---
title: 'Mathematicians Finally Find the Infinite Card Game'
source: 'https://youtube.com/watch?v=1HwKCvsdXiw'
video_id: '1HwKCvsdXiw'
date: 2026-07-27
duration_sec: 1897
---

# Mathematicians Finally Find the Infinite Card Game

> Source: [Mathematicians Finally Find the Infinite Card Game](https://youtube.com/watch?v=1HwKCvsdXiw)

## Summary

This video explores the card game Beggar-My-Neighbor, a deterministic game with no skill element, and details the recent mathematical discovery of an infinite loop within the game. The host explains the rules, the game's half-life behavior, and the combinatorial search for long games, culminating in Braden Cella's 2024 discovery of an infinite game using a regulation deck.

### Key Points

- **Introduction to Beggar-My-Neighbor** [00:00] — The game is introduced, mentioned in Charles Dickens' Great Expectations, and noted to be centuries old. Mathematicians only cracked its major question last year.
- **Game Rules Explained** [01:04] — Players split the deck and take turns playing cards. Number cards are ignored; face cards (Jack, Queen, King, Ace) trigger the opponent to play a number of cards equal to the face card's rank. If they fail to play another face card, the first player wins the round and collects all played cards.
- **Three Key Questions from 1989 Book** [03:06] — The book 'The Mathematics of Games' poses: (a) How long is a game likely to last? (b) Can a game get into an infinite loop? (c) Can we prove it? The video states we now know (b) is yes and (c) is not needed.
- **Deterministic Nature** [03:45] — The game is completely deterministic; no player decisions affect the outcome once the deck is shuffled. The initial arrangement alone determines the winner.
- **Simulation and Long Games** [04:30] — A Python simulation found a game lasting 4,513 cards (630 tricks). The current record is 8,344 cards (1,164 tricks), discovered through random shuffling and simulation.
- **Halflife of the Game** [05:19] — The game has a halflife of approximately 20 tricks: every 20 tricks, there's a 50% chance the game ends. This leads to exponential decay in game length.
- **Number of Possible Starting Arrangements** [06:30] — Because number cards are indistinguishable, the number of distinct starting arrangements for Beggar-My-Neighbor is about 6.54 × 10^20, calculated using combinatorial choose functions for each face card suit.
- **Patreon Championship** [09:02] — The host organized a championship among 2,561 Patreon supporters, where each supporter played every other supporter. The winner, Jeremy, won 1,374 games out of 3,278,180 total games.
- **Search for Infinite Games** [13:18] — Infinite games require a state to repeat. John Conway called finding such a loop an 'anti-Hilbert problem' – important but not worth pursuing. Braden Cella ignored this and started searching.
- **Braden's Approach** [14:40] — Braden analyzed smaller infinite games (e.g., 3 cards per player) and combined them using 'pieces' – runs of cards between Jacks that move independently. He wrote code to assemble these pieces into larger infinite games.
- **Known Infinite Game with 14 Cards Each** [19:21] — An infinite game with 14 cards per player (28 total) was known since at least the late 1990s. Patterns of Jacks and number cards repeated.
- **False Positives: Infinite Games with No Starting State** [22:28] — Braden found about 10 infinite games using a full regulation deck, but none had a starting state where both players begin with 26 cards. The infinite loops existed but could not be entered from a standard deal.
- **Breakthrough: Key State Structure** [24:10] — By analyzing the structure of infinite games, Braden identified a key state: Player 2 holds a Jack and a number card; Player 1's hand is separated by Jacks into 'pieces' that move between players.
- **Discovery of the Infinite Game** [26:40] — Braden found multiple starting hands that feed into a 62-state loop, creating the first known infinite game of Beggar-My-Neighbor with a regulation deck and equal starting hands.
- **Mastodon Bot Playing the Infinite Game** [27:09] — A bot posts moves every 3 hours. The exact same move and deck arrangement repeats every 9 years and 9 months (January 5, 2034).

### Conclusion

The video concludes that the infinite game of Beggar-My-Neighbor has been discovered, ending a decades-old mathematical puzzle. The game's deterministic nature and combinatorial complexity made the search challenging, but Braden Cella's structured approach succeeded.

