A Problem Squared - 120 = Dealing Probability and Healing Gullibility

Episode Date: October 27, 2025

🃏 What are the odds of actually winning Morgan’s very specific, very fun card game?🤯 Are people more gullible now? And what are the best ways to persuade someone they are wrong?👻 And, appropriate to the season, there will be some Any Other BoooooooooooooosnessIf you have some ingenious suggestions of what Matt can do in lunar gravity, head to www.aproblemsquared.com and submit your idea as a ‘Solution’Head to our socials to see Matt’s graph plot and stats on the odds of winning the card game, as well as Bec’s beautifully laid out existence proof of the 52 Card Game!Some further reading on The Gullibility Problem:Why are people so gullible? https://www.bbc.co.uk/future/article/20160323-why-are-people-so-incredibly-gullibleResearch paper on political extremism: https://journals.sagepub.com/doi/abs/10.1177/0956797612464058Nobel Disease: https://rationalwiki.org/wiki/Nobel_diseaseAnd some tetrahedron-shaped packaging - or TetraPaks, if you will:TetraPak: https://www.tetrapak.com/solutions/packaging/packages SunnyBoys: https://australianfoodtimeline.com.au/2016-goodbye-sunnyboy/ See Matt on tour!http://standupmaths.com/shows Here’s how to get involved with Matt’s Moon Pi Kickstarter:https://www.kickstarter.com/profile/standupmaths And here’s how to volunteer for Calculate Pi By Hand with Matt: https://forms.gle/w44THpNJ3jWUPqHy6If you’re on Patreon and have a creative Wizard offer to give Bec and Matt, please comment on our pinned post!  If you want to (we’re not forcing anyone) please do leave us a review, share the podcast with a friend, or give us a rating! Please do that. It really helps. Finally, if you want even more from A Problem Squared you can connect with us and other listeners on BlueSky, Twitter, Instagram, and on Discord.

Hello, Emphasis on the hell and oh! Welcome to a problem-scared, I mean squared, the problem-solving podcast, which is a lot like a spooky story in that what you hear might not be 100% scientifically correct. Despite this, some of you will choose to believe us

without questioning our research, while others will struggle to sleep at night, haunted by the inaccuracies, and eventually driven to madness. And by madness, I mean a problemsquared.com where you'll unleash your corrections. Putting the co-host into co-ghost is Matt Barker. A mathematician, YouTuber, and headless horseman who successfully managed to swap his horse for a head. I think it just makes you a headman. And I am Beck Hill, a comedian and writer who died over 100 years ago, but can still be heard podcasting to this very day. Excellent spooky intro.

Thank you. I figured this is coming out around Halloween time. Good work. And on this episode... I've sorted out Morgan's cards. I do fact checking on fact checking. Whoa. And we'll have any other boo.

Nice. I was really hoping you'd say that. I would have kicked off if you didn't. Matt? Beck. How are you? I'm good. I'm good. It's all go. I'm on tour. Yes. You saw the show.

I did. I saw the show and I listened to our last episode after seeing your show and realize that you talk about the Christmas tree lights in. your show, but you do not yet in that particular show demonstrate. Whoa, whoa. And what's great as in the episode, you're like, well, if you haven't seen me do it, you know it didn't work. So, so, so hang on. You saw, you saw show number four. Yep.

Show number one, no Christmas tree. show number two Christmas tree but not controlled by the audience show number three Christmas tree controlled by the audience show number four no Christmas tree

so we did have the Christmas tree assembled pre-show but then we had so many other tech delays we didn't have time to do the next bit of testing and set up for the tree so at approximately an hour

before the audience were due to come in, I made the call to make everyone's lives easier that we were just going to gently wheel that tree back into the wing and no one would ever notice. So that's what we did. In all honesty, I didn't notice. I didn't notice at all. Even though I'd recorded that episode with you,

had I not listened back, I would have had no idea. It did not occur to me until you were talking about it. And I was like, oh yeah, he briefly mentioned it. And then I don't remember where that led to. But it was such a good show that I didn't feel like I was left wanting. I have a backup show. Like the show is still hopefully good if the tree doesn't work because I knew it would take a while to get it working.

So I just revert back to the non-tree version. I still mention the tree. Hopefully everyone has a good time. But I'm still optimistic I will get the tree working. No guarantees, but I'm going to get the tree out of the van and set it up at home. and do a bunch more testing to see if we can iron out the last few bugs. I like this behind the scenes. Why do I live?

Why do I have my method on display all the time? Because it makes you human. I know. And actually it's one of the themes in the show is I don't just secretly do stuff and then only reveal the bits that worked. I'm very committed to the whole process. So if you are listening to this, the London Day is coming up. 1st of December, that's a bit terrifying because it's the Matilda Theatre in London that

it can fit over 1,000 people and we're well over halfway to filling it, but I would love to get it, you know, fall to the literal rafters. So anyone in or near London, 1st of December, please come along, standupmask.com slash shows. Producer Laura will be attending. Producer Laura will be there. Imagine meeting producer Laura. That's pretty special. The only reason I'm not attending is because I am doing a very silly, surreal Christmas show in Brighton being hosted by a silly character version of Chaucer. Of course.

That old excuse. Only people in Brighton are excused from going to my show. I think they could go to your show if they want. They're two very different shows. If you've already seen Matt and you want some very silly alternative Christmas themed. comedy and you're in Brighton come along to mine. But if you want to see Matt, Brighton's not far from London. No, zip on up, coming to the show. It's going to be emotional. That said, there is something else I would like to discuss in our catch-up if that's okay. Please.

Which is a bit of a request for the listeners. While I've been on tour, one thing I've had to do because, you know, life carries on anyway, I had to go and get a ECG scan done. Yes. That's where they attach a bunch of electrodes to you. I'm fine. This is just one of the hurdles to go on that isa parabolic flight where I'm going to attempt to juggle. And so I've now had the final sign-off for all the medical stuff. I'm all, I'm clear to fly. I've got my juggling balls, but it occurs to me, there's things I could do in lunar gravity that haven't occurred to me. So I just thought I should ask our fantastic listeners, because I brought this up in the juggling conversation. Is there anything else I should be doing a lunar gravity? Now I only get it for like a

minute or two at a time. I get multiple passes though. So if people have, I'm still in full atmosphere. I can't do any lunar experiments that require not having an atmosphere because I, you know, I'll be breathing. But in terms of gravity, if people would like to go to the problem posing page at a problem squared.com and pick solution, I would love to do something in lunar gravity suggested by one of our fantastic listeners. So. Ooh. Yeah. I mean, I instantly am like trying to think of things, but none of them are like scientific. It's just that it'd be quite funny.

