The Surprising Maths of Britain's Oldest* Game Show

⬣ LINKS ⬣

⬣ ABOUT ⬣
First broadcast in 1982, Countdown is iconic British TV. Its numbers game is the perfect balance of challenge and simplicity. In this video, I analyse the hidden mathematics of the game: What are the hardest targets, best numbers to draw, and optimal tactics?

⬣ TIMESTAMPS ⬣
00:00 - Introduction
04:46 - How Many Possible Games?
10:00 - Reachable Numbers from a Given Game Set
14:00 - Results and Tactics: Small Numbers
24:00 - Results and Tactics: Large Numbers
31:00 - Scary Numbers
40:05 - Outro

⬣ CHALLENGE ⬣
So to clarify, I want to see a list of the percentage of solvable games for ALL options of large numbers. Like I did for the 15 options of the form {n, n+25, n+50, n+75}, but for all of them. The options for large numbers should be four distinct numbers in the range from 11 to 100. As I said there are 2,555,190 such options so this will require a clever bit of code, but I think it’s possible! Email me via my website if you think you have it!

⬣ FILES ⬣

⬣ INVESTIGATORS ⬣
I’ve never seen that colour on my screen before. I’m hoping you excel yourself and slug out the solution. Now is the perfect time to join the investigation.

⬣ REFERENCES ⬣
Blog I mention which uses reverse Polish notation:

⬣ CREDITS ⬣
All music by Danjel Zambo.

Images

B Footage
8 Out of 10 Cats Does Countdown, Countdown © Channel 4.
The Chase © ITV.
Have I Got News for You, Only Connect, Pointless, Richard Osman’s House of Games, University Challenge, The Weakest Link, Would I Lie to You? © BBC.
Des Chiffres et Des Lettres © France 3.

[AnotherRoof]

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.

[themattylee]

Looking forward to your Numberwang episode.

[PaulMDavidson]

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.

[fyshman]

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

[malalford]

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

[janemcdonald]

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.

[kevinthurlow]

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...

[cemkesici]

37 on the thumbnail. Veritasium video broke me

[DukeofEarl]

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!

[benwright]

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.

[Scum]

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.

[aviasegel]

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

[moisemust]

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.

[victorenciso]

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

[Alan_Duval]

"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).

[patrickchampion]

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

[cameronbigley]

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?

[WaterCrane]

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).

[mattlingpoo]

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

[Crossark]

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.