## Transcript

you folks want to play a card game and not a&nbsp; fun one one that's provably very boring well I&nbsp;&nbsp; guess it's boring if you care about being able to&nbsp; control your own destiny if you don't care about&nbsp;&nbsp; that could be fun it's a children's game called&nbsp; beg of my neighbor it's been around for a very&nbsp;&nbsp;
long time Charles Dickens talks about it in the&nbsp; 1861 book Great Expectations and it could be older&nbsp;&nbsp; than that it could be centuries older it might go&nbsp; back to when playing cards as we know them first&nbsp;&nbsp;
came to Europe however mathematician were not able&nbsp; to crack this game until well just last year we&nbsp;&nbsp; answered the major question about it and um it's&nbsp; not made the game any more [Music] fun some of you&nbsp;&nbsp;
may have played the game before it goes by a lot&nbsp; of different names camera person Alex what did you&nbsp;&nbsp; used to call it as a kid beat your neighbor Out&nbsp; of Doors beat your neighbor Out of Doors I don't I&nbsp;&nbsp; don't know understand this country so anyway it's&nbsp; got different names you may know it to explain the&nbsp;&nbsp;
rules um so I met up with my friend Mitha in&nbsp; London recently she runs the YouTube channel&nbsp;&nbsp; Looking Glass Universe I said hey while we're&nbsp; catching up in the pub uh do you fancy a card game&nbsp;&nbsp; here's how it went we get half of the deck each&nbsp; your player one so there's the beginning of your&nbsp;&nbsp;
deck the way this works your player one you go&nbsp; first a single card in the middle there a number&nbsp;&nbsp; we don't care number card we don't care number&nbsp; card we don't care and we alternate until someone&nbsp;&nbsp; does a face card the fo has to pop out of the coil&nbsp; okay so aces are face cards in fact Ace is the&nbsp;&nbsp;
fourth face card cuz they go Jack queen king Ace&nbsp; okay and I've now got a number of cards I can put&nbsp;&nbsp; down and I have to get another face card otherwise&nbsp; you win this this this trick all right which what&nbsp;&nbsp;
we call a round for some reason and so Ace is the&nbsp; fourth face card so I have four so ready one two&nbsp;&nbsp; and the problem is three oh an eight now you've&nbsp; got four oh Jack a now Jack's are tough cuz that's&nbsp;&nbsp;
the first face card so I've only got one chance&nbsp; to not lose this round and I got uh number card&nbsp;&nbsp; so you get to pick up all of those all right in&nbsp; order doesn't seem like a good thing whoever gets&nbsp;&nbsp;
the whole deck first wins oh okay so you need to&nbsp; get these cards I need to get those cards whoever&nbsp;&nbsp; wins around goes first on the next round all&nbsp; right mither and I started playing the card game&nbsp;&nbsp; and then we kept playing the card game game and we&nbsp; kept playing the cars going backwards and forwards&nbsp;&nbsp;
neither one of us winning and it kept going this&nbsp; must have taken ages and going and we started to&nbsp;&nbsp; wonder how long can this game go on for one two&nbsp; three yeah is that great we're having fun okay
um okay so hey could we be here forever the first&nbsp; reference to these questions is actually from the&nbsp;&nbsp; 1989 book The mathematics of games and towards&nbsp; the back here it discusses begger my neighbor and&nbsp;&nbsp;
poses three questions how long is a game likely&nbsp; to last can a game get into an infinite Loop if&nbsp;&nbsp; it cannot can we hope to prove this spoiler we now&nbsp; know the answer to B and C it can which means we&nbsp;&nbsp;
don't have to prove it can't but how long can a&nbsp; game go on well all we really need to care about&nbsp;&nbsp; is the starting arrangement of the car because&nbsp; as mither and I were very aware playing this&nbsp;&nbsp;
in the pub you don't have to make any decisions&nbsp; you have no choices nothing you do changes the&nbsp;&nbsp; outcome of the game as long as you don't make any&nbsp; mistakes it's completely deterministic whatever&nbsp;&nbsp;
order the cards are when they start determines&nbsp; the outcome of the game as they very politely&nbsp;&nbsp; say in here this is an automatic game with no&nbsp; opportunity for skill which is why it is so&nbsp;&nbsp;
suitable for family play I mean I guess the point&nbsp; is you don't have to have any skill to play it but&nbsp;&nbsp; it does have all the illusion of the excitement of&nbsp; a game like you've got to avoid going out on each&nbsp;&nbsp;
trick and you can see the decks getting bigger&nbsp; and smaller so it feels like a fun interactive&nbsp;&nbsp; game but it's not it's all predetermined&nbsp; so of course I knock together some terrible&nbsp;&nbsp; python code to see if I could find a really long&nbsp; game by just randomly shuffling a deck of cards&nbsp;&nbsp;
