Minimum impossible score in darts. I was recently playing darts when I began pondering the question of the lowest whole number that cannot be scored in a game of darts.
For clarification, these are the possible scoring options:

*

*Each turn consists of throwing three darts, each with its own score independent of the other two.

*A standard dartboard has the numbers 1-20, along with double and triple scoring places for each number.

*In the center, there is a small ring that gives a score of 25, as well as the bullseye with a score of 50.

*For this problem, scoring 0 by missing the board entirely is completely acceptable.


It is simple to write a Python program to compute the answer to this question. The result is $163$. I cannot think of any mathematical explanation for this.
dart_scores = set([0, 25, 50]) 

for sector in range(1, 21):
    dart_scores.add(sector)
    dart_scores.add(2 * sector)
    dart_scores.add(3 * sector)

three_dart_scores = set([a + b + c for a in dart_scores for b in dart_scores for c in dart_scores])

smallest_unseen = 0
while smallest_unseen in three_dart_scores: 
    smallest_unseen += 1
    
print(smallest_unseen)

