I know that there's a lot of memory mnemonic. That's one way of a mnemonic device is anytime you have to have a list of numbers, just think of patterns or stories to sort of encourage too. Now that actually would be a good idea. Numeric scavenger hunts. I don't know if you want to keep your kid busy for a while, just like give him like a list of 14 numbers that are like four digits and say, find something that these describe.
00:28:02
Speaker
Yeah, I mean, it's just a really simple way. I mean, it has everything that a recreational math problem has. It has a problem, which is data. It has a solution that's non-trivial. It has the feeling of accomplishment. It's pretty much, yeah, it has everything you need.
00:28:21
Speaker
I feel like the way I described that was so lazy and almost mean like, here kid, just go figure out something significant about these numbers, knock yourself out. And then, you know, lo and behold, you've got a math genius who then tells you something amazing. So, you know, there's other ways to keep yourself occupied. But yeah, the goal here is to do it with math.
00:28:40
Speaker
Romano John is really interesting to me because apparently he believed that he was receiving his proofs from the goddess Lakshmi which was his ancestral goddess, like the goddess of prosperity and beauty in the Hindu religion. Yeah, he attributed a lot of his mathematical discoveries to her.
00:28:56
Speaker
Uh, and there's another aspect of this kind of math is that math is, has an emotional component as well as a useful component. If you're just pure boringness, then we wouldn't have like, I don't know. We wouldn't have the festive notation and festive notation. That's the door case thing I've ever said, but we, we don't, we wouldn't have different stuff like that to go along with mathematics. It would just be a purely boring thing, like accounting.
00:29:22
Speaker
You know, there's nothing purer than seeing somebody just raptured by the joy of mathematics and of counting things. I mean, think of the count from Sesame Street. One, ah, ah, ah. Two, ah, ah. You know, I was just saying. I'm not being sarcastic. Not at all. No, the count is a very important figure, both politically and economically.
00:29:46
Speaker
shift the count as a special correspondent on some news news program on economics or something is kind of empire I think that's implied we're not gonna get sued by by HBO
Towers of Hanoi Explained
00:30:03
Speaker
On this next segment, The Towers of Hanoi, I'm going to read this segment here. First, you take three dowels and set them up right, left to right. Next, you place a series of cylinders with holes in the middle on the dowel farthest to your left.
00:30:20
Speaker
each smaller than the other so that they make some sort of cone shape when taken together. The goal of the Towers of Hanoi puzzle is to move every cylinder from that dowel down to the dowel on the right without ever setting a larger cylinder on a smaller one. I think my son has a game of this. It's like it's like these like red and yellow and green rings right?
00:30:41
Speaker
You know what I mean? Yeah, they're like rings that you put on a peg and you have to move. But the thing is with this rule is that you can never put a larger peg on a smaller one. Okay, so the first strategy that I take here is sorting, obviously. Well, not necessarily sorting because there are only three dowels. Okay, I was wrong.
00:30:57
Speaker
Sorry. I was wrong about being wrong. Sorry. I know the solution to this because I was taught it in a class, but I would like to, I don't know, discuss it.
00:31:14
Speaker
Sure. Okay. Okay. So when I'm thinking about my son here, my son is looking at these three dowels and these colors. And then, you know, I don't know that he's really strategizing at that time yet, but he, he could be, you know, hold on one second. I'm going to do a, we're going to do a quick cut and I'm going to get some equip breaking math equipment.
00:31:33
Speaker
And we're back. So right now in front of Gabriel and Adam, there are four cups, each smaller than the other stacked within one another. Okay. Okay. Now keep the cups and together for now. Okay. Got it. Now put two mats so that you have three, three places where you could put the cars. Sure. Sure. Okay. So let me identify three places. Can we do in front of this?
00:31:59
Speaker
And while they're doing that, I'm going to just explain the altered version of towers of Hanoi puzzle. This is called the cups of Hanoi. And I just made it up. You can never put a larger cup inside of a smaller cup because if you did, you'd be a wizard. And, uh, you only have three places where you could set down the cups.
00:32:16
Speaker
Okay, so the idea is to transfer the whole thing over, is that right? Yeah, to Adam's side with only one place in the middle to kind of like be the interim place for cups to go. Sure, so let's do our thinking process out loud here. So all of our cups here, it's like our cups are arranged like Russian dolls, a smaller one inside of a smaller one, you know, one of those things.
00:32:40
Speaker
I think there's a shot glass inside of a whiskey glass inside of a water glass inside of a solo cup. Yep Yeah, a solo cup is the big one here interesting now. I wonder is the solo cup actually bigger than this like I don't know It's taller. I couldn't find four cups. I was You fail or I fail from not having enough cups. Okay, so we basically the first option We only have two options here
00:33:04
Speaker
We are required we're required to use the smallest cup first and either we put it in front of you or in front of me so so so Adam do you have a strong feeling for whether I should put this first with the shot glass in front of you or In the middle here, honestly, I don't have a strong feeling about that I'm not sure how much it would matter, but it probably matters doesn't it?
