CORRECTIONS AND CLARIFICATIONS AND CHALLENGE UPDATE 1. I've seen many comments challenging my method at around 13:20 by saying that numbers greater than 1000 are allowed because we might then divide or subtract to bring them into the 100-999 range. I never say otherwise, but I think my explanation was unclear and I apologise for that. I don't exclude the possibility of generating numbers greater than 1000, just that the only numbers we store are the ones that land in the 100-999 range. For example, if we generate 1500 during calculation, then we continue calculating with it (because we could for example later divide it by 2 to obtain 750) but we don't store 1500 in the solution set, we only store 750 in this case. Hope that clears up that point and sorry if it was unclear! 2. Huge thank you to viewer Trillian who spotted an issue with part of my code. I mention that I sort the set into descending order before applying the various operations. For the section on the "scary" numbers (31:00 onwards), the first operation was performed on an unsorted set, so a very small amount of solutions were missed. It's a minor fault and I believe the conclusions will be the same, but I'm running it again to be sure. 3. Pretty sure viewer Trillian has solved the Countdown challenge. Can't wait to share the results with you in a future video! *Is Countdown Britain’s oldest game show? Strictly speaking, “A Question of Sport” and “Mastermind” have aired for longer; similarly for University Challenge though the modern iteration has only aired since 1994. However, when writing, I categorised these as “quiz shows” as opposed to "game shows" as they are almost entirely based on trivia and knowledge. Countdown isn’t a “quiz” since the contestants aren’t answering trivia questions but are competing in a game. All of these shows are categorised as “game shows” in the world of TV production so you have reasonable grounds to disagree with my classification. Personally, I like the distinction between quiz and game. And also it makes for a snappy title. **On the notation “nCr” for the “choose” function. This isn’t my preferred notation, either. But all other options use superscripts, subscripts, or don’t fit on one line. All the other options reduced the readability, especially in some of the busier diagrams in the video, so I went with the less-standard but ultimately neater option.

Looking forward to your Numberwang episode.

I just remember that one time where the target number came up, and Rachel instinctively said “Oh dear!” immediately, because it was impossible.

8 out of 10 Cats Does Countdown is possibly Britain's greatest gift to the world. A silly yet sublime conjunction of comedy and skill.

As noted elsewhere, it is called Letters & Numbers in Australia. In a very Aussie habit, the contestants started giving the number combinations nicknames. Two large & 4 small numbers were the Family Pack. One Large and 5 small numbers were the Single Parent. The 4 large and 2 small were the Heavyweight and the notorious 6 small numbers were called the Rat Pack. I can't remember what 3 large and 3 small numbers were called.

Your request for improved code has sniffs of Matt Parker's wordle code recently. That got into milliseconds from an original couple of hours. Expect improvements!

I was on Countdown 20 years ago and asked Carol if she used factorization, and she said she normally just aimed near the target, and adjusted as necessary - the "pitch and putt" approach, I guess. People might be interested (or even surprised) to know that if the target is 999, you can get it with 1 1 4 6 6 7...

Richard Ayoade using a 3 & 2 to cube and then square respectively, is some of the best TV ever

I think there's a more obvious reason why 7 and 9 perform so well. Odd numbers allow you to change the odd/even property of your total by addition and subtraction, and even numbers do not. Even numbers multiplied by anything are even. So odd numbers can probably access more target numbers by making it easier to flip between odd and even, or preserve oddness under multiplication.

I love watching 8 Out of 10 Cats Does Countdown. Especially Jon Richardson v. Sean Lock (RIP, 2-time Carrot in a Box Champion). Their chemistry is absolutely incredible. What are your favorite team captains/players?

"Remain Indoors" that's brilliant and also that's Numberwang!

As an American, I discovered British panel and game shows through QI and since then I have completely fallen in love with them. Would I Lie To You and QI being my two favorites, but I have at least watched most of them. The world of these shows is vast and with a recurring cast of comedians, all with the kind of quiet grace of British TV which is so, so much better than the loud and drama-filled US TV. If I had been born in the UK, I imagine I would completely and totally agree with you about how strongly I identify with these shows.

