# 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.
• $163$ ${}{}{}{}{}{}{}{}$ Commented Oct 21, 2021 at 23:11
• I apologize for the misinformation. Two darts with the same score are acceptable. Commented Oct 21, 2021 at 23:16
• @JamesA, could you explain your answer further. I would happily accept an answer if you chose to write one. Commented Oct 21, 2021 at 23:34
• Why not 0? See the last statement. May be you want to edit the question with a claim that at least one dart hits the dartboard.
– Moti
Commented Oct 22, 2021 at 1:36
• What is the relevance of three darts? Are not you really asking what sum can not be made with the values 1-20, 25 and 50?
– Moti
Commented Oct 22, 2021 at 1:38

Score ranges are inclusive of both endpoints, e.g. $$0-60$$ means $$[0,60]$$. Some ranges may overlap (for simplicity's sake). $$\begin{array}{|c|c|} \hline \text{Score(s)} & \text{How to get} \\ \hline 0-60 & a+b+c;\ a,b,c\in[0,20] \\ \hline 61-100 & 3\times20+b+c;\ b,c\in[0,20] \\ \hline 101-120 & 3\times20+2\times20+c;\ c\in[0,20] \\ \hline 121-140 & 3\times20+3\times20+c, c\in[0,20] \\ \hline 141,143,145,147,149,151,153,155,157 & 3\times20+3\times19+2\times c;\ c\in[12,20] \\ \hline 142,144,146,148,150,152,154,156,158,160 & 3\times20+3\times20+2\times c;\ c\in[11,20] \\ \hline 161,\color{red}{164},167,170 & 3\times20+50+3\times c;\ c\in[17,20] \\ \hline 159,\color{red}{162},165,168,171,174,177,180 & 3\times20+3\times20+3\times c;\ c\in[13,20] \\ \hline \end{array}$$ We've done all the possible combinations.

The impossible numbers are: $$163,166,169,172,173,175,176,178,179$$.

• Amazing job! : ) Commented Oct 24, 2021 at 16:17

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):