00:33:23
Speaker
I'm sure, well, yeah, that's part of the strategy. Oh, it totally matters. Yeah, because then alternatively, so I guess in the interest of time, we're going to just do it, right? Okay. Yeah. Okay. I would assume in the middle. Okay. In the middle, in the middle it is, which means that the next step, we have no choice. Yeah. We have to move.
00:33:40
Speaker
this cup so let me just I'll do a play by play update right now the shot glass is in the middle of Adam and all the other cups the big middles and small cups are in front of Gabriel okay so the next one we have no choice so now we've got the whiskey glass in front of Adam and now we do have a choice we can either put the smaller cup
00:33:59
Speaker
in, uh, the cup in front of Adam or we can put that's really all we can do. We don't have choice. We've got one choice here. Okay. There's not in the first part of the game. You can see there's not very many choices. Yeah, definitely. So now the whiskey glass contains the shot glass. Okay. And there's only one thing that can be done. Now the, uh, medium sized glasses in the middle of the two of them. Okay. This is starting to make sense. Yeah. So, and I have, I had no choice about what I do yet. Cause I don't have any, you know, anywhere available. So we're almost done here.
00:34:29
Speaker
Now, oh, guess what? This final choice here, I have all of the cups. Well, oops, you did an illegal move just there. What did I do? What did I do? I forgot to mention one rule. You're only allowed to move one cup at a time. You can't move stacks of cups. Okay. Well, we can still do this if we move the small one into the biggest and then this one into that one. Dude, I forgot that we could move the small cup into the biggest cup completely. Can we do that? Oh, totally. Yeah. Okay. Okay. Do it. Do it. Okay.
00:34:56
Speaker
Here we go. So this guy goes in here. So the shot glasses in the solo cup. Meanwhile, beads of sweat are starting to appear on my floor. And then the medium can go in here. Good. Can we get a medium glasses in the whiskey glasses in the medium glass? Can we get like a golf clap?
00:35:15
Speaker
Okay. Okay. Okay. Like a sound effects. Okay. Now one glass at a time. Should I move the tiny glass into the middle? Uh, okay. Okay. I know our listeners probably have lost track. So, you know,
00:35:29
Speaker
Oh, yeah, but if you're if you're at home doing this you could just try to home with four things of increasing size All right. So hold up, bud. They again now the solo cup is in front of Adam I just have to make sure the listeners do move going on But our listeners are on the edge of their seat Well, just keep going stuff somewhere. I'm not sure we can do this. Can we do this? I don't yeah, we can do that We can we can okay six the whisk glasses in the solo cup. Yes, and then
00:36:00
Speaker
We could put the shot glass in the middle glass I guess yeah choice or that glass in here Yeah, okay, so so I don't know at what point are we gonna you know stick a fork in this and call it Oh
00:36:14
Speaker
Trust me, I'm looking for certain things. It's getting a little bit more confusing than I thought it would be. Now it's whiskey glass, on Gabriel's side, shot and medium-sized glass in the middle and a solo cup on the right. Well, Gabriel is scratching his chin pensively. What's he gonna do? Okay. I think so. Shot glass in the whiskey glass. All right. Water glass in the solo cup.
00:36:41
Speaker
We're almost there. We are. Now the medium glass is in the solo cup and the shot glass is in the whiskey glass and they're on opposite sides. So obviously we have to do this. Now the shot glass is in the middle. Okay and then whiskey glass in the medium glass. We're almost there. We've got one move left.
00:37:07
Speaker
Now you notice that you had four glasses there, but it took quite a long time yeah Another golf clap or something. I don't know some absolutely yeah I think you just added it. I'll just double your voice
00:37:24
Speaker
Now I noticed that you guys Were freewheeling it a few times You're just kind of doing like things that that that that did seem to work out at the end Yeah, and it only appeared that we were freewheeling. It was all calculated Well, let me show you an easier way of doing the puzzle. Okay? All right. Okay. All right, put the all the cups back on Gabriel's side Okay, here we go. Now what we're gonna do is now
00:37:46
Speaker
Actually note, all I want you to do is take out the tiniest two cups and put them on Gabriel side on Gabriel side. Okay. Just the tiniest two. We're going to reduce the puzzle to only two cups. Okay. Now move the cups to get it to Adam side, uh, with the same rules, just the two cups. So one whiskey glass goes in the middle. I'm not, the shot glass goes in the middle. Okay. Whiskey glass goes on the right shot. Glass goes on the whiskey glass.