It'd be quite funny to try and watch you eat porridge or something. Yeah, okay. Okay, let me clarify two things. You have to like make a cup of soup. Yeah, exactly. Okay. So, number one, the main video, I'm going to talk about the maths behind the shape of the flight. So that kind of ticks the serious boxes. So I can kind of mess around for the rest of it. At least that's my theory. Number two, I have to clear with Issa anything I'm taking on the flight. So, and it would be very hard to disguise everything I need to make soup to make it look like regular things I'm taking.

And they probably don't want free floating soup. And also, everything has to be rated for zero G because if there's ever an emergency or a change of course or anything, the moment we deviate from lunar gravity, the policy is to go into zero-g. That's the trajectory we then take. So everything has to be zero-g rated. Wait, hang on. So in emergency, it's going to crash.

That's what they're saying. We just want to make sure it will last just in those moments before you hit the ground. There aren't that many emergencies in life where suddenly switching off gravity isn't going to probably help or at least make things interesting. It does mean I can't take anything with me, which would be a problem. in zero gravity and in double gravity because I can also do things at the bottom of the parabola as such. I want to see you juggle in the additional gravity. I'm definitely going to try. So, although 2G is the point at which there's the maximum risk of motion sickness. So what I'm going to have to do is the first couple, the advice is to remain completely

motionless because your inner ear gets very upset if you're moving around in 2G. But I got permission, if I'm feeling fine, enough of the way through. And I've done all the important stuff. I'm allowed to start trying to do some stuff during 2G. I'm prepared to risk it to see if I can do some intense gravity experiments. But what you're saying is that you might accidentally make soup anyway. I might accidentally make soup and making soup. First of all, under intense gravity and then under very little gravity.

it's probably a worst-case scenario. That's incredible. And what I love about our friendship is that we are very different people. I am so rarely jealous of you because the things that you do are like just like I'm, look, you know me? I'm all for doing a thing. You're going home. I like going above and beyond. I love a challenge.

I've definitely pushed myself. to the limit in many areas. However, I do not have a strong stomach. Like, even you just going to Antarctica on a cruise ship, I was like, no, thank you. Nope, I don't do cold. I don't do boats. Neither of those things. Thank you very much. Oh, I'd hate it. I hate it. So the thought of deliberate, like, I don't like turbulence on flights. Well, yeah. I get nervous flying and I get really bad travel sickness, I couldn't think of a worse thing to do. This is a worst case scenario. Oh, it'd be horrible. But I know that you're going to have a great time and I'm going to get a lot of joy out of watching the footage. I know I would kick myself in the future if I said no.

Like I feel like my commitment to the bit and having the experience. And I'm not questioning your commitment to bits. That's for sure. I have no envy at all at being in your weird Christmas Chaucer thing. What was going on there? You're going to have wonderful time. It's going to be art, people are going to love the show, I'm very happy I'm not involved. Like, we have a similar deal. But it's like this and like being stung by a bullet ant or going on the motorbike or intact, all these things. I'm like, well, I'll kick myself in the future if I don't give it a go now. It's insurance against future regrets. So, we're old. And there's a lot of those things that I would do. But I'm really glad that you're the one being asked to do these things

because it feels like those opportunities are not going to waste. No. And poor producer Nicole, so not producer Laura on the podcast, producer Nicole on my videos, is the backup. Oh, that's great. Because it's such a rigmarole to put these things on. And the word expensive doesn't quite apply because it's, I'm just kind of tacked on to the, a bunch of scientific research and other things. But they don't ever want the opportunity to be wasted because it would be such a shame. It's such a phenomenal thing to happen. So I need to bring a backup person.

So poor producer Nicole has to do all the same medical checks and everything and be ready to fly. On the tiny chance, something means I can't do it. Do you film yourself or do you also have a camera person going with you? I'm not allowed a camera person, but they do allocate a very tall Dutch person who is the official camera person on all of the flights. Do they have to be tall? And Dutch? The Dutch is optional.

The tall is vital. They have to be able to stand on the ground and push against the ceiling with their hand to steady themselves in no gravity. So they do all the flights and they can lock themselves into position because they can always reach opposite sides. Wow. That in itself is a great fact. It's great. And because they've done so many flights, you increase your risk of travel sickness if you're looking through a camera, like looking at a camera. screen looking at everything like your inner ear anything that throws off your sense of

orientation and balance obviously yeah but they've done so many they can happily film all sorts of things and we know in advance they're going to be fine so so you can't bring your own camera person you're allocated a very tall Dutch camera person I love this but again I'm going to attempt to juggle if there's anything listeners would like the very tall Dutch person to film me doing head over to a problem squared dot com let me know Oh, as a brief aside, Matt, while you mentioned the juggling, I got a message from my friend Sonny who did my photos for my poster this year for Amber Fringe and some other great press shots who's a long-time listener.

And Sonny said that he was doing photography at another comedy festival and got chatting to someone there who was a juggler and mentioned one of the facts that they learnt from our previous episode great saying oh juggling is just about throwing more items and you have hands and the person that he was chatting to the juggler was darrell jay carrington who we had on the show who had actually given that fact and he said yeah i was that from beck's podcast he said yeah and he went that's that's that's that's me i said that yeah That's the kind of socially awkward situations. I'm glad we generate. Our first problem comes from Morgan, who went to a problem squared.com.

That is our problem posing page. And they said there is a card game. My family plays. My mother has played her entire life and has never won. My father won the first time he's played and has never played again. I've only won once and played hundreds of times. Morgan then goes on to explain how the game is played, but they end by asking what the odds of winning the game are. They also say, I love the show. I've been a big fan of Matt from Numberphiles slash stand-at maths, calculator unboxings of my comfort videos. Beck has a place in my heart now too. Yay! Oh.