37 on the thumbnail. Veritasium video broke me

as a non-brit that's really gotten into British TV recently, this entire episode and especially that "REMAIN INDOORS" was very much welcome.

Being French myself I have to set the record straight. The French game « Des chiffres et des lettres (numbers and letters) » is actually a merger of two older games, « le mot le plus long » (the longest word) and « le compte est bon » (countdown). Or so I thought. I don’t know the old formats as they merged long before I was born. According to Wikipedia « le mot le plus long » was renamed « des chiffres et des lettres » following the addition of the countdown part.

"We would also go through another two mathematicians." Brilliant! I wish this had been around before I was on Countdown :D The only real preparation I did before going on was I decided to consistently choose three large and three small, which is still a relatively unusual choice. Choosing 1, 2 or 4 is more common overall, and that was one of the reasons I went with three, my opponents were immediately wrong-footed by it. Additionally, I noticed that a lot of players forget that they can use two large numbers to create a small, so I tended to treat three large and three small as one large and four small (in effect), selecting the most utilisable large number for the "pitch" and then using the other two as a small number in disguise :D On one occasion, n the semi-final, I had to do hat in reverse, using two sevens to get close to 50 and then multiply that up to the target number, unfortunately I overthought it and ran out of time before completing the solution (which would've got me five points, IIRC).

I was once a contract programmer for Channel 4, and they used Java for some of their back-end. To practise it, since I wasn't overly familiar with the language, I decided to develop a Countdown numbers game solver, because hey, it's Channel 4! You select your 6 numbers and target and used a brute force approach using Reverse Polish Notation, since it translates well into programming via a stack. I actually used the "famous numbers game" of 25 50 75 100 6 3 with a target of 952 as a test case, and it came up with quite a few interesting results, such as making 927 first (which is convenient as it's a multiple of 3) then adding 25 at the end (in this case, the solutions tended to be permutations of either making 927 and adding 25, or making 23,800 and dividing by 25).

Numberwang-adjacent reference earned my subscription. Well done, sir.

I think it’s also worth noting the time limit and the sheer familiarity contestants have with the traditional set of large numbers when considering what makes a set of large numbers “challenging”. Like you said, you can always divide by 25 in the original set to get a 2 or 3, but that’s not possible in any other set. Those rules of thumb go out the window when returning contestants are faced with new sets of large numbers. It’s also generally more tricky to multiply something by 19 or 64 on the fly than by 25, 50, or 100. I think that’s also why that guy couldn’t figure out what his partial solve would be when he multiplied it by 75 on the fly. Most people generally can’t rely on rules of thumb for multiplying three-digit partial solves by large numbers other than 25, 50, or 100. Doing all that comparatively complicated arithmetic in 30 seconds would, indeed, challenge the champions, I think.

The theme tune to Countdown was written by, amongst others, Brian Bennett who also played on the recording. The source of this information was from Brian himself when he appeared on 'Face to Face' with Rick Wakeman (it's on You Tube). For those who don't know who Brian Bennett is, he replaced Tony Meehan in October 1961as the drummer of The Shadows.

reverse polish notation is what I thought of immediately for finding the solution spaces so I'm glad it was mentioned at the end

Oh I'm so hyped for this, always have been intrigued by this game and also other similar arithmetic number puzzles like 24 or the Four 4s and I haven't seen a ton of exploration from them so this is cool to see

The fact that countdown has been going on for 41 years (at the time of the video) and the video just so happens to be 41 minutes long is awesome