00:38:12
Speaker
There we go. That was easy, right? Now, if you, instead of considering that as being moved, now remember the moves again, we're a small glass in the middle, then big glass to the target square, then a small, then small glass in the big glass. Think about it this way. You start by moving all of the glasses in the big glass to the middle.
00:38:37
Speaker
Then you move the one big less to the right square. Then you move all of them from the middle to there. Okay. So you sense an algorithm here. You divide and conquer.
00:38:46
Speaker
Yeah, divide and conquer. So now, so basically stack all the four cups together again. All right. Okay. Or take the modulus and conquer. Watch lesson to episode 35. Please be discreet. Okay. Okay. Now your first task obviously is to move all three cups from the big cup to the middle one. Okay. So middle square. We're going to start by moving the smallest cup all the way over, right? Well, first of all, well, if the first task is to move all three to the middle,
00:39:16
Speaker
Now let's take that task as one task. Oh, got it. Got it. Ah, I see. Now, if you want to move three glasses to the middle, how do you move three glasses to the middle? Ignore the big glass. I got this. I got this. I'm going to move the shot glass into the middle and then I'm going to put the next glass over on the far right. And then I'm going to stack the small cup in Gideon's cup.
00:39:37
Speaker
And then I'm going to put the big cup in the middle. And then I'm going to move the smaller cup and the smallest cup, the Russian doll thing here in, into the middle. So now all three are in the middle. Yeah. But you move two glasses at the same time, but you can see how that could be done in three moves. Oh, yeah. We'll just have to do one after the other there.
00:39:57
Speaker
Yeah, you have to move the small cup to the so the shot glass to the solo cup. Okay, then do all that But basically you just did three moves and I cheated hang on. Yeah. Yeah, I'll do it right here You know just so that we know that I'm not a cheater. Okay there Okay, and of course and now you move the shot glass to the middle Yes, and look you're on step two of the algorithm. You just moved three from one to the middle. Yeah, so the next step is obvious, right? Move the big one. Okay, cool. Okay
00:40:23
Speaker
I feel like we did a lot of extra steps when we did this the first time. So now the next step is to move the three glasses in the middle to the one on the right, right? So how do you move the two in the middle to the one on Gabriel's side? Because remember, that's the middle step.
00:40:43
Speaker
I would, I would put the tiny shot glass in a solo cup. Yep. Okay. Cool. Yay. And then I move the, the whiskey glass over here and then I'd move the tiny glass, um, to the whiskey glass. Yeah. Or would I do we know we don't, oh yeah. And then we put this one in here. Yep. Cool. There. And then the last three steps are obvious. One, two,
00:41:11
Speaker
I love that sound. Don't you just love that sound. That's a great sound. Now what's interesting about this is that it's proven that this takes exponential time. And the kind of algorithm we just use is a called recursive one. But if we had 64 cups, for example, this would take much longer than the universe has been around to do. Oh, wow. Oh, wow. Jeez. That's a whole lot of cup moving. Wow. We aced that, dude. High five. Yeah. What else you got for us? What else? What else you got?
00:41:42
Speaker
Large numbers. That's it, these large numbers? Yeah. Like 30. That's pretty large. Imagine having 30 houses. It is a large number. It's a large number of houses. Yeah, exactly. But a number that's actually larger than 30, you'd probably agree, is 100, right? I would agree. I would agree with that, yes. And that only has two zeros after it.
Exploring Large Numbers
00:42:06
Speaker
And as many of you know, Google, G-O-O-G-O-L, has how many zeros after it? 100.
00:42:11
Speaker
Yeah. 100 zeros. Okay. That's what a Google is. What number has a Google zeros? That is called a Googleplex. Oh. A Googleplex. I didn't know that. Yeah. It's next on our outline. Every eighth grader's like, yeah, I knew that. So.
00:42:31
Speaker
ever powerful comeback so you don't even know me but a Google is already so big that it's more than the number of particles in the universe probably or I think it's like maybe a million times less or something
00:42:43
Speaker
A little hand-wavy, but okay. It's around there. But this is, imagine we took every particle in the universe and turned it into a zero, then made a new number of it. That'd be gigantic. That's pretty gigantic. There's just no reference for that. That's unfathomably huge. Dude, this is going to cause an existential crisis among all of our listeners who are trying to fathom that. Well, keep your therapist handy because we're going to talk about Graham's number. Okay.
00:43:08
Speaker
Okay, so basically this is how it's stated on Wikipedia, and then we'll do it in English. Connect each pair of geometric vertices on an n-dimensional hypercube to obtain a complete graph on 2 to the n vertices. Color the edges of this graph either red or blue. What is the smallest value for n for which every such coloring contains at least one single colored complete subgraph on four coplanar vertices?
00:43:33
Speaker
And this is just part of Ramsey theory, which is saying like, I have a shape and how big do the shape has to be for this thing to hold? That's the definition of Graham's number. No, no, Ramsey. Well, that's the definition of Ramsey Graham's number is the thing I read at the first part. And I'm going to simplify that. We're going to read that in like Gordon Ramsey's voice.