Nothing specific, just a place. I've just got a place there. It's nice. Thanks. I like it. It's comfortable. You're all the best. Blah, blah, blah.

Back, after they were so nice, you just blah-blad over the rest of them. Just blah-blad all over them. Now, the reason I didn't read out the rules to the game is because, Matt, I think you had a plan. Well, I thought maybe, because I think I've got my head around the game, maybe if you want to play a round, I'll talk you through it, and then the listeners will get a sense of how the game works. So I believe you've got a deck of cards. I love this, because I am a practical learner. You would do a learner. Yeah, yeah, yeah, yeah.

Okay, so I have a deck of cards. You have a deck of cards. 52 regular cards, pretty straightforward. So the goal of the game is to deal all the cards out. and at the end have zero cards remaining in front of you because there are certain situations in which you can then throw cards away. So you've got to deal them all down, but then you can get rid of some of them as you go.

And the way you get rid of them is you start by dealing four cards in a row. So just deal four cards. Starting on the left, just come across to the right. What do you mean by dealing? So actually, I use the word deal to mean you're holding a deck of cards where you can't see them and you turn them over one at a time and put them down. that would be my definition of deal and when I say left to right I'm imagining you put one down and then you put another one immediately to its right and then another one immediately to the right of

that one then to the right of that one so you're building up a row of cards as you put them down so I'm going to reveal four cards in a row yes in a row okay I've got the four of hearts yep these seven of clubs the six of spades and the three of spades okay interesting now the game, you always look at cards that are four apart or have two in between them. Because you've got four cards in front of you, you look at the last card you put down, the first card you put down. If they're the same suit, you can remove both of the cards in between them.

Are they the same suit? I don't think they are. No. No. If they're the same value, like if they're both a jack or both a four or something, then you can remove all four cards, including the, beginning in the N1. Right. But they're not the same, are they? No. So what you do now put down a fifth card. So it's the eight of clubs. Eight of clubs. You're now comparing it to the one four before it,

which would be the second card in. Oh, which is the seven of clubs. Matching suit. You can remove the two in between them. So chuck those two out, just in a discard pile. They're dead to. Get out of here. Yeah, exactly. And now slide the N1 down because you've taken those two out. The line collapses down. now it's a bit too early in the game but technically you should check again the end card and the one now i'm going to say three before this is a fence post problem issue there's always the card you're looking at two in between and then the one three before or four before depending on if you count the first one so there are batches of four let's go with that so you always look at the end one and the other end of the batch of four yeah but you've only got three cards left so you can't do

that so you have to put another card down okay and that's the three of clubs and it does not match a suit or number because my other one is four of hearts okay and then put down the next one so then we sort of keep going like that so it's a king of spades which doesn't do anything because you've got to look at the one four before okay then keep going and you just keep adding wow so seven of spades yeah annoyingly not useful ah boo one more so one more nine of clubs okay that's That's the same suit, because four before is three of clubs. So you can take the two art in between and collapse it down. Okay.

And then check again, the end one, double check four before it as well. It matches suits. Then get rid of those two. Oh, this is very pleasing. It is pleasing, isn't it? Yeah, it's got a sort of solitaireish feel about it. Yeah, when you get several suits in a row that all just collapsed down and down and down and down, a very long row of cards can get very small very quickly.

Very satisfying. So I'll do another fourth one. And then you just keep going. And you just keep doing that. Yeah. Okay. So I've done the five of clubs, which doesn't match my first one. Four of spades, which doesn't match the four before it.

Five of hearts, which doesn't match the four before it. Four of clubs, which matches the five of clubs for before it. So I'll take out those two middle cards, collapse it down. I've just had another two collapse one. Because you get the double collapses. Having a number match is good because all four go. I'm playing while you're talking. You should be.

When the values match or four of them go, but then it can't cascade because it can't carry on. Because you've just taken all four off the end. Yeah. You still going? Yep. So the question now is, when you run out of cards to deal,

how many will still be left in your row? Oh, another double down. It's very satisfying. It's kind of a fun game. I'm off the table, guys. How long can I get? We're running out of the table. Okay, it's now going on to the couch.

Uh-oh. One, two, three, four, five, six, seven, eight, nine, ten. Eleven. But I finally had a matching suit. Oh, so I got matching, so I got two queens. Like I got a queen and then the four before it is another queen. So you take all four of them out, including the queens. Lovely.

Look at them going. Okay. So I was lucky with my very last play to get a king that matched a king four before it, which meant I could take all four out. But I still have four remaining cards and I've run out of cards to deal. Oh, so you're down to four. That's pretty good. Now, the only way you can end with zero cards is if

you happen to have dealt the very last one was the matching number of the very first one that's still on the table there and you got pretty close right but i guess you cancelled down to the final four and then it didn't collapse any further yes that's correct so when i did the dice previously about rolling snake eyes yeah i worked it out the long way just like thinking about the probabilities and doing the calculations for some of the easy situations but then for the more complicated ones i just wrote some code to simulate it because I was too lazy to do it the long way. In this case, I don't think it would be possible to do it the long way. Like the exact probabilities would be so difficult to calculate. I don't think it would be achievable. If we can just park that thought

for one second, I was going to say achievable by us. By us. I were potentially by anyone. So the other thing I want to flag up is, this game is entirely deterministic. If you're given a deck of cards, it's already decided what's going to happen. Because there's just that many cards in that order. You're turning them over. As long as you don't make a mistake following the rules, for the same starting deck, you'll always get the same outcome. There's no decisions for you to make. Yeah.

