Numbers on the Dartboard

You may have noticed that the numbers on a standard board for the game of darts have the following order, going clockwise starting at the top: 20, 1, 18, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12, and 5.

Why this particular order?

A possible criterion for designing a dartboard is to penalize poor shots as much as possible. That can be done, for example, by maximizing the sum of the absolute value of the difference between adjacent numbers. The larger this sum, the more a poor shot is penalized.

To achieve this maximum, the numbers 11 to 20 should interlace 1 to 10. In this case, the differences between adjacent numbers total 200.

A standard board with 20 sectors has a total of 19! (or 19 ✕ 18 ✕ 17 … ✕ 1 = 121,645,100,408,832,000) possible arrangements. Of those, 10! ✕ 9! (or 1,316,818,944,000) give the maximum possible penalty sum of 200.

Surprisingly, even though there are so many possible arrangements that give the maximum penalty score of 200, the standard dartboard in everyday use has a difficulty of only 198. The "flaw" lies in the placement of 11 next to 14 and 6 next to 10. You could correct this imperfection simply by inserting 14 between 6 and 10 to achieve the maximum difficulty.

Of course, the sum of the differences between adjacent numbers represents just one possible way of defining the difficulty of a dartboard. You might consider, for example, the squares of the differences of adjacent sectors or not only immediate neighbors but also near neighbors.

Moreover, the dartboard also includes areas representing doubles, trebles, and the inner and outer bull. These features, which play an important role in many games, considerably complicate any analysis of the board.

Expert players often concentrate on the treble 20 when aiming for a high score. Players who throw with a much lower accuracy may be wise to aim at sector 14, which has 9 and 11 as adjacent sections. That's the area of the standard dartboard that happens to deviate from the requirements for maximum difficulty, at least according to one criterion.

It would be interesting to delve into the history of dartboards to see how the standard numbering scheme came about.

Originally posted May 19,1997

References:

Cohen,G.L., and E. Tonkes. 2001. Dartboard arrangements. Electronic Journal of Combinatorics.

Everson, P.J., and A.P. Bassom. 1994/5. Optimal arrangements for a dartboard. Mathematical Spectrum 27(No. 2):32-34.

Selkirk, K. 1976. Redesigning the dartboard. Mathematical Gazette 60:171-178.