When I was doing my first year studying Maths in university, one of our modules, the statistics one, had a piece of coursework where we had to use what we'd learned up to that point to analysis a probability distribution of our choice. Some examples of ones we could use were given, which most people used, but I went and did an analysis on how many different Numbers rounds could be made for Countdown. Because it was just an analysis of the probabilities of each set if numbers and not if any were possible, I didn't go too far, but I had been told that if I'd wanted to do a further analysis for a dissertation in my third year, then a few of the lecturers would have interest in working on it with me. Didn't end up doing that, but it was something I wished I had done. EDIT: Because I did my coursework on a notepad that exportedd what I wrote digitally, I was able to find what I did in the cloud and checked my numbers with what you had in the possible games portion of this video and had the same as you, aside from the bit at the end where I multiplied 13,243 by 899, due to 100 not really being a possible target in the main show. This meant I had 11,905,457 total possible games.

I live in the US. I went to the UK in vacation in 2015. My British friend introduced me to Countdown and 8 out of 10 Cats does Countdown. I still watch 8 out of 10 and discovered several dozen panel shows hosted by comedians and then all of them guest on the others shows. I also have the Countdown clock on my phone.

I can't believe you explained that famous numbers round (@29:16) and now I completely understand how he did it. It just takes some mathemagic, and you look like a genius.👌

The intuitive explaination behind why 1 and pairs of small numbers don't achieve many results really tickled my brain, really elegant and, well, intuitive interpretation.

Honestly, it's completely unsurprising that the multiples of 25 are the worst performing set of large numbers, there is a lot of redundancy in that set. That said, there is a massive difference between "possible for a computer to do" and "easy for a human to do in 30 seconds".

Recently went down the rabbit hole of 8/10 cats does Countdown (I'll let you guess why). So much more fun and thought provoking than the trivia shows I grew up with. Thanks for the awesome explanation to all my questions I didn't know I had!

In Spain, there was a similar show, also based on the french one, called Cifras y Letras. It ran between 1991 and 2012.

Thanks so much for this! Always when I watch countdown I wonder these things but then forget to look into them! You've cured me!

The University Challenge clip you used was from the season I was on. I played against that SOAS team, they were excellent!

I'm not into maths at all, but this was an accessible, interesting and engaging video. Subscribed!

One concern I have with your calculations is the difference between total count of sets and the probability of each set. I'm not sure if it's accounted for somewhere (maybe combinations vs permutations?) and I missed it, but a set like 100 75 50 25 2 1 is much more likely than 100 75 50 25 1 1, as the 2 1 can be drawn as 2 1 or 1 2, which is drawn at a 2.1% likelyhood, where 1 1 is drawn at a 0.05% likelyhood. in this case with 4 large numbers, any duplicate occurs 5.2% of the time (1/19), which is FAR lower than the expected 10% found using simple permutations.

First time on this channel and what an excellent video! I have long thought about analysing Countdown but you have excelled with your approaches. Fantastic :D *subscribed*

As both a fan of Countdown and a mathematician, I really enjoyed this video! Great content and it was a super interesting watch! You have a new subscriber!

Great video! I think there is also an important non-mathemaitcal reason for choosing the big numbers, which is - human intuition (or at least, decimal-based culture raised human's intuition). Though there are more sollutions with n=19, as shown in the video - we think in 5s and 10s. And by "we" I mean both contestants at the studio and viewers at home, who need to follow the game and feel it is relatable for them.

Thank you SO much for this. I’ve finally realise why the Australian version is called “Letters and Numbers”, rather than the more famous “Countdown”. If you’re going to license a show, you might as well only pay one rights holder (ie French), rather than paying two (French and British) when the second one is just a name.

I've only seen two videos of this channel and I already love it ! I should go on an Investigation to find other channels like these... To Focus on a subject and give intuition about it is the best way to learn !

Your editing keeps getting better, love watching ur videos :D

90 minutes for one set of numbers? Well, if we get some of Matt Parker’s viewers to tackle the problem, they should be able to come up with a program that runs in a few milliseconds, considering how they solved a problem he needed a whole month for in less than a second. Simply switching to a different language (or maybe even just using pypy) would probably give a speedup of 10 to 100 times.

