## 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 partitions of 501 - 50 - 8 * 50 = 51 into 0s, 1s, 4s, 7s, and 10s, with at least one 0:

```
( 8 ) ( 8 ) ( 8 )
(2 1 0 0 5) + (2 0 1 1 4) + (2 0 0 3 3) = 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 throwing three 167 “finishes” in a row, repeating triple 20, triple 19, and bulls eye. The bulls eye is harder to hit than any triple, and a 9-darter can include at most three of them. I will likely never witness that in my lifetime, but then who could have imagined all of these incredible rarities?