Numbers on a dartboard - we know why they are in that order but how was it calculated without computers?

Question

The arrangement of the numbers around the circumference of a standard dart board is as shown below

  20 1 18 4 13 6 10 15 2 17 3 19 7 16 8 11 14 9 12 5

Oddly enough, no one seems to know for sure how this particular arrangement was selected. ... it's clear that the numbers are ordered to mix the large and small together, and possibly to separate numerically close values as far as possible (e.g., 20 is far from 19), no one seems to know of any simple criterion that uniquely singles out this particular arrangement as the best possible in any quantitative sense.

http://www.mathpages.com/home/kmath025.htm

Question

This appears to be an unsolved problem. How did the inventor of the standard dartboard come up with the order of the numbers in such a way as to minimise scores that are produced by inaccurate throws?

Can anyone see a pattern or was it just trial and error?

Given that computers weren't available then (pre 1900), can anyone suggest a pencil and paper method that produces a near-optimum result (and specifically this result) in a reasonable time?

Answer 1

The numbering system on a standard dartboard is designed in such a way as to reduce ‘lucky shots’ and reduce the element of chance. The numbers are placed in an order to encourage accuracy and punish inaccuracy. The placing of low scoring numbers either side of large numbers e.g. 1 and 5 either side of 20, 3 and 2 either side of 17, 4 and 1 either side of 18, will punish poor throwing. If you shoot for the 20 segment, the penalty for lack of accuracy is to land in either a 1 or a 5. That is basically it.

Answer 2

It's trivial to find a "near-optimum" result in reasonable time

I've never played dart outside of the casual pub going, so I don't know any of the real intricacies of the game. However, it seemed intuitive to me that, when designing the layout of the dart board, we'd want as high of a penalty for missing a shot as possible -- that is, we'd want to maximize the difference between neighboring shots.

But I'm a 19th century dart player. I don't care much for mathematical proofs, or rigid brute force optimizations. Without thinking too deeply about it, I just... gave it a go?

First try

I started out by trying to just interleave the available numbers in reverse order: every second number would decrement from twenty, the other numbers would increment from 1. This gave the sequence of 20 1 19 2 18 3 17 4 16 5 15 6 14 7 13 8 12 9 10.

Both intuitively and by applying the measurement from the question's linked article, it was obvious that this wasn't very well balanced. Intuitively is seemed like the penalty would decrease as we moved towards the end of the sequence. The given measurement describes this in a slightly less intuitive way: the trios that started with a high square (i.e. those with the "small" numbers in the middle) would decrease as we moved towards the end of the sequence. The trios that started with a low square (i.e. those with the "big" numbers in the middle) would increase as we moved towards the end. This indicates a smaller variation overall near the end of the sequence than at the start.

Regardless, this sequence didn't feel nice, which just wouldn't do. No rigid mathematical proof or anything. I want to throw pointy sticks, not think more than I need to.

Second try

For the second go I tried to mitigate that by placing numbers that are close to eachother as far apart as I could. At every second step I tried to find a number that "balanced" the previous trio of numbers as well as possible (i.e. 20-1-? would have 10 added as the last number), but wasn't particularly rigid about it.

The process looked a little like this:

Index:     1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20

Sequence: 20 __ __ __ __ __ __ __ __ __ 19 __ __ __ __ __ __ __ __ __
Sequence: 20  1 __ __ __ __ __ __ __ __ 19  2 __ __ __ __ __ __ __ __
Sequence: 20  1 10 __ __ __ __ __ __ __ 19  2  9 __ __ __ __ __ __ __
Sequence: 20  1 10 18 __ __ __ __ __ __ 19  2  9 17 __ __ __ __ __ __
Sequence: 20  1 10 18  3 __ __ __ __ __ 19  2  9 17  4 __ __ __ __ __
Sequence: 20  1 10 18  3  8 __ __ __ __ 19  2  9 17  4 11 __ __ __ __
# the thinking starts becoming less rigid, since no real good numbers are available any longer
# I try to just... balance things out as I feel like
Sequence: 20  1 10 18  3  8 16 __ __ __ 19  2  9 17  4 11 12 __ __ __
Sequence: 20  1 10 18  3  8 16 __ __ __ 19  2  9 17  4 11 12  5 __ __
Sequence: 20  1 10 18  3  8 16 __ __ __ 19  2  9 17  4 11 12  5 13 __
Sequence: 20  1 10 18  3  8 16 __ __ __ 19  2  9 17  4 11 12  5 13  6
Sequence: 20  1 10 18  3  8 16  7 __ __ 19  2  9 17  4 11 12  5 13  6
Sequence: 20  1 10 18  3  8 16  7 15 __ 19  2  9 17  4 11 12  5 13  6
Sequence: 20  1 10 18  3  8 16  7 15 14 19  2  9 17  4 11 12  5 13  6

This gives a final sequence of 20 1 10 18 3 8 16 7 15 14 19 2 9 17 4 11 12 5 13 6. Notably I didn't even try to optimize it after placing all the numbers -- this layout felt good enough, so I wouldn't want to spend any more time not throwing arrows at walls.