This game has existed in France since 1972. One of the winners of multiple shows in 1975 became one of the show co-hosts and stayed until... 2022!

I love how just that rhythm of the last few seconds of countdown music can be tapped on a table and it has its own meaning. It gets used a lot when playing scrabble in our home...

loving this way of visulisation. explains way better than my thought process. i thank you

I have both a degree in Applied Mathematics, and Computer Science. But couldn't be bothered to solve your ending conundrum without being able to win a teapot!

Nice vid. I actually would have done 20 choose n for my small nos, because that's weighting the sets by their probability of coming up, which is what you care about as a player.

What a great find your channel is for me. I recently searched KZfaq for videos of the show Countdown, so the algorithm put this video on my start page. Sometimes the algorithm isn't so bad...

This Investigation into the mathematics behind the Countdown game is beautifully done! I must say the production quality of your videos is extraordinary - supernatural, even...

Hyped for the video! Your content has been nicely evolving

mind blown. I love University Challenge, I spent a few months watching a lot of TV and Quiz night on BBC2 was something I always looked forward to. Whenever the camera pans to the team conferring, I always imagine them saying "Venezuela Venezuela" to each other, I think it was in a sketch show from some point in the last 30 years where on team just continually gives Venezuela as the answer. I'm just made up that somebody actually did it on TV

I appreciate the effort this video took, and loved it. Thanks!!

This was a LOVELY video!! Sure I enjoyed the math breakdown, but you are SO adorably British, and I learned so much about y'all's culture from this video!! Keep up the great work, mate!!

You are in my top 20 of all youtubers and 4 in math. Mathologer, 3b1b, and numberphile in the top 3. And numberphile is swingy based on guest. You are better then many of his guests but some guest top even Mathologer. I get excited every time you drop a video. Please keep it up. For reference, I actively watch 483 channels.

Another thing to take into account with the large numbers change is the time limit, although mathematically it maybe that the numbers are technically better, those nice round numbers are much simpler to work out quickly for most people.

What a great video, I got lost when you started talking about champion of champions numbers but other than that I was hooked

Excellent video! Fun to watch and interesting

REMAIN INDOORS Brilliant! I chowed down on my protein fudge as soon as I saw that.

"Upon solving a problem, you realize you weren't the first person to do it." Was it Euler again?

Excellent video and explanations. Thanks!

I am Italian and I also love TV quiz shows. In the 1980s we had our own version of Countdown: it was called "Paroliamo." In this show, there was the game with the numbers that was showed in the video and it was called "Contiamo" ("Let's Count"). I remember being a child and trying to do the math operations with paper, pen and calculator to make it faster!

You should try for a collab with Matt Parker. He has done similar types of videos for other numbers problems. He would love this

Just stumbled upon this randomly and am mightily impressed - I don't generally comment on videos. Thanks very much for this! I'm just diving back into Python, and I'm going to enjoy following your thought process!

Fun fact, the guy at 29:07 (James Martin) was my maths tutor at uni; he was so modest about it but you could tell how proud he was deep down that this achievement went viral and he became some recognised for it

I have not watch the whole video yet, but to consider the number of possible combinations of scoring numbers, I segment this by number of pairs. There are 10C3 = 120 ways of having 3 pairs of numbers (all small). There are 10C2 = 45 choices for 2 pairs, and 12C2 = 66 choices for the other 2 singles 45*66 = 2970. There are 10 choices for 1 pair, and 13C4 = 715. 715*10=7150. Finally there are 14C6 = 3003 possibilities for 6 unique numbers. So in total we have 120 + 2970 + 7150 + 3003 = 13243. Though this was done quickly, and may contain a mistake.

He stayed so calm through this! I was getting stressed trying to figure out how 3,3,2,2,1 and 1, could sum up to 81.

I have a sudden urge to rewatch that episode of IT Crowd hehe Thanks, awesome video!

