Some Remarks on Darts

How simple mathematics can improve your score

Vijay Tymms March 2013

Background and pre-empting some reactions

• This was some “work” I did when a PhD student at the University of Liverpool

• Possible question which might be at the back of you minds: “How did he manage to get a PhD whilst doing stuff like this?”

• Possible slightly incredulous reaction: “Jeez, what a geek!”

It was a major chunk of procrastination that really delayed the write up Guilty as charged!

Aims of this work

• To show, using probability arguments, the best way to get a good score in a game of darts

• Demonstrate that casual players can improve their dart game

• Show that most amateur players have their tactics wrong

• Provide some simple instructions on how to improve to players at all levels

The dartboard

The standard game of darts

• Attain a score (usually 501) in as few darts as possible

• Must finish with a double or the bullseye • Throw in turns with three darts per turn (max

score 180) • Two stages (i) amass enough points to get an

“outshot” on a double (ii) finish the game by hitting a double

• This work concentrates on stage (i) how best to accumulate a high score to get within range of an outshot

Other variants

• Round the clock

• Killer

• Cricket

• These are all popular pub games

• One reason is that darts has a degree of luck to it!

Why darts is a popular pub game

• One reason is that because of the points distribution there is an unpredictable nature to a player’s score

• On a single turn a poor player can score higher than a better player

• This makes the game popular and addictive. Like gambling, a poor player can win now and again against a better player

• Concentric circle darts wouldn’t be as popular!

Why darts is a popular pub game

A slightly silly graph Su

itab

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as

a p

ub

gam

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Luck/Skill ratio, or, likelihood one player can beat a better player

An issue with casual players’ tactics

• Players often aim for the treble 20 to maximise their score

• The 20 is flanked by the 5 and the 1; the player risks getting the classic “bed and breakfast” score of 26 or worse

How the tactics may be improved

• An elite player is best off going for the treble 20

• An absolute beginner is best off merely trying to hit the board – they should aim right for the centre to try and get a score at all

• An intermediate player may be best off aiming somewhere else

Possible alternative locations

An idea to investigate the problem

• Write a computer program to investigate what average score a player will get if they aim for any part of the board

• It should cater for different levels of skill

• It should predict excellent players should aim for the treble 20 and poor players for the centre

• It should make testable predictions about where intermediate players should aim

An assumption: normal throwing distribution

• The model assumes that a player’s shot follows a normal distribution in 2D

• The better the player the narrower the distribution

• The width of the curve can be calculated from the percentage of darts that land in the bullseye when aimed for

Some evidence for this

The “Badger’s Arse Effect”

Creating a pixellated dart board

• Splits a square into 200 x 200 pixels with a standard dart board in the centre and creates a contour plot:

Scanning the normal distribution over every pixel

• A player enters what percentage of darts land in the bullseye when aiming for it

• The program calculates the width of the normal distribution curve

• Program calculates what score is accrued by aiming at each pixel on the board

Some results

Some results

Some results

Some results

Some results

Some results

Some results

Some results

Assessing your own level 1: measure % in bullseye

• Two ways to see where it’s best to aim

1. Throw lots of darts at the bull and see what percentage land in it

• The more darts you throw the more meaningful the percentage will be

• Let’s try it anyway with just ten darts – volunteer?

Test chart

Test chart

AND LET’S SEE WHAT YOU COULD

HAVE WON…

It’s a speedboat!

Assessing your own level 2: taking an average when aiming at the bull(*)

2. Aiming at the centre of the board, throw 10 darts and record your average score

• Refer to the test chart – which matches you bullseye average?

• Volunteer?

(*) Or anywhere actually, the same principle applies

Test chart

Test chart

Conclusions

• It is worth going for the treble 20 if you can get 3/10 shots or more in the bull

• If you get 1 or 2 out of 10 in the bull it is best to aim just below the treble 20

• If 6-10% land in the bull there is a sweet spot around the treble 7 and 16 area that gets the highest score

• Less than this in the bull and you’re better off aiming for the centre

• Most casual players probably fall into the latter category!

Some caveats

• The throw distribution may differ from the assumed one • If you want a chance at beating a significantly better player

than yourself the best option may be to aim for the highest score

• Aiming for the treble 20 may be a more enjoyable way of playing the game – the pleasure of hitting it once in a while may override getting merely decent average scores

• Darts is a pub game and involves drinking. Does the best place to aim for change with increasing booze comsumption? You get worse with lots of drink but do a couple of beers actually help?!

A thoroughly unhealthy experiment

Ave

rage

sco

re o

ff 5

0 d

arts

Cans of beer downed

A thoroughly unhealthy experiment

Possible further work

• Look at the possibility of skewed distributions or cater for an individual player

• Include bounce outs?!

• Consider out shots on doubles?

Acknowledgements

I would like to thank Jim Bowen. Super smashing great