playing the game forward and keeping track of&nbsp; how long it went for and I did okay I got a game&nbsp;&nbsp; that went for 4,513 cards that 630 tricks which I&nbsp; thought was pretty impressive the current record&nbsp;&nbsp;
however is a game that went for 8,344 cards which&nbsp; was over 1,000 1,164 tricks and actually here is a&nbsp;&nbsp; plot of over time the current world world record&nbsp; for longest game played and what you'll notice&nbsp;&nbsp;
is that they're roughly linear as we've had more&nbsp; processing power to check more random games we've&nbsp;&nbsp; just happened to find longer ones it's pretty&nbsp; much exactly random and actually very early on&nbsp;&nbsp;
even in the 1989 book people noticed that as&nbsp; you're playing Begg of my neighbor it's kind&nbsp;&nbsp; of got a halflife so roughly every 20 tricks&nbsp; there's a 50% chance the game will have ended&nbsp;&nbsp;
and you just have an exponential decay because&nbsp; of that which is quite neat because it's such&nbsp;&nbsp; a neat exponential decay like radiation with a&nbsp; halflife we can just use an off-the-shelf formula&nbsp;&nbsp; to predict the longest game we will see for the&nbsp; number of Starting Games we've checked now the&nbsp;&nbsp;
250 in that equation is just the average number of&nbsp; cards per game that's just the mean and then the&nbsp;&nbsp; exponential drop off predicts that I checked well&nbsp; I checked around 1 and a half billion games and if&nbsp;&nbsp;
you put in a billion you see yeah you should be&nbsp; expecting somewhere around 5,000 is cards in the&nbsp;&nbsp; maximum game and as computer power gets better and&nbsp; better we can check more and more starting games&nbsp;&nbsp;
we will predictably find longer and longer games&nbsp; which are still finite in nature but why can't&nbsp;&nbsp; we just use a computer to check them all how many&nbsp; starting Arrangements can there be well famously&nbsp;&nbsp; there's a lot of ways to shuffle a deck of cards&nbsp; in this case it's not nearly is bad because once I&nbsp;&nbsp;
take out all the face cards all of these cards are&nbsp; fungible these are all indistinguishable we don't&nbsp;&nbsp; care which is which these are just IR relevant&nbsp; padding we just care about where these are so&nbsp;&nbsp;
if I was to put in let's say the Jacks first I've&nbsp; got here 36 boring cards plus these four exciting&nbsp;&nbsp; cards that will be 40 cards and so where these&nbsp; go is just 40 choose four because we don't care&nbsp;&nbsp;
like we don't distinguish between the Jacks so&nbsp; if we did 40 choose four imagine they shuffled&nbsp;&nbsp; in that would give us all the combinations we've&nbsp; now got 44 cards so 44 choose four to put in the&nbsp;&nbsp;
Queens 48 choose four to put in the Kings finally&nbsp; 52 choose for once the aces go in so as far as&nbsp;&nbsp; begar my neighbors concerned there are only around&nbsp; about 6.54 * 10 20 possible starting decks we've&nbsp;&nbsp;
only checked at the the moment up until about&nbsp; 10 to the 15 games so 10 to the 20 is still a&nbsp;&nbsp; way off because the 10 to the 15 checks take&nbsp; take a long time we're not there yet one day&nbsp;&nbsp;
we will have checked every single possible&nbsp; game of Begg of my neighbor I don't know what&nbsp;&nbsp; year that's going to happen in we could try and&nbsp; extrapolate from the plot maybe it's Mor's Lorry&nbsp;&nbsp; at some point we will know what the longest game&nbsp; is in the meantime it's still an open challenge&nbsp;&nbsp;
see if you can find a longer game if you want&nbsp; more information about the history of the records&nbsp;&nbsp; I will link to a website below made by someone&nbsp; called Richard man who is now a mass professor&nbsp;&nbsp; at the University of Leeds although they started&nbsp; the site well before that and it accidentally&nbsp;&nbsp;
became the central Focus point for current world&nbsp; record for very long games of beggar my neighbor&nbsp;&nbsp; and actually Richard has been very helpful putting&nbsp; this video together he kindly answered all of my&nbsp;&nbsp; questions so huge thanks to Richard by the way&nbsp; if you want to try solving a puzzle of your own&nbsp;&nbsp;
that's not quite as involved as the ones in the&nbsp; video this video is sponsored by Jane Street&nbsp;&nbsp; fantastic financial company who sponsor lots of&nbsp; my videos they are running the Academy of math and&nbsp;&nbsp; programming soon amp applications open details&nbsp; in the description and at the end of the video&nbsp;&nbsp;
I have one of their decommission puzzles you can&nbsp; try it's I had a lot of fun solving it so that's&nbsp;&nbsp; at the end however it occurred to me because&nbsp; beggar my neighbor is completely deterministic&nbsp;&nbsp; once you've shuffled the cards that's it you could&nbsp; have a a championship where people competing don't&nbsp;&nbsp;
