120 = Dealing Probability and Healing Gullibility

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 Marker.

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 booze.

Nice.

I was really hoping you'd say that.

I would have kicked off if you didn't.

Matt?

Back.

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 realized 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 is 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.

Yep.

Had I not listened back, I would have had no idea.

It did not occur to me until you're talking about, 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.

Oh, 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 date is coming up, 1st of December.

That's a bit terrifying because it's the Matilda theatre in London.

It can fit over a thousand people, and we're well over halfway to filling it.

But I would love to get it, you know, full to the literal rafters.

So, anyone in or near London, 1st of December, please come along.

Standupmass.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.

Come and see 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 electros to you.

I'm fine.

This is just one of the hurdles to go on that ESA 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 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 in 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'll be breathing.

But in terms of gravity, if people would like to go to the problem posing page at a problemsquared.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 try to think of things, but none of them are like scientific.

It's just that

it'd be quite funny to try and watch you eat porridge or something.

Yeah, okay.

Okay, let me let me clarify two things.

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 ESA 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 0G 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 0G.

That's the trajectory we then take.

So everything has to be 0G rated.

Wait, hang on.

So in emergency

it's gonna crash.

That's what they're saying.

We just want to make sure it will light up

just in those moments before

you hit the frown.

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.

I love 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.

No, 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.

Wow.

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, whatever's going on there.

You're going to have a 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 Antarctica, 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 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

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 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 problemsquared.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 Ember Fringe and some other great press shots, who's a longtime 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 than you have hands.

And the person that he was chatting to, the juggler, was Daryl J.

Carrington, who we had on the show,

who had actually given that fact.

And he said,

yeah,

was that from Beck's podcast?

He said, yeah.

And he went,

that was me.

I said that.

Yeah.

Brilliant.

That's the kind of socially awkward situations I'm glad we generate.

Our first problem comes from Morgan, who went to a problemsquared.com.

And 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 number file slash standup maths.

Calculator unboxings of my comfort videos.

Beck has a place in my heart now too.

Yay.

No, 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.

Beck, after they were so nice, you just blah, blah, blah over the rest of them.

Just blah, blah, blah 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 around, I'll talk you through it and then the listeners will get a sense of how the game works.

So i believe you you've got a deck of cards i love this because i am a practical learner i'm not a 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 car deck of cards where you can't see them and you turn them over one at a time and put them down Ah, 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 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 to reveal four cards in a row.

Yes, in a row.

Okay, I've got the four of hearts,

the 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 and the end one.

Right.

So 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.

Get out of here.

Yeah, exactly.

And now slide the end one 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 they're in 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.

Yep.

Annoyingly, not useful.

Ah, boo, one more.

So one more.

Nine of clubs.

Okay, that's the same suit, because four before is three of clubs.

So you can take the two out 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 collapse down and down and down, down, a very long row of cards can get very small very quickly.

Very satisfying.

So then you just fall through.

You 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 four before it.

So I'll take out those two middle cards, collapse it down.

I've just had another two collapse ones.

Because you get the double collapses.

Having a number match is good because all four of them are.

I'm playing while you're talking.

You should be.

When the values match, all 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?

Another double down.

It's very satisfying.

It's kind of a fun game.

I'm off the table, guys.

How long can it get?

We're running out of the table.

Okay, it's now going onto the couch.

Uh-oh.

One, two, three, four, five.

You never want to go

eight, nine, ten,

eleven.

But I finally had a matching suit.

Ooh, 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 go.

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 yep 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, this is a thing to bear in mind.

Achievable by us.

By us.

Well, 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 deck of 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

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 Wordle 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 Wordle 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 way.

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 Wordle 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 to do.

It's doable.

And the question is: how much would you reduce the space of possible, like the collection of possible games?

Would it then become small enough to be searchable, or 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 see what happened so that's okay 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 it's smart

Okay, I've shared a plot with you.

Ooh.

Thank you.

That's the correct 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.

So 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 stripey 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 really pleasing.

And so,

in general, most of the time you're going to end up with around 10, 10 to 14 in there cards left.

Is it possible to arrange this graph 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 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 it 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 2 cards left.

Oof, that's got a sting.

And then, you are absolutely right.

The 17th

most likely outcome with a probability of 0.71%

is 0 cards.

It's higher than I expected.

It's about two-thirds of the way down the leaderboard.

The following games are all less likely.

34 cards, 36, 38, 40, 42, 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, 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 you'd clubs heart spades diamonds clubs heart space diamonds all the way down those and then again for the offsets So I'm pretty sure you've got 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 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 have had one should have had one match the fact that Morgan's dad got it the very first time at just under one percent 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 setup 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, well, it depends on your family relationship.

That could be argued to be not very nice.

Setting the decks up so everyone else apart from your mom keeps winning all the time.

Now, that's hilarious.

So I would say set up a bunch of of decks that are all guaranteed to win, and then your mum'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 you're pulling and then reveal to your mum that you've got a whole bunch of ones that work and you're just 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 it once,

one, and refused to ever do it again.

That's the funniest move for someone who knows how to code and then find a...