00:43:52
Speaker
connect each pair of two metric vertices on an end dimensional hypercube to connect a complete graph. That was funny. But what that means is a hypercube is like, you take two cubes and you connect the corresponding vertices. That's a four dimensional cube. You do it with two four dimensional cubes. It's a five dimensional cube, et cetera. That's a hypercube. We have an app on our website, breakingmathpodcast.com hypercube.html.
00:44:18
Speaker
And basically, what this is saying is how many sorry to answer your question. Yeah, that app is a hypercube app, which we've mentioned before. Yeah. And so it's saying like if we have a hypercube where you connect every corner to every other corner. So like if you if we were a three cube, you know, just a normal cube, it would be like all the corners are connected to all the corners, all the edges are there. OK. But so like let's say we did this with like a hundred dimensional cube or something. I don't some big dimension.
00:44:45
Speaker
And you're asking the question, how many dimensions does this have to be in so that if we colored every line that we drew, red or blue, would there be anywhere a square with an X in it where the square is red or the square is blue? And it turns out that the number of dimensions is very huge that is a lower bat for the number of dimensions. Oh, wow. Okay. And we're going to have to talk about two things to actually talk about how gigantic this huge number is.
00:45:15
Speaker
So addition, addition is just, I don't know what, like two plus three is what? Yeah. Five, obviously. Yeah. I got that. And so two times three would be a repeated addition, right? Two plus two plus two, right? Yes. And then two cubed to two, the three would be two times two times two, right? Yes. Yeah. Which would be basically repeated multiplication, which in multiplication itself is repeated addition. Yep.
00:45:44
Speaker
And so exponentiation is repeated multiplication. Multiplication is repeated addition. But what is beyond exponentiation? Repeated exponentiation. Yeah, and then repeated that. Interesting. So imagine if you have four up arrows. So that's, this is called Knuth's up arrow notation. So one up arrow is exponentiation. Two up arrows is one step beyond exponentiation. So four up arrows is three steps beyond exponentiation.
00:46:12
Speaker
Oh wow. So you could imagine three with four up arrows, three would be a gigantic number, right? Yeah, that would add up pretty fast. It would be like gigantic. It would be much bigger than Google, right? Yes. Wow. That's step one.
00:46:24
Speaker
Step two is to take three, four up arrows, three, and then draw that many up arrows between three and three. And that's step two. Okay. So this is what's beyond exponentiation of exponentiation. Yeah. Now imagine you did that 64 times. Good God. That's Graham's number. So can we like make a bigger number for breaking math? Like a 65 times breaking that's number.
00:46:49
Speaker
You heard it first, breaking math's number is four, with five up arrows to the four, if n equals one, but then four to that many up arrows, four, 65 times. It's bigger than Graham's number. Yeah, take that Graham. Take that Graham in your crackers.
00:47:08
Speaker
Yep. Wow. And why was Graham wanting to do this? Oh, that's how many dimensions the cube would have to have to satisfy that property. That's right. I'm pretty sure that they think that the real number of dimensions is around six or something, but it could be as high as that. That would be a lot of dimensions. You take the count a long time to get that far. It weighs a Google. A Google in one.
00:47:35
Speaker
Especially if he has to laugh between each one. Fortunately, he's immortal. He's a vampire. That's what he does. He's immortal, so he just counts. How long would it take to count to Google? I don't know, like a Google seconds. Oh, that's it. Sorry, that wasn't a good math problem. My bad. Assuming that each laugh, you know, is... Okay.
00:48:08
Speaker
Sanity and awareness are forsaken in favor of the black mathematical arts. As you go forth, shame the unbelievers of the gravity of math by forcing them to move about 64 rings from one tower to the other with not any ring on top of another that's smaller than it. Treble. Fear it. Fear it.
00:48:32
Speaker
I'm Baka. And I'm Gabriel. And this has been Breaking Math. On this episode, we had Adam. And Adam, anything you want to plug? No, not really. Just glad to be here with you guys. Awesome. Very good. And any plugs that we got? Yeah, I had mentioned earlier, actually, that we are about to launch a brand new podcast. We are teaming up. Sorry, I cracked my knuckles as you were saying that. Yeah, I'd like to mention, I had mentioned this earlier, that we are about to launch a brand new podcast in which we had partnered with a local speech and debate coach.
00:49:02
Speaker
It's going to be a podcast all about speech and debate. It's going to have stories of great victories and horrible defeats and about the art of arguing in communication. We are very excited about it. The podcast is called The Soapbox with Trey Smith. You guys should find it on Facebook. We have a lot of updates on Facebook. There's also on Twitter, which is at soapboxpod1 on Twitter. And on Anchor, it's going to be anchor.com.