The only brain power that's going into it is making sure that I'm remembering to check the right card. Correct. So there would be a way to analyze this analytically, to do it exact, because there's only a finite number of ways the Decker cards can start. You could just check them all. Or you could do some kind of very complicated network of ways the game can change. But the number of possible starting positions is so big. Now, it will come down a little bit because if you think about when you play a game, when you got rid of two cards in between it doesn't matter which way around they were so subtly different decks would have equivalent games because the ones you throw out in fact the ones you throw out right could be of any order you could totally shuffle the throwouts because they have no impact on the game other than being removed and potentially not have caused a match along the way so you can't have every single possible shuffle but there's still a lot of equivalent decks that you could get rid of it's making me think of the word or

problem. Yes. For any new listeners, this is when Matt looked into whether you could, in five guesses, guess five completely different words with different letters. Yes. And in that one, I use the word space to mean like the collection of. Like I say the space of possibilities. I mean like the collection of all possibilities. Yeah. The collection of all possibilities for word all was huge, but I figured it was just small enough that I could check them all, which is why my code took a month. Now, other people did that in a much more clever ways. They did some clever shortcuts. They still got the correct answer, but they didn't just naively check every single possible one like I did, because I'm a doofus.

But it still happened within a month. This version, the naive way of doing it, I suspect, is like approaching the heat death of the universe. I suspect there's so many combinations you can't do it that way. Because when you were talking about throwing out certain cards, that made me think, the reason I thought of the The wordal problem is because you'll sometimes get words that use the same letters but in a different order and that might add to the number of combinations you could have of different words, but it also doesn't affect it. Yes, correct, correct.

And it was easy in wordle to get rid of words that are anagrams of each other, but you'd have to have a bunch of thinking about removing decks that were equivalent games to each other. Like that's a slightly more difficult thing. It's doable. But yeah. And the question is, how? How much would you reduce the space of possible, like the collection of possible games? Would it then become small enough to be searchable?

Would it still be too big? The intermediate step is there might be a very clever way to do it. I say analytically, like properly, like work it out exactly. Here, I'm not sure if there is a faster way than the terrible way. Which is a long way to say, I just wrote some code to play 10 million games and see what happened. So that's what I did. Do you want to see, do you want to see a graph? I can show you a plot.

I do. While you do that, I'm going to play another game. I think I'm going to be very similar to Morgan's father, where if I do win, I will never play again. Smart. Okay, I've shared a plot with you. Ooh. Thank you.

That's the great response when I show you a plot. so what I've done is I ran 10 million games and for every single game because it's deterministic I just take the deck and convert it to how many cards are left afterwards I then just keep track of how many cards are left and have like a running tally of like how many times there were 10 cards left how many times there were 12 cards left you only get even number of cards remaining because you always remove two or four and you've always got you start with four It's always an even value, which you can see in the plot, and we'll share this on our social media. If people want to check it out, you can see on the plot, I've left in where the odds would have been,

so you kind of get a stripy plot because there's only a bar for all the even values, including zero as an even value. Spoiler. Now, you can see the plot doesn't quite reach 10%. And that's because there's a whole bunch in the middle, which are all around about 10%. there's a 9.3% chance you end up with 14 cards there's a 9.9% chance you've got 12 cards left there's a 10% chance you've got 10 cards left that's very pleasing and so um in general most of the time you're going to end up with around 10 10 to 14 in their cards left is it possible to arrange this graph but it by order of frequency only yeah I can sort that

You know what? I can sort by frequency. Because what I've noticed here is that there appears to be more of a chance that you'll end on zero than there is that you will end on 40. Correct. Excellent observation. So some games do end in zero, which we knew because Morgan's dad has won the game and Morgan's won at once. So I've just given you the rankings of the likelihood of being left with a certain number of cards. And coming in, top of the top number one, most likely is 10 cards, followed by 12, 8, 6, 14, 16, 4 all the way down. That was mine.

The 11th most common is two cards left. Ooh, that's got a sting. And then you are absolutely right. The 17th most likely outcome with a probability of 0.71% is zero cards. It's higher than I expected. It's about 2,000 away down the leaderboard. the following games are all less likely 34 cards 36 38 40 42 44 44 46 48 and 50 now interestingly my simulation never came back with all 52 cards dealt out I think that is possible whenever I'm researching something or

investigating I'll have like a little you know bonus questions or things that are you know little distractions or tangents I could go on And one of the things I was like, that's interesting. In the 10 million games, none of them ended at 52 cards. I suspect that's because it's incredibly unlikely. But I do think it's possible. I reckon you could manually arrange a deck of cards. Like, imagine just putting out the row all 52 in front of you. It's such that none of them can cancel.

Is there a way to mathematically prove that? Oh, I think you just do what's called an existence proof. You just do it. You would have to rotate. every fourth card would have to be a different suit. So you'd club's heart spays, diamonds, clubs, heart spays, dime, all the way down those. And then again, for the offsets. So I'm pretty sure you're going to arrange that.

And then you just got to make sure you've shuffled the values around such that no two of them are four apart. I'm pretty sure that's possible. I reckon you could come up with an algorithm that would generate the ones that would do that. I didn't end up going down that tangent, but I'm almost certain if you wanted to be a jerk, you could pre-arrange a deck of cards such that none of them would ever cancel the entire time you're playing this game. Yeah, you do know that that's exactly what I want to do now. Well, there you go.

That's your open question, Beck. You can report back, see if you can get that to work. All right, I'm on it. Don't expect me to do anything else for the rest of this podcast. No, once this podcast is done. Yeah, so don't work on that now. There are other tangents I didn't take because they went outside what Morgan asked. but I'm going to leave them dangling in case some listeners want to pursue them.

So, to answer Morgan's exact question, you will end up with zero cards left approximately 0.71% of the time, which is roughly one in every 141 games, which kind of checks out. Morgan said they've played it hundreds of times. it's only happened once and it happens roughly every 141 games it does make it seem very unlikely Morgan's mom

I guess they're in the US Morgan's mother has played this her whole life and has never happened because if you've played hundreds of times you should have had one should have had one match the fact that

