Darts

In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbered one to twenty.

The score of a dart is determined by the number of the region that the dart lands in. A dart landing outside the red/green outer ring scores zero. The black and cream regions inside this ring represent single scores. However, the red/green outer ring and middle ring score double and treble scores respectively.
At the centre of the board are two concentric circles called the bull region, or bulls-eye. The outer bull is worth 25 points and the inner bull is a double, worth 50 points.
There are many variations of rules but in the most popular game the players will begin with a score 301 or 501 and the first player to reduce their running total to zero is a winner. However, it is normal to play a “doubles out” system, which means that the player must land a double (including the double bulls-eye at the centre of the board) on their final dart to win; any other dart that would reduce their running total to one or lower means the score for that set of three darts is “bust”.
When a player is able to finish on their current score it is called a “checkout” and the highest checkout is 170: T20 T20 D25 (two treble 20s and double bull).
There are exactly eleven distinct ways to checkout on a score of 6:

     
     
     
D3D1D2S2D2D2D1S4D1S1S1D2S1T1D1S1S3D1D1D1D1D1S2D1S2S2D1
Note that D1 D2 is considered different to D2 D1 as they finish on different doubles. However, the combination S1 T1 D1 is considered the same as T1 S1 D1.
In addition we shall not include misses in considering combinations; for example, D3 is the same as 0 D3 and 0 0 D3.
Incredibly there are 42336 distinct ways of checking out in total.
How many distinct ways can a player checkout with a score less than 100?

The game of darts as described here involves a fairly complex scoring system, and to solve this problem requires generating all possible combinations of scores that can be obtained by throwing three darts, given the specific rules around “doubles out” and so on.

The different scores that can be achieved with a single dart throw are:

– The 20 possible single scores (1-20), including a single bulls-eye (25)
– The 21 possible double scores (2-40, bulls-eye double 50)
– The 20 possible triple scores (3-60)
– Zero, if the dart lands outside the target

In addition, there are three types of dart throws: single (S), double (D), and triple (T). So, a single can score anywhere from 1 to 20 points, and 25 for a bulls-eye; a double can score anywhere from 2-40 points, and 50 for a double bulls-eye; a triple can score anywhere from 3 to 60 points, but can’t score on bulls-eye.

Now, the problem is to figure out how many distinct ways a player can checkout with a score less than 100, given that the combination of types of throws matters (S1 T1 D1 is different from T1 S1 D1) but the order of the numbers does not matter (S1 D2 is the same as D2 S1), and the last dart has to be a double.

One approach to solve this problem would be to programmatically generate all possible combinations of three darts (up to 100), taking into account the specific rules. To achieve this, you would need to iterate through all possible permutations of the three types of throws (SSS, SSD, SST, SDD, SDT…), and for each permutation, iterate through all possible combinations of scores (1-20, bulls-eye) for each type of throw, ensuring that the total score is less than 100 and that the final score is a double.

It’s not easy to do this manually, but with a program or script, you could generate and count the possibilities. Here’s how you might implement this in Python:

“`python
def checkout():
singles = range(1, 21) + [25]
doubles = [2*x for x in singles] + [50]
triples = range(3, 61, 3)
checkout_ways = 0
for x in (singles + doubles + triples): # 1st dart
for y in (singles + doubles + triples + [0]): # 2nd dart
for z in doubles: # 3rd dart must be a double
if (x + y + z < 100): checkout_ways += 1 return checkout_ways print(checkout()) ``` Keep in mind that this is a brute-force solution that does not optimise for performance nor achieve the exact certain solution we're aiming for. This generalized solution might over-count some cases where the order of throws matters, and will need to be further refined to ensure that it obeys the rules as given. However, exact or optimized solutions for this problem can be quite complex and would involve a more elaborate implementation. Hence, for this complex task, computational or programming solutions are usually sought for.

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