even have to show up I can just tell them oh&nbsp; by the way here's the shuffle deck you were&nbsp;&nbsp; allocated for your game and here's who won cuz&nbsp; my code can work that out very quickly and then&nbsp;&nbsp; I realized wait I could have a championship where&nbsp; people don't even have to know that they were in&nbsp;&nbsp;
the championship so at the moment I have 2,561&nbsp; supporters on patreon and I thought it be very&nbsp;&nbsp; funny to have a standup Mass patreon wide champion&nbsp; ship where everyone plays everyone else and so&nbsp;&nbsp;
I've written some code to do that I'm going to&nbsp; set it running right now and if you support me&nbsp;&nbsp; on patreon you're in the mix and you're about to&nbsp; play everyone else okay here we go and start it's&nbsp;&nbsp;
happening the first ever official standup math&nbsp; World beg of my neighbor card Championship 2025&nbsp;&nbsp; happening right now at the rate of just over&nbsp; 10,000 games a second uh we'll come back when it's&nbsp;&nbsp;
done and we're done so all 2561 patreon supporters&nbsp; played each other for a total of three, 278,000&nbsp;&nbsp; 180 games it actually took a little bit longer&nbsp; than it should have so it should run in under five&nbsp;&nbsp;
minutes but at the same time I was exporting the&nbsp; details of every single game into a separate text&nbsp;&nbsp; file for every single patreon supporter so if you&nbsp; do support me on patreon I will email you a text&nbsp;&nbsp;
document and it will have your complete breakdown&nbsp; and we'll start by having a look at some of those&nbsp;&nbsp; so I'm going to pick I've got a folder here with&nbsp; 2 and a half th000 text files and here's what the&nbsp;&nbsp; text files look like this is the one for Lis flow&nbsp; so hello Lis flow thank you for your support here&nbsp;&nbsp;
are your standup Mass official 2025 Begg of my&nbsp; neighbor World card championship results and&nbsp;&nbsp; look at this it's got every single other patreon&nbsp; supporter cuz you played everyone and it's got&nbsp;&nbsp; exactly what happened in the game pick a game at&nbsp; random here you lost against Craig F oh sorry to&nbsp;&nbsp;
hear that in a game that lasted 13 tricks required&nbsp; 87 cards to be played quite a quick game and then&nbsp;&nbsp; I give the starting hands for both players zero&nbsp; that's just like the cards that could be anything&nbsp;&nbsp;
and then 1 two 3 and four are for Jack queen&nbsp; king Ace so the number just represents how many&nbsp;&nbsp; cards the other person can play you can swap in&nbsp; you can recreate the exact hands if you want and&nbsp;&nbsp;
play any of these believe me when I wrote the&nbsp; code and I was testing all of this I did then&nbsp;&nbsp; go through and manually play some of these games&nbsp; to make sure they worked oh you won against ASDF&nbsp;&nbsp; I didn't censor one-word names and you know who&nbsp; has one of them three blue one brown yes Grant&nbsp;&nbsp;
supports me on patreon which means every other&nbsp; patreon supporter you've now played three blue&nbsp;&nbsp; one Brown in a game of beggar my neighbor now in&nbsp; here everyone's names are censored they won't be&nbsp;&nbsp;
in yours though so you can search through and&nbsp; see did you win or did you lose when you played&nbsp;&nbsp; Grant but what you all want to know is who won the&nbsp; championship let's find out I'm going to pull in&nbsp;&nbsp;
everyone's results and then rank everyone from&nbsp; most games one right down to Fest and Fest is&nbsp;&nbsp; Connor sorry Connor you only won 1,25 games you&nbsp; were just beaten out by Don Luigi who W 1,26 and&nbsp;&nbsp;
then you can see it gradually goes up this is&nbsp; like very very fast patreon credits I'd like to&nbsp;&nbsp; thank everyone that you're watching them fly by&nbsp; oh and who's at the top Jeremy congratulations&nbsp;&nbsp; Jeremy with 1374 games six games out before&nbsp; second place you are the official winner of&nbsp;&nbsp;
the 2025 World beg of my neighbor standup Mass&nbsp; Championship honorable mentions Johan and David&nbsp;&nbsp; so that's finite games speaking of which we can&nbsp; check back in with myself and Mitha playing in the&nbsp;&nbsp;
pub past Matt and past Mitha are still going and I&nbsp; can guarantee they are playing a finite game isn't&nbsp;&nbsp; that convenient however we don't care about finite&nbsp; games we need to move on to infinite games we want&nbsp;&nbsp;
a game that's going to last forever and when this&nbsp; was discovered last year in 2024 while someone&nbsp;&nbsp; named Braden Cella I was so excited everyone was&nbsp; because we didn't think this was was going to be&nbsp;&nbsp;
possible for a very long time people have been&nbsp; looking for this Loop even before the mention in&nbsp;&nbsp; the book from the80s back in the 60s we believe&nbsp; the mathematician John Conway came across this&nbsp;&nbsp; game realized that an infinite Loop would be super&nbsp; interesting couldn't find one and they speculated&nbsp;&nbsp;
