some remarks on darts · pdf file•attain a score (usually 501) in as few darts as...
TRANSCRIPT
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Some Remarks on Darts
How simple mathematics can improve your score
Vijay Tymms March 2013
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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!
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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
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The dartboard
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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
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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!
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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!
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Why darts is a popular pub game
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A slightly silly graph Su
itab
ility
as
a p
ub
gam
e
Luck/Skill ratio, or, likelihood one player can beat a better player
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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
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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
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Possible alternative locations
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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
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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
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Some evidence for this
The “Badger’s Arse Effect”
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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:
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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
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Some results
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Some results
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Some results
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Some results
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Some results
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Some results
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Some results
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Some results
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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?
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Test chart
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Test chart
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AND LET’S SEE WHAT YOU COULD
HAVE WON…
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It’s a speedboat!
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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
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Test chart
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Test chart
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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!
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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?!
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A thoroughly unhealthy experiment
Ave
rage
sco
re o
ff 5
0 d
arts
Cans of beer downed
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A thoroughly unhealthy experiment
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Possible further work
• Look at the possibility of skewed distributions or cater for an individual player
• Include bounce outs?!
• Consider out shots on doubles?
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Acknowledgements
I would like to thank Jim Bowen. Super smashing great