Morgan's dad got it the very first time at just under 1% that's plausible I mean one in every 141 people will have it happen the very first time they ever play. So that's very plausible. I mean, to make sure my code was working correctly, I got it to spit out a bunch of setups that resulted in a zero card remaining game. And then I manually stepped through them. I set the deck up in that order

and then I played the game to make sure it worked because I wanted to verify my code was playing the game correctly. So I then wanted to play one of the zero games. And I played some non-zero games as well to make sure it all matched. And then it occurred to me that I can now generate a whole bunch of set-up decks where this works. So there's two things to bear in mind here. First of all is, if Morgan wanted to play a very funny practical joke on their mother. Now, I would not suggest that you set the deck up such that your mother wins,

because I feel like that's mean. to make her think she finally did it and you rigged it is not I don't think what depends on your family relationship that could be argued to be not very nice sitting the decks up so everyone else apart from your mum keeps winning all the time now that's hilarious so I would say set up a bunch of decks that are all guaranteed to win and then your mom's not going to be looking for a you know magic trick so just pretend to shuffle and swap out a different deck and then everyone else in the family have a zero card game in a row

maybe two and then reveal to your mum that you've got a whole bunch of ones that work and you're just, you're being particularly sneaky. I would actually say the more important point is does your dad do any programming? Because if that was me, I would have done exactly what I've just done,

set the deck up, done at once, one and refuse to ever do it again. that's the funniest move for someone who knows how to code and then find it or maybe if your dad doesn't program but it's still like Beck easily distracted to try and do something because I can see Beck is already trying to arrange a deck in such a way that it won't collapse I don't think I haven't noticed you told me not to and I'm doing it yep so the question is Morgan a A is your dad a mat in which case they probably wrote some code and then just did it once to refuse to it again. B, is your dad a Beck? Because you probably could work out a zero

collapse game if you spent long enough reverse engineering what the deck has to do. So I also think that's possible. Or C, is your dad a perfectly normal human who happened to get very lucky? That's possible. I've currently got 17 cards as I'm going. I think I've got a way of doing it, but we'll find out. I believe in you. But to literally answer Morgan's question, 0.71% and in fact it's harder to have like the real challenges to play a game such that you've got as many cars left as possible. That's a harder game. If you look at the distribution, it's got a long tail to the right. Like the main peak is off to the left towards lower numbers. It's actually easier to have fewer left or indeed zero than it is to have lots left. That's the

skinny tail. So if you really want to make it a challenge, it's to not have them cancel out, see how long you can go. The other thing I want to say to listeners who have not been nerd sniped by this, can you arrange a deck such that takes 52? When I ran the simulations and I got the probability back for it collapsing all the way down, and I looked at the number, I'm like, that's odd. 0.007705. I'm like, that's, I recognize those digits. And then when I did the inverse to work out how many games you'd have to play on average to have a zero card game and it came out to be 141.4. I'm like, I recognize that. I'm like, wait a minute. Those are the digits of the square root of two. Pretty much exactly

the probability of having a zero-sum game is one over root two. Huh. It's terrifyingly close. Like, I suspect it's exactly one over root two. Why on earth with the probability of playing this game and winning be one over Route 2. There must be a reason. That can't be a... Could it be a coincidence? So that's my return challenge to our listeners.

Why on Earth is the probability one over root 2? Is it just a bigger cosmic coincidence than Morgan's dad winning the first time? Or has the universe stack the deck on us? Is there a reason for this? Oh, imagine if this was the... the key. Imagine if like this was our 42 moment, life, the universe and everything. This was like the film contact. Everything is unraveled. Yeah, exactly. Yeah, yeah, yeah.

In the digits of this ridiculous card game. Well, Matt, I think you've answered that not only perfectly well. I love a graph. I love a visual representation. I'm very satisfied with that answer. So from me, I'm going to give you a ding. And I did it. She did it. Let it be known. That was just based on pure instinct. That's how you approach all problems in your life. I didn't have to change tactic. I knew that at one point I would probably hit a hiccup and I was able to easily overcome it. I'm very proud of myself. I'm going to send a photo. It is like a metaphor for life. For you? Yeah, it really is. I'm going to get you guys to check it for me though because I am quite prone to the uh look i'm a bit of a parker square person myself oh come on let's stop

turn to re contextualize your success levels oh wow that is a great photo back thanks we'll obviously put this photo up on socials this is a good system you've done opposite colors as well which makes it easier to track if it's the same or not visually yeah yeah really nice back excellent work not that i doubted you other than that win time where I was doubting you. Yeah, I think you've done it. Yeah, it's lovely and systematic. It's a little bit, you mix it up a bit towards the end there, but it works.

Well, you think it does, but actually, so what happens, for the listeners, I'm going to explain what my thing was, is I did ace to three hearts, and then obviously I couldn't keep on with a suit, otherwise you'd have a matching suit, and I couldn't do like ace of a different suit because then the aces would match. So then I went two to four of spades, three to five of diamonds, four to six of clubs, five to seven of hearts, et cetera. And then I kept going until I had a little bit of a rollover moment when I got to queen, king, and ace of clubs. And then obviously the next one is starting with the hearts in my order of going via suit. But then I hit the king of hearts.

and then I'd already used the ace and the two previously. So then I went, oh, I haven't used the ace of spades yet. I haven't used the second of diamonds yet. So if you look, it looks a bit weird, but actually it's counting up one by one, which means you get no repeat of the numbers and the suits keep into changing. So then I'm just using basically the remaining cards one at a time,

and it's just counting sequentially up. Yep. So then it's ace of spades, two of diamonds, three of clubs, four of hearts, five of spades, etc. Yeah, they're all little batches of three and they're just incrementing up the whole way around. Excellent work back. You have completed an existence proof of a length 52 game. The rarest of all the games, I'm very prepared to say. I wonder if that's the only solution or if there are other ways.

I predict there might be other ways. I mean, your one will be a whole family of equivalent solution. because you could swap values and suits all over the place. I didn't need to do it in those suits and those orders, yeah. Yeah. But actually, maybe if you included all value swaps, maybe you have found. Maybe it's just the one family. Wow.

Okay, I don't want people getting distracted by that. I want the root two thing solved. And if this spare time, you can look into the rest of this. Is it linked? That would be very satisfying. Oh. Well, obviously, I gave you a. ding for the way that you solve that map, but we'd like to hear back from Morgan as to whether

you ding this, and we'd like to hear from any listeners who have decided to have a crack at the problem that matters since posed. And it can do that by going to a problemsquare.com and selecting solution in the drop-down box. Our next problem was sent in by Alyssa, who went to the problem posing page at a problemsquare.com and type the following words. Hello. I have noticed gullible people in my life getting more gullible lately. I don't know if I believe that. However, Alyssa would like to know what else from clever ways to encourage introspection

without anyone realizing that I think they are gullible. Alyssa would at least like to convince people that, yes, some people are brave enough to lie both on TV and on the internet. Do you think people would just go on the internet and lie? Is that a thing? Surely not. that's me being gullible. Beck, what do you go for us?