that it would be very difficult to do it and they&nbsp; named it an anti-h Hilbert problem and the Hilbert&nbsp;&nbsp; problems were problems which were very important&nbsp; and everyone should start working on them in math&nbsp;&nbsp; itics an anti-h Hilbert problem Conway said was a&nbsp; problem which is very difficult but no one should&nbsp;&nbsp;
be bothered doing like it was so pointless&nbsp; people shouldn't even bother dealing with it&nbsp;&nbsp; Braden however didn't listen to them Braden gave&nbsp; it a go and here's what he found my name is Bren&nbsp;&nbsp; cassella I graduated with a degree in computer&nbsp; science I really was curious about what made some&nbsp;&nbsp;
problems difficult because you know not really&nbsp; understanding a lot of math you think we can do&nbsp;&nbsp; so much with it and understand so much you know&nbsp; what really can't we do and why can't we do it so&nbsp;&nbsp;
I looked up unsolved problems in mathematics and&nbsp; found one that was very um you know approachable&nbsp;&nbsp; and understandable Bren very kindly had a call&nbsp; with me so he could talk me through the details&nbsp;&nbsp; of what he managed to achieve and I think I've got&nbsp; my head around it now the first key point is that&nbsp;&nbsp;
long but finite games actually give us no Insight&nbsp; whatsoever into Infinite games because those long&nbsp;&nbsp; games they just happen to have the right amount&nbsp; of noise or whatever so that they can survive&nbsp;&nbsp;
the half-life process for a very long time but&nbsp; every single state they're going through so at&nbsp;&nbsp; the end of each trick once the players have their&nbsp; cards back we can consider that a snapshot of the&nbsp;&nbsp; game that's the current state of the game and&nbsp; every single state it goes through is different&nbsp;&nbsp;
until eventually one player wins so the final&nbsp; state is that one player has the null pile for&nbsp;&nbsp; an infinite game we need to have a state reappear&nbsp; because there's only a finite number of states and&nbsp;&nbsp;
because it's completely deterministic from one&nbsp; state to the next when you're playing the game&nbsp;&nbsp; if you hit a state that's happened before you'll&nbsp; be in an infinite Loop and that Loop is probably&nbsp;&nbsp; not going to be as long as these you know hundreds&nbsp; over a thousand different games so actually we're&nbsp;&nbsp;
looking for some different properties so actually&nbsp; analyzing the long games not going to help what&nbsp;&nbsp; will help however is analyzing smaller easier to&nbsp; find infinite games here's an example of a tiny&nbsp;&nbsp;
game where I'm going to give the both players&nbsp; only three cards so there's one deck there and&nbsp;&nbsp; the other one you know I'm going to use Reds&nbsp; for the unimportant cards and two black jacks&nbsp;&nbsp;
so I've put them this way up so you can kind of&nbsp; monitor what's happening in each player's deck&nbsp;&nbsp; this player will go first puts a jack this person&nbsp; will then play and then lose so they will put it&nbsp;&nbsp; over here then they will play they'll play they'll&nbsp; lose so that goes over there and if you're paying&nbsp;&nbsp;
close attention we've now got the situation where&nbsp; it's you know uh unimportant unimportant or number&nbsp;&nbsp; number jack number jack number but as soon as we&nbsp; start playing again play play play one is exactly&nbsp;&nbsp;
the same state that swapped and we can keep doing&nbsp; that play play play back down there and so this&nbsp;&nbsp; tiny like toy version of the game will carry on&nbsp; infinitely because you're just swapping from the&nbsp;&nbsp;
same state well basically the few cards are going&nbsp; backwards and forwards and you will stay in this&nbsp;&nbsp; Loop infinitely this is an infinite game here's&nbsp; another game that repeats but instead of three&nbsp;&nbsp; cards each we're going to have four cards each so&nbsp; we're gradually getting closer to where we have&nbsp;&nbsp;
26 cards each that's going to take a while okay&nbsp; so we play the first trick that's the result of&nbsp;&nbsp; the first trick very familiar Jack and a number&nbsp; card over here this player won though so all of&nbsp;&nbsp;
these get put onto their pile and that's worth&nbsp; remembering it goes number number King number jack&nbsp;&nbsp; number and again that could be any face card apart&nbsp; from a jack up next short boring round they've won&nbsp;&nbsp;
the trip brick and that's suspicious it's exactly&nbsp; the two hands we started with they have swapped we&nbsp;&nbsp; could carry again they'll swap back and that will&nbsp; keep repeating over and over and the kind of the&nbsp;&nbsp;
key thing to take away from this other than these&nbsp; games repeat is that you've got this suspicious&nbsp;&nbsp; Jack in a number that keeps showing up and then&nbsp; you got this other big chunk of cards that seems&nbsp;&nbsp;
