Darts and Diophantine equations

Darts is a popular spectator sport in the UK. I had access to cable television recently so I was able to watch a few games.  What I find interesting about professional darts is that the players must solve a Diophantine equation to win.  For those who know nothing of the game, it involves throwing a small pointed projectile at an enumerated target board that looks like this:


A dart that lands on a given sector on the board obtains that score.  The center circle of the board called the bulls eye is worth 50 points.  The ring around the bulls eye is worth 25 points.  The wedges are worth the score ascribed by the number on the perimeter.  However, if you land in the inner ring then you get triple the score of the wedge and if you land in the outer ring you get double the score.  Hence, the maximum number of points for one dart is the triple twenty worth 60 points.

There are a few variations of the game but in the one the professionals play, each player starts with 501 points and the goal is to get exactly to zero by adding up the points of the thrown darts. Each player throws three darts per turn and they alternate.     The maximum number of points per turn is then 180 points.  In addition, they must hit the double ring or bulls eye at least once on the last turn to win.

In the first few rounds they aim for the triple twenty so in principle they could get down to 141 in two turns.  However, it is quite rare to get three triple twenties in a row and on average they seem to get about 100 points per turn.  Once they get below 180 points then it is possible to win in one turn but it requires solving a Diophantine equation.

Suppose that they need to score n points to win. Then they need to solve the equation 2 X + aY +b Z = n, where a,b \in\{ 0, 1, 2, 3\} and X, Y, Z \in \{1, 2, \dots, 20\}.  For example, if they need 141 points they could do it with a triple 19 to get 57 points, a triple twenty, which gets to 117 points, and finally a double 12 to get to 141 points.   Nine darts is the minimum number to win, since the maximum score with 8 darts is 480.  Hence, a perfect darts game is over in three rounds.

I wonder if there is a winning solution for all scores below 180.  My guess is that there is but I don’t have a proof. It would be a good exercise for a middle or a high school student.  I also wonder if the players solve the Diophantine equation in their heads as they play or if they just remember the necessary combinations.  They seem to do it instantly, and much faster than I could do it.  They also adjusted on the fly because if they missed their intended target on a toss, they would solve the remaining two variable Diophantine equation.  Finally, I wonder if darts players are better at arithmetic then non darts players?   If so, then maybe we should start encouraging people to play darts in the US.

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