|
| 1 | +""" |
| 2 | +In the game of darts a player throws three darts at a target board which is |
| 3 | +split into twenty equal sized sections numbered one to twenty. |
| 4 | + |
| 5 | +The score of a dart is determined by the number of the region that the dart |
| 6 | +lands in. A dart landing outside the red/green outer ring scores zero. The black |
| 7 | +and cream regions inside this ring represent single scores. However, the red/green |
| 8 | +outer ring and middle ring score double and treble scores respectively. |
| 9 | +
|
| 10 | +At the centre of the board are two concentric circles called the bull region, or |
| 11 | +bulls-eye. The outer bull is worth 25 points and the inner bull is a double, |
| 12 | +worth 50 points. |
| 13 | +
|
| 14 | +There are many variations of rules but in the most popular game the players will |
| 15 | +begin with a score 301 or 501 and the first player to reduce their running total |
| 16 | +to zero is a winner. However, it is normal to play a "doubles out" system, which |
| 17 | +means that the player must land a double (including the double bulls-eye at the |
| 18 | +centre of the board) on their final dart to win; any other dart that would reduce |
| 19 | +their running total to one or lower means the score for that set of three darts |
| 20 | +is "bust". |
| 21 | +
|
| 22 | +When a player is able to finish on their current score it is called a "checkout" |
| 23 | +and the highest checkout is 170: T20 T20 D25 (two treble 20s and double bull). |
| 24 | +
|
| 25 | +There are exactly eleven distinct ways to checkout on a score of 6: |
| 26 | +
|
| 27 | +D3 |
| 28 | +D1 D2 |
| 29 | +S2 D2 |
| 30 | +D2 D1 |
| 31 | +S4 D1 |
| 32 | +S1 S1 D2 |
| 33 | +S1 T1 D1 |
| 34 | +S1 S3 D1 |
| 35 | +D1 D1 D1 |
| 36 | +D1 S2 D1 |
| 37 | +S2 S2 D1 |
| 38 | +
|
| 39 | +Note that D1 D2 is considered different to D2 D1 as they finish on different |
| 40 | +doubles. However, the combination S1 T1 D1 is considered the same as T1 S1 D1. |
| 41 | +
|
| 42 | +In addition we shall not include misses in considering combinations; for example, |
| 43 | +D3 is the same as 0 D3 and 0 0 D3. |
| 44 | +
|
| 45 | +Incredibly there are 42336 distinct ways of checking out in total. |
| 46 | +
|
| 47 | +How many distinct ways can a player checkout with a score less than 100? |
| 48 | +
|
| 49 | +Solution: |
| 50 | + We first construct a list of the possible dart values, separated by type. |
| 51 | + We then iterate through the doubles, followed by the possible 2 following throws. |
| 52 | + If the total of these three darts is less than the given limit, we increment |
| 53 | + the counter. |
| 54 | +""" |
| 55 | + |
| 56 | +from itertools import combinations_with_replacement |
| 57 | + |
| 58 | + |
| 59 | +def solution(limit: int = 100) -> int: |
| 60 | + """ |
| 61 | + Count the number of distinct ways a player can checkout with a score |
| 62 | + less than limit. |
| 63 | + >>> solution(171) |
| 64 | + 42336 |
| 65 | + >>> solution(50) |
| 66 | + 12577 |
| 67 | + """ |
| 68 | + singles: list[int] = [x for x in range(1, 21)] + [25] |
| 69 | + doubles: list[int] = [2 * x for x in range(1, 21)] + [50] |
| 70 | + triples: list[int] = [3 * x for x in range(1, 21)] |
| 71 | + all_values: list[int] = singles + doubles + triples + [0] |
| 72 | + |
| 73 | + num_checkouts: int = 0 |
| 74 | + double: int |
| 75 | + throw1: int |
| 76 | + throw2: int |
| 77 | + checkout_total: int |
| 78 | + |
| 79 | + for double in doubles: |
| 80 | + for throw1, throw2 in combinations_with_replacement(all_values, 2): |
| 81 | + checkout_total = double + throw1 + throw2 |
| 82 | + if checkout_total < limit: |
| 83 | + num_checkouts += 1 |
| 84 | + |
| 85 | + return num_checkouts |
| 86 | + |
| 87 | + |
| 88 | +if __name__ == "__main__": |
| 89 | + print(f"{solution() = }") |
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