I liked this question because I think, yes, not only is misinformation an issue and media literacy is a very, you know, important thing. And so I thought not only is it interesting because of that, but because it is something that no matter how much we think it affects others, it does also affect ourselves. Oh, yeah. So I looked into why we believe things even. even if they've been proven otherwise and found some really fascinating things. First of all, have you heard of Nobel disease? I'm now on high alert because I think you're going to make one of these up and then point out how gullible I am that I believed you.

It was a real thing. Oh, you're putting too much weight in my ability to organise. That's what you would say. Is this where people who win a Nobel Prize will then say some really dumb stuff in a different subject area? A hundred percent. Sometimes not even a different subject area. Sometimes it's kind of along the same lines. Everyone listening who's won a Nobel Prize is now really nervous. Oh no, her back doesn't pick on me. For instance, Louis J. Ignaro, who got a Nobel Prize in Physiology or Medicine in 1998, also had a contract with Herbalife. Oh.

Which is one of those multi-level marketing schemes. shifting them herbs yeah there's no evidence to say that you're more likely to support suedo science or something like that if you win a Nobel Prize it's it's nothing to do with that it's just interesting how many then also went on to support other things one of the things that they pointed out was when by the time that a Nobel Prize is handed out it it can often be decades after the original discovery or invention and by that to time, you know, that person is a little bit older and we have found that with age... There's some cognitive decline, yeah.

Or we just get more set in very specific ways. We don't question things as much as time is long and it's a lot of brain power to do those things. We've been doing this podcast for over five years now and I'm pretty sure my brain slowed down in that time. I don't think that's a causal link. I think that's just the passing of time. Yeah, yeah, exactly.

fortunately those of our listeners who have been listening along with us this entire time have also declined at the same rate yeah exactly so they can't deceive a difference yeah yeah we're all in this together guys we're going down with the ship it's also obviously being stated that it could that if they win an award it could bolster their own confirmation bias because they've now been proven that they know a lot in a very particular area and so they're less inclined to do as much research despite the fact that, you know, in order to win, you should be able to show that you've gone about a very scientific way of doing it. People seem to be more gullible doesn't

necessarily mean that they are less intelligent. It just means that there are certain factors that might play into it. I don't necessarily think intelligence and gullibility. If we're using gullibility in a very general kind of fallacies of the human mind sense, I don't think there's necessarily that much of a correlation. Yes. I will also flag up, and this is kind of parallel to what we're talking about, what I'm going to call the Morgan's Dad effect, where one in every 141 people will just win that game the first time.

And you could then read a lot into it, but there are things that are vanishingly unlikely, but they will still happen at random. And as far as you know, you're not aware of the million other. people who it didn't happen to when they shuffled or rolled a thing or stepped out or whatever. But throughout your life, things that are incredibly unlikely will happen to you because just you're going to do a lot of things in your life. Occasionally, they're randomly going to happen. But it's very easy to then read too much meaning into that because it's hard to remember, or at least to factor in the many, many other things you or other people did that didn't lead to that

because it's all happening outside your awareness at the time. Well, it's interesting to say that because, yeah, you hear more about those things because they are interesting because it's the thing you weren't expecting. But when you hear more about something like that, you start to think, well, it must happen more often just because you know about it. Yeah. You know, if you were to know about every time someone has played that and not one first time, then you would be less like, it wouldn't stand out as much, but because, you know, we only heard about a few people playing that game and of that few people, Morgan's dad won first time, that's going to subconsciously make you think, oh, it's like a one in four chance. You know, like, you're going to think, oh, it's much higher chance than it is. I got into a discussion with, and I'm going to try and

anonymize this. People try to second guess how the YouTube algorithm works and things it does, and someone got pretty convinced they'd found like not a conspiracy but they'd found something in it and they're like and these other people all told me they had the same thing and I'm trying to say yes but if you ask a lot of people has this happened to you the people for whom the answer is boring and no are way less likely to reply than the people for whom the answer is interesting and yes and I'm like yeah I'm like it's still good to collect lots of data and you should do that but you've got to bear in mind if you've got a selection bias

so you're not necessarily getting all the results positive and negative if you're more likely to get positive than negative you've got to factor that in. Yeah, yeah, exactly. I'm going to put a pin in this because I want to circle back to this when I fully answer

I don't believe you. Alyssa's problem. But I wanted to mention one more thing. Have you heard of cognitive misers? cognitive misers as in like you're stingy with your cognitive load or I'm guessing I have no idea exactly that so most people are cognitive misers oh love it we save time and energy by using intuition rather than analysis so because we're cognitive misers it means that we sometimes will overlook stuff so I found an article which asks these two questions and you're

encourage to answer these questions as quickly as possible. Okay, I'm ready. Snap decision. I'm ready. How many animals of each kind did Moses take on the ark? Two. Some were seven. And Margaret Thatcher was a president of what country?

She was the prime minister of the UK. Okay, so correct. I had the same reaction to these two questions, by the way. I had the same thing where I went, she wasn't president, she was prime minister. Okay, good, good. Which is the point. Oh, really? Did you see any problem with the first question?

How many of each type of animal? I mean, I'm assuming you're referring to the story. Did I skip to some kind of assumption? How many animals of each kind did Moses take on the ark? How many animals of each kind? I said two. Okay, so this is called the Moses illusion. Right.

Because Moses did not build the ark. Oh, right. Oh, all those old Bible guys look the same to me. Right. And the thing is, is like, we're not, now, that's not say, oh, you're wrong, you're an idiot, you've been tricked or whatever. It's pointing out what we've done is we know what question is being asked. So the fact that there is something a little amiss there, our brain is overlooking it because we're going, well, that's not

important. We understand the question is how many animals of each kind. So we're focusing on the question. We're not thinking about the specificities. Even if you knew immediately, like you did with Margaret Thatcher and you went, well, she's prime minister, but also you answered the question. because you knew that was the question. But Moses didn't jar. Yeah. Moses didn't jar, but at the same time, that's not really important in this situation. So your brain kind of did the right thing.

