Impossible Dart Scores

 

If you throw a single dart at a dartboard, what’s the most you can score? I think most people know that it’s 60 points, which can be obtained by hitting the triple-20. The least you can score is zero (by either missing the board, hitting the board in a non-scoring region, or having the dart fall out).

There are a plurality of scores between these two extremes that can be obtained. For instance 25, from hitting the outer bull, or 34 from hitting the double-17.

Some scores are obtainable in more than one way, for instance, 15 can be obtained with a triple-5 as well as a natural 15. Some scores, like 6, can be obtained three ways!

There are some scores, however, in the range [0-60], that it is impossible to achieve with a single dart. For example, it’s not possible to score 59 with a single dart, nor 49.

It's not hard to calculate the impossible scores. Here is a list of all the impossible one dart scores:

Obviously, these impossible scores are in the range [0-60].

Graduating to two darts, it’s easier to fill the gaps lower down, but the maximum possible score also doubles, doubling the possible range of scores that need to be hit. It’s not quite as bad as this, however, because, with the first dart, there are lots of possible starting bases that you can use before scoring the second dart (For instance, starting with the first dart in triple-20, or triple-19, gives you a wide range of high scoring bases before you apply the score of the second dart).

Here is a list of all the impossible two dart scores:

These are the impossible scores in the range [0-120]. Remember, either of the darts can score zero, so you instantly inherit all the solutions from the section above.

A standard turn at darts is a throw of three projectiles. What are the scores that it is impossible to achieve?

Here is a list of all the impossible three dart scores:

We can see that 163 is the minimum score it is impossible to make if you throw three darts at a board.

Here is a table of the impossible numbers for the first 25 darts. Do you see a pattern quickly emerging?

A related set of puzzles are the infamous Frobenius Number problems, more commonly referred to as the Chicken Nugget Numbers.

In this puzzle, the goal is to try and find the minimum number of chicken nuggets it is impossible to buy using combinations of the quantized offerings that McDonalds® has on its menu.

The principles also have use in (theoretical) optimizations for coin or stamp denominations to minimize the number of different coin values that might need to minted, or different stamps printed.

 

In American football (NFL rules), any score is possible except one*. The only way to score one point is by a single point conversion after a six point touchdown. As two points are awarded for a safety and three points for a field goal, all other scores apart from one are possible.

Update: A couple of people have pointed out that it is technically possible to score a single for a safety during a point after attempt. It’s never happened, but it is possible. However, the single point gained is not isolated, and could only be obtained after the other team had scored six points, so you could not stack an arbitrary number of single point scores and get any desired result without the other team having scored the same multiple of six points. But yes, a score of 6–1 it possible (if I understand the rules!)