RE: The point around 37:00 regarding Scary Numbers. Something I would have looked at is how divisible the large numbers are by each other. Given that a big part of the 'Pro Strat' seemed to rely upon the fact that [100/25 = 4; 75/25 = 3; 50/25 = 2]. The Countdown challenge is, crucially, a timed one. The difficulty doesn't revolve around a solution being possible; but a solution being divineable in 30 seconds. So relieving the Competitors of one of their chief tools would seem to have quite the effect.

I've got it down to ~90 seconds per deck (100 75 50 25, etc) with C# and Multithreading... I'm sure there are some other ways to improve it but that's where I'm at for now. Pretty much just implemented exactly what you talked about. I considered RPN for awhile as I'm well versed, but I can't see how it would help, and any memoization seems fruitless too since it's usually quicker for the CPU to just do the 5 math operations rather than look up a value amongst billions of others... But I'll think on it some more! I also used the fact that all the smalls can account for 2850 of the 13243 gamesets, so I never have to calculate those again.

I love Pointless in particular; the fact that zero points is the best, it requires Pointless trivial knowledge and the presenter and dynamics. Also love eggheads, university challenge and QI. You have a productive country in terms of quizzes and game shows 🤣

Really great presentation and a very informative video!

Remain Indoors! You are now my favourite mathematical youtuber. We don't talk about the event! Mitchell and Webb were my childhood!

Fascinating video!! B28 I34 F30

So I don’t have an answer for the challenge (yet at least) but I did recently code a program which can solve combinations with limited repetition, which would make counting the number of possible games easier. I also came up with a formula involving repeated sums, but my code, thankfully, does it a lot more efficiently than that. I found it was a problem which I was facing pretty regularly and so I made a program which can take in any number of objects and specify how many repeated objects of each type are allowed and spits out the number of combinations. I don’t want to explain my whole thought process and intuition of how I made it if no one’s interested, but if anyone is, let me know and I’d be happy to! Good luck to anyone tackling the challenge!

1:41 - yeah, I'm with you on this. UK game shows (or panel shows) got me throught covid lock-down

amazing analysis. Who ever knew this was what was going on in my head when I solved most of the numbers games before the clock expired...

The universally acknowledged Matt-Parker's-Law of "I'm just a mathematician" programming challenge clearly states: for every Python script whose runtime is over a minute, there exists a C programmer, whose sub-millisecond solution is hard to be measured precisely.

The best way to chose your set of numbers is always 4 little and 2 big.. just as Sean would have wanted. Rest in Power, Sean. Forever a cheeky bastard. ❤

I loved this show so much, but alas, being across the pond without cable, I haven't seen it for years. Thanks for tackling it here.

A complete nutter. An utterly unnecessarily deep analysis of a parlour game for no reason at all. A perfect example of how a mathematician thinks, if you want to call it that. A great video!

I would think part of the challenge with those scary numbers would be how uncommon it is for someone to work with those numbers. People are quite comfortable with working with multiples of 5, 25, and 100 all the time. When was the last time someone asked you to do something with a 37, especially with the pressure of going against a clock. So regardless of whether those "weird" numbers would help or hinder the contestants, at least from the audience's perspective it would seem more challenging.

13:30 I hope I understand correctly that you put the number into the solution set if it is in {100,...,999}, but still continued the process with it, since you can go well above 999 and still come back down to the acceptable range with division and subtraction

Loved the explanation (in Aus I think we derive the French name for the show - "Letter and Numbers") (and I spotted the edit for the 452 example around 22:00 :P )

I love the maths round of Countdown, and my favourite game set to work with is 3 large 2 small. [Apparently, The hardest game set] but your breakdown of this game has given me a better understanding of its system. Thank you.

definitely appreciated the Mitchell and webb reference

Fantastic video, while your on a roll I was wondering if you could explain the other big British number game; Numberwang! As a newcomer to the show I've struggled to reliably guess what is and is not Numberwang! and I'd love to hear your thoughts on how to improve ones Numberwang!ery.

