mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Countdown probability, pt. 2

 2014-04-11 
As well as letters games, the contestants on Countdown also take part in numbers games. Six numbers are chosen from the large numbers (25,50,75,100) and small numbers (1-10, two cards for each number) and a total between 101 and 999 (inclusive) is chosen by CECIL. The contestants then use the six numbers, with multiplication, addition, subtraction and division, to get as close to the target number as possible.
The best way to win the numbers game is to get the target exactly. This got me wondering: is there a combination of numbers which allows you to get every total between 101 and 999? And which combination of large and small numbers should be picked to give the highest chance of being able to get the target?
To work this out, I got my computer to go through every possible combination of numbers, trying every combination of operations. (I had to leave this running overnight as there are a lot of combinations!)

Getting every total

There are 61 combinations of numbers which allow every total to be obtained. These include the following (click to see how each total can be made):
By contrast, the following combination allows no totals between 101 and 999 to be reached:
The number of attainable targets for each set of numbers can be found here.

Probability of being able to reach the target

Some combinations of numbers are more likely than others. For example, 1 2 25 50 75 100 is four times as likely as 1 1 25 50 75 100, as (ignoring re-orderings) in the first combination, there are two choices for the 1 tile and 2 tile, but in the second combination there is only one choice for each 1 tile. Different ordering of tiles can be ignored as each combination with the same number of large tiles will have the same number of orderings.
By taking into account the relative probability of each combination, the following probabilities can be found:
Number of large numbersProbability of being able to reach target
00.964463439
10.983830962
20.993277819
30.985770510
40.859709475
So, in order to maximise the probability of being able to reach the target, two large numbers should be chosen.
However, as this will mean that your opponent will also be able to reach the target, a better strategy might be to pick no large numbers or four large numbers and get closer to the target than your opponent, especially if you have practised pulling off answers like this.
Edit: Numbers corrected.
Edit: The code used to calculate the numbers in this post can now be found here.

Similar posts

Countdown probability
Pointless probability
Big Internet Math-Off stickers 2019
World Cup stickers 2018, pt. 3

Comments

Comments in green were written by me. Comments in blue were not written by me.
 2016-07-20 
@Francis Galiegue: I've pushed a version of the code to https://github.com/mscroggs/countdown-numbers-game
Reply
Matthew
 2016-07-20 
@Francis Galiegue: Sadly, I lost the code I used when I had laptop problems. However, I can remember what it did, so I shall recreate it and put it on GitHub.
Reply
Matthew
 2016-07-20 
If you could, I'd love to have the code you used to do this exhaustive search?

I'm a fan of the game myself (but then I'm French, so to me it's the original, "Des chiffres et des lettres"), but for the numbers game, this is pretty much irrelevant to the language and country :)
Reply
Francis Galiegue
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "m" then "e" then "d" then "i" then "a" then "n" in the box below (case sensitive):

Archive

Show me a random blog post
 2019 

Jul 2019

Big Internet Math-Off stickers 2019

Jun 2019

Proving a conjecture

Apr 2019

Harriss and other spirals

Mar 2019

realhats

Jan 2019

Christmas (2018) is over
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

fractals latex speed sound mathsjam chess inline code countdown accuracy menace bubble bobble go data weather station stickers probability a gamut of games matt parker error bars cross stitch palindromes nine men's morris python electromagnetic field manchester raspberry pi bodmas football braiding interpolation christmas craft game of life polynomials propositional calculus ternary map projections wool oeis london arithmetic martin gardner trigonometry graph theory pac-man games logic reddit plastic ratio rugby the aperiodical light sorting programming people maths gerry anderson flexagons books estimation london underground mathsteroids asteroids binary coins hexapawn pizza cutting draughts reuleaux polygons php pythagoras tennis harriss spiral golden spiral realhats game show probability triangles royal baby misleading statistics folding paper puzzles platonic solids captain scarlet video games dates statistics christmas card european cup world cup machine learning javascript news folding tube maps mathslogicbot final fantasy approximation big internet math-off golden ratio dataset radio 4 chalkdust magazine geometry hats national lottery noughts and crosses curvature sport manchester science festival dragon curves chebyshev rhombicuboctahedron frobel twitter

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2019