to just get passed around by going to these super&nbsp; small versions of the game it makes it very easy&nbsp;&nbsp; to find games which repeat infinitely however&nbsp; they're so small there's very little insight into&nbsp;&nbsp;
why they're repeating we want to find a middle&nbsp; ground with enough cards that there's some insight&nbsp;&nbsp; to be gleaned but not so many that we can't find&nbsp; any and a lot of people go straight to the middle&nbsp;&nbsp;
of the deck in fact this is where Braden started&nbsp; looking for hands which have 13 cars each and&nbsp;&nbsp; lead to infinite game that however is impossible&nbsp; there are no cuz there's small enough options that&nbsp;&nbsp;
we can check them all there are no infinite games&nbsp; using exactly half the deck but there is one very close so here's an infinite game that was&nbsp; known about at least by the late '90s there&nbsp;&nbsp;
a paper from 99 I link to below and it's two&nbsp; more cards than half the deck so there are 14&nbsp;&nbsp; each total of 20 8 I'm going to play through a&nbsp; trick or two and we'll see if we can spot any patterns it's the one thing to&nbsp; notice at this point we played&nbsp;&nbsp;
for a little while player one oh Lion&nbsp; Share of the deck player two it's got&nbsp;&nbsp; a nothing card a Jack and a nothing&nbsp; card but what's going to happen next
okay we a little bit later on and now we've swap&nbsp; sides player one is down to the something Jack&nbsp;&nbsp; something and player two has all of the cards&nbsp; in a very familiar order and this is going to&nbsp;&nbsp;
keep repeating and it's very interesting we've&nbsp; kind of got a Jack and a single one over there H&nbsp;&nbsp; anyway leaving that for now what Braden thought&nbsp; they would do is see if they could get smaller&nbsp;&nbsp;
games that repeat and combine them into bigger&nbsp; ones and you can just kind of mush them together&nbsp;&nbsp; so that as you're playing along you kind of have&nbsp; one repeating and then the other one repeating&nbsp;&nbsp; and so you can take smaller repeating games and&nbsp; build up into bigger and bigger repeating games&nbsp;&nbsp;
and that's exactly what Braden did the issue was&nbsp; you need to build them up in such a way that you&nbsp;&nbsp; eventually hit exactly 52 cards so they have to&nbsp; be the right sizes obviously have to be combined&nbsp;&nbsp; in a way that they work and you need to make sure&nbsp; you've got the correct distribution of the Court&nbsp;&nbsp;
cards or the face cards you have to have four of&nbsp; each and not every combination of ones that repeat&nbsp;&nbsp; that produce a repeating pattern necessarily have&nbsp; those properties so Bren had to write some code to&nbsp;&nbsp;
search for all the ways you can put the different&nbsp; repeating bits together such that you then spit&nbsp;&nbsp; out another repeating pattern and then check that&nbsp; it's a standard deck check that it's the correct&nbsp;&nbsp;
number of cards and check that you've got four of&nbsp; each of the four different face cards so I end up&nbsp;&nbsp; finding all of these standard deck games none of&nbsp; them ever become balanced so n none are ever valid&nbsp;&nbsp;
you can never actually get to those ones Braden&nbsp; had found an infinite Loop of game States about 60&nbsp;&nbsp; or so game States all the way around none of them&nbsp; were perfectly balanced none of them had two packs&nbsp;&nbsp;
of 26 cards each so then Braden took each of those&nbsp; States and revers them and while forwards you&nbsp;&nbsp; always go to one specific State afterwards on the&nbsp; way back there actually multiple previous states&nbsp;&nbsp;
that could have fed into that so Braden took each&nbsp; of the different states in the loop reversed them&nbsp;&nbsp; all as far back as they could go for where there&nbsp; were valid states that could lead into them and&nbsp;&nbsp; discovered none discovered absolutely no balanced&nbsp; set so they had found the infinite game there was&nbsp;&nbsp;
just no way to start it so like fine they&nbsp; found another infinite game with a regulation&nbsp;&nbsp; deck no balance starting points they ended up&nbsp; finding about 10 different infinite games with&nbsp;&nbsp;
a regulation decks worth of cards but across all&nbsp; 10 or so of them there were no starting points&nbsp;&nbsp; Braden was Finding infinite games but there were&nbsp; games that could never begin so what's a computer&nbsp;&nbsp;
scientist to do Braden was finding all these&nbsp; infinite gains that there was no way to start&nbsp;&nbsp; them I have to say what an arbitrary constraint&nbsp; that is these are all perfectly valid Solutions&nbsp;&nbsp; almost indistinguishable from the correct solution&nbsp; we just happen to have a game where you deal equal&nbsp;&nbsp;
