mscroggs.co.uk
mscroggs.co.uk
Click here to win prizes by solving the mscroggs.co.uk puzzle Advent calendar.
Click here to win prizes by solving the mscroggs.co.uk puzzle Advent calendar.

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.
@Francis Galiegue: I've pushed a version of the code to https://github.com/mscroggs/countdown-...
Matthew
                 Reply
@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.
Matthew
                 Reply
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 :)
Francis Galiegue
                 Reply
 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 "y" then "-" then "a" then "x" then "i" then "s" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Nov 2020

Christmas (2020) is coming!

Jul 2020

Happy √3+1 Approximation Day!

May 2020

A surprising fact about quadrilaterals
Interesting tautologies

Mar 2020

Log-scaled axes

Feb 2020

PhD thesis, chapter ∞
PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

game of life nine men's morris frobel dataset estimation gaussian elimination binary interpolation data visualisation signorini conditions finite element method bodmas latex mathslogicbot tmip talking maths in public go logic map projections palindromes national lottery mathsjam hats radio 4 statistics dragon curves martin gardner dates a gamut of games propositional calculus big internet math-off realhats approximation simultaneous equations squares coins determinants computational complexity python asteroids braiding pac-man hexapawn captain scarlet golden spiral numerical analysis exponential growth golden ratio puzzles video games data javascript rhombicuboctahedron trigonometry geometry matrices sorting speed stickers countdown oeis pi matrix multiplication inverse matrices logs manchester curvature tennis ternary hannah fry news london underground harriss spiral noughts and crosses weather station game show probability royal baby advent calendar platonic solids european cup weak imposition light graph theory craft twitter geogebra wool error bars graphs menace matrix of cofactors sobolev spaces gerry anderson wave scattering electromagnetic field plastic ratio programming folding tube maps convergence chebyshev reuleaux polygons rugby matrix of minors chess reddit games flexagons preconditioning books raspberry pi people maths machine learning christmas card cambridge folding paper bubble bobble bempp mathsteroids manchester science festival ucl accuracy royal institution sound sport php matt parker fractals pi approximation day cross stitch draughts boundary element methods chalkdust magazine london arithmetic the aperiodical inline code triangles probability christmas misleading statistics final fantasy polynomials pythagoras quadrilaterals pizza cutting world cup phd football

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020