The scoring mechanism from the article linked in the question gives this a sum of squares of

$$(20 + 1 + 10)^2 + (1 + 10 + 18)^2 + \dots + (13 + 6 + 20)^2 + (6 + 20 + 1)^2 = 20410$$

Which makes it even more optimal than the "London" layout, according to that measure.

So... by just doing it, I think it's become clear that reaching a layout that's at least as "near-optimum" as the classic one is possible by hand ;)

Reframe

Of course, this is only "near-optimum" if we consider a) the measurement used in the article to be a valid approximation of what we want from a dart board, and b) we consider the $20478$ score of the London arrangement to be "near-optimum".

Given that the article goes on to find many, many arrangements that have scores closer to $19950$, I'd challenge that last point. As such, I don't think there's very much value in the question, given that the London arrangement (and mine) isn't near-optimum.

Answer 3

This is more an observation of the pattern than a method to get it, but if we assume a shooter has a spread of one space, meaning for example if aiming at 20, there is an equal chance of hitting 20,5,or 1, then we get these expected values for each target.

20 1 18 4 13 6 10 15 2 17 3 19 7 16 8 11 14 9 12 5
8.6 13 7.6 11.6 7.6 9.6 10.3 9 11.3 7.3 13 9.6 14 10.3 11.6 11 11.3 11.6 8.6 12.3

The expected values range from 7.3 to 14, a pretty big spread. But if we order targets by expected value, we get

17 13 18 20 12 15 6 19 10 16 11 2 14 4 9 8 5 1 3 7

This is semi-close to being ordered. Basically, if you evenly hit the target you're aiming at or one of it's neighbors, the best places to shoot for are actually 1,3 and 7, while the worst are 17, 13, and 18. There are still a couple inconsistencies, such as 14 being so high on the list, but this gives a general framework.

Other observations

Even spread is impossible: Consider 20. With value $a$ on its left and $b$ on its right, the expected value is $(20+a+b)/3$. Now consider spot $a$. 20 is one neighbor, call the other neighbor $c$. So if we have $(20+a+c)/3 = (20+a+b)/3 => c = b$ which is impossible, because there are no repeating values.

Smallest possible spread: If we order the scores 20,1,19,2... I think we get the smallest difference in expected values, from 17 = 8 to 10 = 13.66

Answer 4

There is a somewhat regular pattern to the ordering, as shown below.

The following is a series of observations that may help explain their positions on the board. The goal will be intuition rather than mathematical optimisation, which would align with the anecdotes about its origin. Indeed, this arrangement is not mathematically optimal despite being very close.

a) The criss-cross pattern in the arrangement of the numbers 20 to 13 is an intuitive attempt at distributing the largest values evenly. Shown in blue lines above.

The pairs are 20-19, 18-17, 16-15 and 14-13, with a consecutive difference of 1. It is also intuitive to place the largest pair 20-19 as near to the diameter as possible, while allowing space for the smallest values to be placed. The placing of 18-17 is parallel to that of 20-19 and allows six values to be placed to its right; for instance, compared to only four values to the left of 12-16. The pair 16-15 is placed horizontally in the bottom half of the board, which is desirable since the complementary placement of 14-13 in the top half enables fair distribution. Of these eight numbers (20 to 13), the top half sums to 65 and the bottom half to 66.

b) The parallel pattern in the arrangement of the numbers 1 to 7 (except 6) attempts to distributing the smallest values evenly. Shown in yellow lines above.

The pairs are 1-3, 4-2 and 5-7. It is intuitive to place the smallest pair 1-3 as close to the largest pair as possible, and allows 4-2 and 5-7 to be placed on either side of the chord. The reverse arrangement of the pair 4-2 may allow fair distribution; of these six numbers, the top half sums to 10 and the bottom half to 12. The consecutive difference of 2 explains the omission of 6, but why it does not remain as 1 is unclear.

c) The arrangement of the numbers 6 to 12 (except 7) may be a continuation of the pattern in b). Shown in pink circles above.

It can be argued for three reasons that there is still a pattern if the sequence from b) were continued as 8-6, 9-11 and 12-10. First, the arrangement agrees with the left-to-right and top-to-bottom direction of each pair on the board. Second, these three pairings give rise to a symmetrical triangular arrangement with 14 at the midpoint of the base. Third, the inner sum of 9 and 11 equals the outer sum of 8 and 12.