Or maybe if your dad doesn't program, but is still like Beck, easily distracted to try and do something because I can see Beck is already trying to arrange your deck in such a way that it won't collapse.

Don't think I have a notion.

You told me not to.

And I'm still doing it.

Yep.

So the question is, Morgan, A, is your dad a Matt?

In which case, they probably wrote some code and then just did it once and refused 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 challenge is to play a game such that you've got as many cards 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 it takes 52?

When I ran the simulations and I got the probability back for it collapsing all the way down, I looked at the number, I'm like, that's odd.

0.0070705.

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 was 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, it, I suspect it's exactly one over root two.

Why on earth would the probability of playing this game and winning be one over root two?

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 two?

Is it just a bigger cosmic coincidence than Morgan's dad winning the first time?

Or is the has the universe stacked the deck on us?

Is there a reason for this?

Ah, imagine if this was the key.

Imagine if like this was our 42 moment.

Life, the universe, and everything.

This was like the film Contact.

Everything is unraveled 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 was very proud of myself.

I'm going to send a photo.

It is like a metaphor for life.

For you.

Yeah, it really is.

All right, listen.

I'm going to get you guys to check it for me, though, because I am quite prone to the.

Look, I'm a bit of a Parker Square person myself.

Oh, come on.

Let's stop trying to recontextualize 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 window of time where I was doubting you.

Yeah, I think you've done it.

Yeah, it's lovely and systematic.

It gets a little bit, well,

you mix it up a bit towards the end there, but it works.

Well, you think it does.

But actually, so what happens for 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 the 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, etc.

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 was 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 this 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 interchanging 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.

Yep, they're all little batches of three and they're just incrementing up the whole way around.

That's right.

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 solutions because you could swap values and suits

all over the place and serve the same thing.

And you can steward 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.

Well, okay, I don't want people getting distracted by that.

I want the root tooth thing solved.

And if there's 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 solved 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 you can do that by going to aproblemsquared.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 problemsquared.com and typed 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 are some 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 got for us?

I liked this question because I think, yes, not only is misinformation an issue and media literacy is a very

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 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?

100%.

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, I hope Peck 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 pseudoscience or something like that if you win a Nobel Prize.

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

by the time that a Nobel Prize is handed out,

it can often be decades after the original discovery or invention.

And by that 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.

In fact, we've been doing this podcast for over five years now, and I'm pretty sure my brain slowed down in that time.

And 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 perceive a difference.

Yeah.

Woo.

Yeah.

We're We're all in this together, guys.

We're going down with the ship.

It's also obviously been stated that it could, 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 you've gone about a very scientific way of doing it.

People seeming seeming 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, 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 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 you 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.

You know, if you were to know about every time someone has played that and not won 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 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.

Oh.

When I fully answer

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.

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 encouraged 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?

Huh?

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.

Yeah.

And the thing is, is like, we're not.

Now, that's not to 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've, we know what question is being asked.

Yep.

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.

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 if you knew immediately like 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 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

all of the things that we have to,

yeah, we need to build this reality in order 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 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 brainpower 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 laugh.

Occasionally, we'll look at each other.

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 it 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 doofuses 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 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 the illusion of understanding something.

Yeah, it could 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 you're 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 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 cow?

How does that work?

Yeah, how does that work?

How did the cheese get there?

And they're like, oh, I don't know.

It just is.

Like, I was reading this article and it is.

Yeah.

Okay, but like.

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 to tell them they're wrong.

Feels like an

actionable bit of...

It's not the whole answer, obviously.

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.

Alyssa, give us a ding or otherwise over at the problem posing page at a problemsquared.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 Apples-ness.

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 Rod R-U-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 TetraPack.

Makes sense.

Hence, Tetra, tetrahedron, tetrapak, 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 wrote in to say the fact that it's a tetrahedron is why Tetrapac is called Tetrapak, and that's where the shape originally came from.

The reason we picked Rud to represent everyone who said that was they went to one of the tetrapak 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 there you are.

Less packaging for the same, 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 Sunnyboys,

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 Sunnyboy

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 polar 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.

Thomas got in touch.

Listening to the same episode, said it was brilliant, blah, blah, blah.

Said they saw me at Edinburgh.

Wow, wow, wow.

Okay, here we are.

They want to point out, first of all, that PG Tips did a pyramid-shaped teabag, and they've correctly used the past tense.

The entire maths community was sad, or 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 teabag you know the tetrahedral shape was advertised because it had more volume there's more space for the tea to infuse compared to a regular teabag whether or not 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 um in terms of tetra pack the classic volume to um surface area thing and i guess most teabags now are flat whereas this is an inherently 3d shape and we also heard from daniel Daniel regarding the group of shapes with cross-sectional perimeters.

May I suggest the name stectagles?

I feel it recognises 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 are unaware, we have a Patreon, which is a 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

Joan

Neoff

Larity

Elijah.

tear.

He re to fo.

When I say bo, selector.

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,

out of 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 flips.

Who gets lucky first time?

Yeah.

Something the first time.

And

doing a Dexter.

Yeah.

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.

Ah,

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.

Nah, it's a miss.

Ah!

You know, you heard me for like half a second.

Oh, I got got tricked, not treated.