#
Useless Darts Trivia

##
Which scores are finishes?

One can throw 36, 40, or 50 plus any triple up to 120, and
these numbers are 0, 1, and 2 modulo 3 respectively.
In a picture:
... 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170
triple+36 * * *
triple+40 * * * *
triple+50 * * * * * * *
finishes: ... 150 151 152 153 154 155 156 157 158 160 161 164 167 170

##
How many possible 9-darter games are there?

This seems rather dificult to count until we start to distinguish cases
based on final double and minimum of the first 8 darts.
The 50,50 entry for instance is calculated as
( 8 ) ( 8 ) ( 8 )
(2 1) + (2 1 1) + (3 2) = 168+840+560 = 1568.
\dbl 24 30 34 36 40 50
min\ --------------------------------
34 56
40 672
45 8
48 56
50 56 672 1568
51 8 224
54 56 448
57 8 56 56

Giving a grand total of 3944 possible 9-darters, as confirmed in
this
report, which also states the number of essentially different
solutions (as a multi-set of first 8 dart scores) as 22.
Note that a finishing double of e.g. 38 is not possible, as there
is no way to partition 501-8*60-38 = -17 into the numbers
-3, -6, -9, -10, -12, and -15.

In practice you'll only see 9-darters starting with two 180s,
leaving a 141 finish. The number of ways to finish 141 is calculated similarly:

\dbl 24 30 34 36 40 50
min\ --------------------------------
34 2
40 2
45 2
48 2
50 2 2
51 2 2
54 2
57 2
That's 20 ways.

##
What is the most impressive 9-darter?

That would be the one leaving the maximum number possible at each step.
If the first two darts total 91 points then a 9-darter requires
all remaining darts to hit triple 20 until the final bull's eye.
Since 91-60=31 is impossible to throw, the next hardest is
a starting throw of double 17, leaving triple 19 for the 2nd dart:

34+57+60=151, 60+60+60=180, 60+60+50=170.

So, if you ever see someone opening with a double 17, you know they
either have very pooor aim, or are being very ambitious.

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john.tromp@gmail.com