4:00 Jimi hanging off the telephone wire and a shirtless telephone repair-man is what those seconds bring to my mind, IDK what your talking about xDDDDDD

Great job on explaining the game, as for your code. I love it .. thank you so much

I think uneven numbers make sense, because you can get to an even number easily with uneven numbers, but it's much harder to get to an uneven number with even numbers.

Enjoyed it, thank you. My mum was on Countdown in the 80's.

Many thanks for this excellent analysis.

as a canadian, i'm surprised there is no north american countdown. i've always loved countdown. great game :)

Holy shit I got the university challenge question right about Eigen values. That was deep storage knowledge awaked from the depths from when I was a game dev student in uni. I've never had cause to use that knowledge in my life.

I legitimately grinned at the flashing "Remain Indoors"... one of my favourite Mitchell & Webb sketches.

21:58 - Some slick editing - the 6x4 inside the parentheses magically corrects itself to 6x5 as the blackboard is placed away from view behind the counter. Play it in slo-mo and you'll see what I mean. Anyway, the content is superb, and I learned a huge amount. Many thanks for uploading. Please take care and stay safe.

I was really surprised by one episode where the target was unobtainable. For many years I thought CECIL was choosing some random series of operations on the given numbers that would lead to a number from 100 to 999. But it turns out it's just a random number generator, and the number of possible targets for most sets is so big that it usually just works out anyway!

I'm not a programmer, but for the random set of large numbers I'd extend your idea: If you have calculated the reachable targets for 11, 12 and four smalls, you can add that to any set of large numbers that contains 11 and 12. Similar for three large and three smalls, and four large and two smalls.

Great video, have watched countdown since I was little. The perfect amount of nerd + maths = genius content ❤

Interesting that a British institution such as Countdown was imported from France.

so i wrote what i would consider a rather efficient solver. takes around 1.5 seconds to check each set of 4 large numbers. so far i've checked every set containing the number 100: the worst of these is 100, 20, 16, 12 with 8686851 winnable games in total, or about 92.87%. then i started checking from the bottom up, with so far the worst set being 24, 18, 16, 12 with 8557017 winnable games, or 91.48%. for comparison, the standard 100, 75, 50, 25 set can win 95.21% of possible games (not 95.12%, which is still shown in the spreadsheet linked in the description). i'm pretty confident the 24, 18, 16, 12 set is going to end up being the worst one overall, due to how much small numbers keep showing up in the worst sets, but i've only checked the sets with 100 and then all sets with numbers up to 30. doing the rest of it would take around a month of computation. so unless someone manages to write a significantly better solver, i don't think i'm going to check all of the intermediate sets myself. in case anyone is curious, here's the 10 worst sets so far (and most likely the 10 worst overall): (24, 16, 15, 12) : 8587061 91.80% (24, 18, 14, 12) : 8586167 91.79% (20, 18, 14, 12) : 8584224 91.77% (20, 16, 15, 12) : 8582572 91.76% (24, 16, 14, 12) : 8569981 91.62% (18, 16, 14, 12) : 8566699 91.59% (20, 16, 14, 12) : 8562742 91.54% (24, 20, 16, 12) : 8560412 91.52% (20, 18, 16, 12) : 8559824 91.51% (24, 18, 16, 12) : 8557017 91.48% edit: further optimization got it down to 0.1 seconds per set, which allowed me to check all sets. (24, 18, 16, 12) is the worst overall, at least by this metric. if we consider the actual likelyhood of a game being winnable with each set, instead of how many of the total possible games are winnable, the worst set becomes (80, 64, 48, 32) with a probability of 93.45%. if anyone is interested then the code is on github. my username there is randomdude999, the repository is called countdown. youtube seems to hate me posting direct links to it though.

I watch all of those game shows. I watch Richard Osman's house of games for fun and only connect when I'm feeling too smart because I can't get any of it. From the us but travel to the UK a few hours pretty much every night. Thanks for the cool video