It was focusing on the right thing. You can't have all processes going at once because you'll burn out very quickly. But it's an example as to why sometimes without even realizing it, you will miss certain facts or things because you are making assumptions about what the answer should be. and it might mean that you're missing other things that will help you denote whether the rest of it has been affected by that, so to speak. Look, our brains are just wet assumption-making computers. Yeah.

And that's what I write in every anniversary card I give Lucy. Yeah. And that's the key to a solid marriage. Exactly. From one wet assumption-making computer to another. now cognitive fluency is another reason why we tend to believe things this very much goes back in line with what we're saying before that confirmation bias because if we hear something enough and if it nicely bridges a gap that we had in our brains then our brain goes oh that piece fits perfectly now it all makes sense and we'll put it in there because of that we're far less likely to change our minds because it ties everything together really nicely. And in another article I was reading, they put it really well where they said, and this

comes now down to answering the problem of how to change people's minds or get them to think more carefully about what they're reading and believing. If you take out a page from someone's book, from a bound book, it then affects the integrity of the rest of the pages that have been bound in there because the way that the pages are connected and bound together. So if you take out one page, it might be

the wrong page, but then the corresponding page might fall out and then that means it gets loose. The other pages start falling out. Your whole concept of what everything is built upon can start falling apart and that is terrifying. And we need all of the things that we have to

leave my book alone. We need to build this reality in order to to feel sane. And as soon as we start picking away at it, that can very quickly and easily feel like madness and a way of sort of descending out of reality. And so our brains to protect ourselves hold on to certain things because we just don't want to lose our footing on what we understand life to be. So if you are trying to talk to someone who has believed something on the internet, that you believe not to be true. Bearing in mind, we're all susceptible to this.

So, Alyssa, you might think that people are being very gullible, but it's very possible that you are also being gullible. No one of us doesn't have some incorrect assumption or a bit of false knowledge in our brain somewhere. Yeah. Because honestly, the smartest people I know are the people who can sometimes be the most gullible. Matt, you and Lucy, I have flat out told lies to, I haven't during this.

I have not had the brain power to do it. But you and Lucy, I have flat out told lies to as a joke and you'll go, really? And I go, no, and laugh. And then you guys are. Occasionally, we'll look at each other and I want to say, you know, for so-called smart people, we're not very intelligent. Just, oh my goodness. But also, that's because you trust me that I might have a piece of information that

sounds a bit weird, but you're like, oh, that probably is true because Beck goes down rabbit holes and probably has researched that a lot. Okay, now it's true. On one hand, yes, you are a terrible friend. But the more important thing we're trying to get here is Lucy and I are in professions that people perceive to be linked to intelligence and we're both visibly good at maths and physics correspondingly, but yet we are absolute dupuses in every other regard. I think it's linked. And I actually, this is why I think more people should have some level of gullibility. I'm going to say not gullibility. I'm going to put it as trust. I think people need to be more open-minded. And I guess that's the other thing. Open-minded can also include

skepticism. But the fact that you are so open-minded to hear new things, you will change your mind. You know, you are open to changing your mind and it not affecting your sense of reality of it not becoming your undoing. You don't attach it. to your personality or your mental well-being, you just want to know the truth. And so you're much more open to facts or things that might go against your belief. And in fact, that was the thing that they noticed with the so-called Nobel disease was they said a lot of those people are used to thinking against the grain.

They discovered things and invented things because they were able to think outside of the box. So it makes sense that if that proved itself to be true, perhaps other things outside of the box could also be true. And so I think it's in some ways very noble when people are sort of open to hearing things like that. However, it's very important to balance it out with a level of skepticism and research to double check if it is actually correct. And I think we're all guilty of that. We are all guilty of not doing that. We're all guilty of being cognitive misers.

And I think this is what it comes down to. and I thought that this was very interesting. There was a paper released in 2013 from the University of Colorado that was titled Political Extremism is supported by an illusion of understanding. And during that research, what they did is they got a very large group of people

and split those groups into two. And the first group, they got them to talk about their sort of extreme political views and explain why they believe those views. And then they looked into it a bit more to work out how open they were to questioning those views after being asked to explain them and why they believe in them and stuff. The second group were asked about their extreme political views and then asked to explain how those policies work.

So not why do you believe in them, but to try and explain how you would understand the policy. So someone could easily argue why they think it is. They could say, oh, if you implement this policy, this will be the outcome. But if you say to them, okay, what is the process? Like how does that policy come into effect? And basically what they found is a lot of people, well, pretty much everyone, feels that they have a better understanding of something through familiarity. But you're saying that being familiar with something can give you the illusion of understanding something. Yeah.

It can be very easy to believe something because it feels familiar and you feel you know about it. If you were trying to encourage someone to do further reading, rather than sort of pointing out your wrong or, you know, that website is questionable or whatever. Ask them to explain it in more detail to you. And what they found is that when people were asked to explain something more, they did start to realize, well, I don't actually. actually know as much as I thought I knew about this. And they became more moderate in their beliefs. They became a lot more not necessarily letting go of those beliefs, but they became a lot more likely to hear other beliefs without reacting as strongly or putting it off, especially if you are able to explain how something works. A really good example was, again, this is about

taking the page out of the book and it, you know, affecting the integrity of the entire binding of the pages. It helps if you can replace that page. So if someone says, oh, I read this article, The Moon is Made of Cheese. And you said, oh, I thought something else. But like, how big was that count? How does that work? Yeah. How does that work? How did the cheese get there? And I'm like, oh, I don't know. It just is. Like, I was reading this article and it is. Okay. How did we work out? it was cheese. Yeah, yeah, what's the process there? But you could explain why it's rock and how it became rock, then that is offering something to fill in that gap so that the integrity of the book is less likely to fall apart. Now, obviously, what you want to do is make sure that you are filling

that in with something correct. So that's why it is also up to you to know what you're talking about. I think the key here is that we all need to be much more honest about the ways in which we are all gullible, it's much easier for someone to admit that they've been gullible when you have already admitted that you have been gullible. If you are more willing to say, I don't understand this thing, it's okay for you and your friend to both say, do you know what? I actually don't know what the moon is made of. Let's look it up together. I think it might be this. You think it might be this, but neither of us can explain why. I think it's rock. You think it's cheese. But neither of us can explain how it got there. Why don't we look it up? Why don't we do some research?