number of cards to each player when you start if&nbsp; we didn't have that constraint these would all&nbsp;&nbsp; work in fact if we didn't have this distribution&nbsp; of cards like the certain number of different&nbsp;&nbsp; types of face cards in a deck then it would be&nbsp; different it's just what we discover in terms&nbsp;&nbsp;
of the structure and just putting in zeros&nbsp; and ones and twos and threes and fours into&nbsp;&nbsp; different orders to see which repeat according&nbsp; to these rules the fact that one particular&nbsp;&nbsp; group of constraints on that leads to a card&nbsp; game that people actually play I think is very&nbsp;&nbsp;
interesting but there's no reason for those to be&nbsp; the rules and Braden was finding great solutions&nbsp;&nbsp; they were just falling apart at this one last&nbsp; hurdle but it meant that the ones that they&nbsp;&nbsp; were finding were on the right track they had&nbsp; to find one with a single Balan state in there&nbsp;&nbsp;
somewhere and the wonderful side effect of finding&nbsp; all these uh false positives if you will is that&nbsp;&nbsp; Braden was learning a lot about what these games&nbsp; look like when they are an infinite game and so&nbsp;&nbsp;
what they did was pull apart these games look at&nbsp; the structure underneath them and then find a new&nbsp;&nbsp; way to build up a new infinite game here's what&nbsp; bradden noticed during the course of an infinite&nbsp;&nbsp;
game there would be a state at some point which&nbsp; was kind of like the key State and that's where&nbsp;&nbsp; player two would a Jack and have some kind of&nbsp; unnecessary number card on the end players one&nbsp;&nbsp;
hand would have a bunch of cards at the beginning&nbsp; which I'm just going to represent as a mash of&nbsp;&nbsp; cards here some number of cards then a Jack&nbsp; and then another pile of cards here and then&nbsp;&nbsp; a Jack and then another pile of cards and then&nbsp; a Jack and then a number card and these piles of&nbsp;&nbsp;
Cs or these runs of cars between the Jacks Braden&nbsp; referred to as pieces so you'd have one piece here&nbsp;&nbsp; Jack piece Jack piece Jack number card and&nbsp; these pieces these runs of cards between the&nbsp;&nbsp;
Jacks pretty much operate independently when I was&nbsp; saying before you could combine games which repeat&nbsp;&nbsp; together by like smooshing them together this&nbsp; is the same idea but these don't individually&nbsp;&nbsp; have to be games that would repeat on their own&nbsp; these are just units pieces as such of cards&nbsp;&nbsp;
that move together during this infinite game and&nbsp; the cards interestingly during the course of the&nbsp;&nbsp; game these pieces would kind of move over over to&nbsp; the other player and then move back like this was&nbsp;&nbsp; like storage for the pieces and they'll transfer&nbsp; over then transfer back and eventually the whole&nbsp;&nbsp;
system would repeat I will link to the paper below&nbsp; it's actually still a work in progress I will link&nbsp;&nbsp; to the current version on archive they're still&nbsp; fleshing out exactly how they're going to explain&nbsp;&nbsp; it but in short instead of searching for other&nbsp; games that repeat infinitely and trying to put&nbsp;&nbsp;
them together they could just try and find these&nbsp; individual pieces and then make sure initially&nbsp;&nbsp; that they're the right size and distribution of&nbsp; cards to get a standard deck and then check all&nbsp;&nbsp; the states they go through and all the feed in&nbsp; states to find one that's perfectly balanced and&nbsp;&nbsp;
to simplify their life slightly they didn't even&nbsp; bother with the third one CU you can force the&nbsp;&nbsp; third one to just be two nothing cards so that&nbsp; simplifies what we're looking for and then over&nbsp;&nbsp; here once they locked one in like one that six&nbsp; long that greatly reduces the permutations for&nbsp;&nbsp;
the possible cards that you would need to find&nbsp; another one and they came up with a very clever&nbsp;&nbsp; wave using filters to be a to search for these&nbsp; pieces pretty much in isolation and then run their&nbsp;&nbsp; code to combine them all together and eventually&nbsp; it worked and here it is the infinite game any&nbsp;&nbsp;
one of these starting hands with little Dogg ear&nbsp; means the top player goes first plane boxes bottom&nbsp;&nbsp; player goes first these are all perfectly balanced&nbsp; regulation deck games they all feed into that same&nbsp;&nbsp;
Central Loop of 62 different states which will&nbsp; repeat forever a phenomenal achievement to find&nbsp;&nbsp; and for those of you wondering the game mither and&nbsp; I were playing was not the infinite game it was&nbsp;&nbsp; a finite one we were playing the longest no and&nbsp; finite game if that disappoints you my good friend&nbsp;&nbsp;