Why don't we try and find out together? I think there's some really nice ways of going, I think we just see things as a fight all too often. Rather than trying to disprove something by mentioning the thing, this is where you can get back into the cognitive fluency where if you keep repeating the thing, even if it's wrong, people will start to believe that thing anyway, just because they've heard it more. I hope that that helps. I would say, first of all, the moon is clearly made of potato. Secondly, I suspect your double prong of stay humble, bear in mind, you definitely have misconceptions and mistakes in your own life.

But it's better to engage people to discuss what they believe and why than it is telling they're wrong. It feels like an actionable bit of, it's not the whole answer, obviously. even if we've not fixed misinformation in our modern online media age. But it feels like that's a nice actionable bit of advice that we can take back to Alyssa. And Alyssa can decide if they believe you or not and report back. So I don't want to ding you out of the gates, but I feel like, you know, you've convinced me to try these approaches. So I reckon we'll go back to Alyssa.

Lisa, give us a ding or otherwise. Over at the problem posing page at a problemsquare.com. Where everything on that page is factual. Yeah. And if you found any inaccuracies in what I just said, then that was intentional. And now it's time for any other Bob for Applesness. So we have a theme to this show's blobness, which is when I talked about the frozen treats. And discussion about the name of that shape, and many, many people got in touch.

And good old producer Laura has pulled out some representative things people sent in. And we're going to open with the poster child for something a lot of people mentioned is Rudd, R-U-D, who loves the show, blah, blah, blah. They wanted to point out that the packaging with the tetrahedral shape is the original packing shape invented by tetrapak. Makes sense. Hence tetrahedron, tetra pack, named after the shape. The Swedish packaging company. And that's where the name tetrapak came from, the tetrahedron.

And a lot of people pointed this out. So thank you so much to everyone who wrote in to say the fact that it's a tetrahedron is why tetrapak is called tetrapak. And that's where the shape originally came from. The reason we picked Rudd to represent everyone who said that was they went to one of the tetrap factories on a school trip way back in the mid-90s and that's where they were told that the reason for this packaging shape is it's easy to make because you're sealing in alternate orthogonal directions and has a very good surface to volume ratio so they are less packaging

for the same way it's no sphere but less packaging for the same amount of actual product inside there so well done thank you for everyone who pointed out the link to Tetrapak the company. Yes. We also had a lot of fellow Aussies who were furious with us for not recognizing the shape from sunny boys, which have the same shape. But I did have a look at the packaging and it did send me down nostalgia lane. Maybe they weren't in WA or maybe I just totally missed them because I haven't got,

there's no sunny boy corner of my brain. I feel nothing looking, gazing. upon the Sunny Boy Pack. I remember them and they have a very pleasing sort of, or at least back in the day had a very pleasing sort of nostalgic logo. I do remember, I never had them because I was more of a, we call them icy poles, you know, there's like, it's just a plastic stick of syrupy water. That was, that was, I either had those or I was like a proper, give me an ice cream. Did you have a favorite icy pole or ice cream growing up? My favorite was the drumstick that's like proper ice cream though

basically a cornetto yeah it's a cornetto Peter pointed out that as a Victorian primary school student in the 60s they got free school milk in the same packaging often left in a steel shelter

in the sun by the delivery drivers oh Thomas got in touch listening to the same episode said it was brilliant blah blah blah said they saw me in Edinburgh one one okay here we are

they want to point out first of all that PG tips did a pyramid-shaped tea bag and they've correctly used the past tense the entire maths community was sad well at least I was when they stopped doing their tetrahedra

well it was the same packaging the stamp stamp orthogonal stamp packaging and at the time the tea bag the tetrahedral shape was advertised because it had more volume there's more space for the tea to infuse compared to a regular tea bag whether or that's true I don't know

that is what they claimed back in the day but then they stopped doing it. So, I mean, but that's the same property that we were just discussing in terms of Tetrapak, the classic volume to surface area thing. And I guess most tea bags now are flat, whereas this is an inherently 3D shape. And we also heard from Daniel

regarding the group of shapes with cross-sectional perimeters. May I suggest the name Stectagulls? I feel it recognizes Dr. Katie's contribution as well as Beck's infallible notion that all shapes are essentially rectangles. Yeah, I don't know. We'll have to check with Katie,

but it is a funny word. I'm happy with it. It's very pleasing. That's how naming shapes works. And as we come to the end of this terrifying tale, we would like to thank all of you for listening. And especially those of you who support us financially and allow this podcast to happen. If you were unaware, we have a Patreon, which is patreon.com forward slash a problem squared.

You can support us, choose your tier level, you'll get all sorts of bonusy things, as well as an opportunity to potentially have your name mispronounced at the end of an episode. That's right. We choose three names at random from our Patreon supporters. And on this episode, we would like to thank. Yon, Neof, Larty. Elie. Noor.

Nore like this. Nortatat. He reads. two, foe re. When I say bo, selecta. Finally, I would like to thank my co-host, Matt Parker, who has been doing this entire podcast episode via Ouija board. That's true. It's taken a very long time.

It's taken a very long time. But thankfully, we've also got the final person that I want to thank. If you look into the mirror and say, Laura Grimshaw three times, a producer appears, and reminds you to send her the images you wanted to post to the social channels. Laura Grimshaw. Laura Grimshaw.

Laura Grimshaw. The Grimshaw Reaper, we call her. Good boo. Okay, Beck. Last time, unbelievably. nowhere you hit me yeah it's such a beck thing it's a classic morgan's dad maneuver it really is i also do want to get this going we've got the morgan's dad which is the phenomenon when someone flukes you get lucky first time yeah something the first time and doing a dexter yeah which is which is

essentially the answer to the second problem is that people need to be encouraged to do a dexter more often that's very true including themselves anyway you hit me D2 Hit Oh nuts I don't think I'm going to get as lucky as you but I'm going to stick to my system

and say J10 No it's a miss You heard me for like half a second Oh I got tricked not treated I Ha ha ha.