Matthew scroggs has made a masteron bot which&nbsp; is playing the infinite game you can see here&nbsp;&nbsp; every 3 hours it posts the next move between who&nbsp; iuma Alice and Bob playing the infinite game and&nbsp;&nbsp;
it started in April last year once the infinite&nbsp; game had been discovered now Tech technically&nbsp;&nbsp; it's already looped six times about to Loop&nbsp; seven times according to the pure math version&nbsp;&nbsp; but we don't care what the number cards are but&nbsp; as you can see Matt showing the exact cards and&nbsp;&nbsp;
they take a lot longer to repeat if you want every&nbsp; single card to get back exactly where it was and&nbsp;&nbsp; we care about what they are see this post here on&nbsp; the 5th of April 2024 that's the first one in the&nbsp;&nbsp;
official cycle and this exact move will happen&nbsp; again with identical decks on the 5th of January&nbsp;&nbsp; in the year 2034 although Matt scrs is pretty&nbsp; convinced the bot will break around 2029 and&nbsp;&nbsp;
he won't bother fixing it if you've enjoyed this&nbsp; journey using math and computer science you first&nbsp;&nbsp; of all will enjoy the puzzle I promised coming&nbsp; in a second and if you're graduating from high&nbsp;&nbsp; school this year 2025 you might enjoy Jane streets&nbsp; amp that's their Academy of math and programming a&nbsp;&nbsp;
five-week residential Course in New York City for&nbsp; high school students who are going into University&nbsp;&nbsp; either later this year or next year and you want&nbsp; to do some more math and computer science for&nbsp;&nbsp; five weeks it's so good every day you're solving&nbsp; puzzles like the puzzle I gave to Mitha when we&nbsp;&nbsp;
met up before we played the card game in the pub&nbsp; the puzzle setup is that you're trying to guess a&nbsp;&nbsp; number someone is thinking of and they're saying&nbsp; higher or lower except if they say the same thing&nbsp;&nbsp; three times in a row three highs or three low&nbsp; was you lose this turns what would have been a&nbsp;&nbsp;
standard binary search puzzle into something much&nbsp; more intricate interesting and as we discovered&nbsp;&nbsp; difficult well let's lock in binary search 10 all&nbsp; right that's your second guess mhm and it's still&nbsp;&nbsp;
three it's still three consecutive yeah okay so&nbsp; when I say lower okay now you're like uhoh okay&nbsp;&nbsp; I'm in trouble well then I have to guess one cuz&nbsp; you have to go one yeah cuz I don't want to L you&nbsp;&nbsp; still got two more guesses up your sleeve yeah so&nbsp; that's a problem but because you got six guesses&nbsp;&nbsp;
there's enough risk you might have to go racing&nbsp; back up again so this cuz you're coming up has&nbsp;&nbsp; to favor up Y Cool okay okay yeah oh nice that's&nbsp; kind of interesting that you have to like favor&nbsp;&nbsp; the direction that you like yeah and so that's all&nbsp; six guesses now what if we'd said lower here yeah&nbsp;&nbsp;
and we got here I say high oh no yeah I'll put&nbsp; the exact wording of the individual questions you&nbsp;&nbsp; have to solve below but in general you're trying&nbsp; to work out a strategy and here we're trying to&nbsp;&nbsp; crack the maximum range of numbers you can always&nbsp; correctly identify the guest from if you have six&nbsp;&nbsp;
guesses as you can see we've picked 20 as the&nbsp; midpoint and we're assuming the first guess is&nbsp;&nbsp; lower by symmetry you get the top range as well&nbsp; and it was fun I love standing around a whiteboard&nbsp;&nbsp; trying to solve a mass problem coming up with&nbsp; new notation bouncing backwards and forwards on&nbsp;&nbsp;
different ways to solve it and if you enjoy this&nbsp; sort of thing you're going to love amp based in&nbsp;&nbsp; New York City James Street cover all the cost of&nbsp; attending amp as well as a $5,000 scholarship for&nbsp;&nbsp; students who face some kind of barrier in their&nbsp; education it's a lot of fun and get to meet&nbsp;&nbsp;
me I come and give a talk that's pretty exciting&nbsp; applications for amp close on the 12th of March so&nbsp;&nbsp; either you apply or tell someone you think should&nbsp; apply thank you J Street huge thanks to myth and&nbsp;&nbsp; for being so tolerant and playing that game with&nbsp; me doj a channel Looking Glass Universe I'll link&nbsp;&nbsp;
to it below and thanks to you for watching&nbsp; and if you're still watching this late in&nbsp;&nbsp; the video you either are a patreon supporter or&nbsp; you should be a patreon supporter come on if you&nbsp;&nbsp; supported me on patreon you would have been in the&nbsp; championship who knows what I'm going to do next&nbsp;&nbsp;
probably this again that was fun thank thanks&nbsp; for joining me for this yeah no this was great&nbsp;&nbsp; so what do we do next well hey if you enjoyed the&nbsp; excitement of this have I got a card game for you
okay e
