A question the internet answers badly

Twenty Between One and Five

The most valuable number on a dartboard, the 20, is wedged between the 1 and the 5, two of the smallest. Whether or not anyone meant it, the board is arranged so that a shot which just misses the number you wanted lands you somewhere much worse. Below is the whole idea, made operable: rearrange the numbers yourself and watch a single figure — the price of a near miss — rise and fall. The real board scores near the very top of it. Then, at the bottom, the seven scores you can never finish on.

The standard board — 20 at the top, clockwise

Every large number has small neighbours. Aim for the fat 20 and drift a hair to either side and you collect a 1 or a 5, not the 18 or 12 a lazier layout would give you.

1 · The price of a near miss

Suppose you wanted a way to measure how cruel a numbering is. Walk once around the ring and, for every pair of touching numbers, add up the gap between them. A board that keeps big numbers away from small ones has small gaps and a low total; a board that jams extremes together has a high total — it punishes the wobble that sends your dart one sector over. Mathematicians call this the L₁ penalty; it is exactly the number below.

Rearrange the board · the near-miss penalty updates live

L₁ · sum of neighbour gaps

198 / 200 max

L₂ · sum of squared gaps

2478 / 2642 max

Tap a number, then another, to swap them. (The positions stay put; the numbers move.)

Two things are worth sitting with. First, the standard board scores 198 — and the best any arrangement of 1–20 can do is 200. It is two short of perfect. The maximum is reached by any board that strictly alternates a small number (1–10) with a large one (11–20), and every such board ties at 200; press Maximise L₁ to see one. The real board is not quite alternating — the 6 sits beside the 10, and the 11 beside the 14 (two small numbers touching, then two large) — which is exactly why it lands two short of the maximum instead of reaching it.

Second, change the measure and the verdict changes. If you think a big near miss should hurt more than a small one — square each gap before adding, the L₂ penalty — the standard board scores 2478 against a possible 2642. Now it is visibly off the peak. So the honest answer to “is the dartboard optimal?” is: nearly, for one natural measure of cruelty; less nearly, for another. It is a compromise, not a theorem — which fits what little we know of where it came from.

Who arranged it, and did they do the sums?

The layout is traditionally credited to Brian Gamlin, a Lancashire carpenter, in 1896, but the record is thin: the darts historian Patrick Chaplin found no contemporary mention of Gamlin and thinks he never existed, giving the strongest claim to Thomas William Buckle of Dewsbury, who by Chaplin's account made the board in 1913. We have no surviving design notes. So “the board was engineered to punish near misses” is an inference from how it behaves, not a fact from an archive. What is solid is the behaviour itself: the numbers really do sit near the top of the near-miss penalty, whoever put them there and for whatever reason.

2 · The finish you can never hit

A leg of darts does not end when you reach zero — it ends when your last dart lands in a double (the thin outer ring) or the bullseye (the double of 25). That one rule, plus a three-dart turn, decides every checkout. Pick a target and see it worked out — or find one of the totals that simply cannot be done.

Checkout finder · 2 to 170

Finish 170

There are exactly seven totals at or below the maximum that no three darts can finish — each shown red below. The very top checkout, 170, has exactly one route: two treble-20s and the bull.

3 · How many ways to score it

Set the finishing rule aside and just ask: in how many distinct ways can three darts total n? Almost nobody makes 3 (only 1 way: three ones); the count climbs to a broad peak around 70 (191 ways, at both 68 and 70) and then falls away toward the 180 maximum. The shape below is exact — it is OEIS sequence A242678, recomputed here in your browser.

Ways to score n with three darts · A242678

Hover or drag across the chart.

Show the check: 42/42 green

The single dart reaches 43 distinct values across 62 areas (singles, doubles, trebles, two bulls); the top is 60 (treble-20).
Scoring reproduces OEIS A242718 / A242681 / A242678 (ways to score n with 1 / 2 / 3 darts) term-for-term, counting distinct values.
Checkouts reproduce OEIS A338551 (ordered darts, finishing on a double). The impossible finishes ≤ 170 are exactly 159, 162, 163, 165, 166, 168, 169; the maximum checkout is 170 = T20, T20, Bull.
Arrangement. The near-miss penalties D_p follow Cohen & Tonkes (2001). The closed forms for the maxima (proved in the paper) are confirmed by exhaustive search on small even boards, n = 4, 6, 8. At n = 20: standard L₁ = 198 (max 200, any alternating board); standard L₂ = 2478 (max 2642, Selkirk's formula).
One correction. Cohen & Tonkes print the standard board's L₂ as 2374; recomputed from the board they display it is 2478 — a difference of 104, and an apparent typo (their L₁ = 198 and both maxima reproduce exactly). A second slip, off the real board: for odd n the paper gives the L₁ maximum as 2⌊n/2⌋²; exhaustive search gives (n² − 1)/2 (at n = 3 every arrangement scores 4, not 2).
Runnable: node research/dartboard-order/verify.mjs.

Sources & apparatus

G. L. Cohen & E. Tonkes, “Dartboard Arrangements,” Electronic Journal of Combinatorics 8(2) (2001), #R4. Defines the penalty D_p(A) = Σ|i_j − i_j+1|^p and proves the maximisers. The printed value D₂(A_d) = 2374 is corrected to 2478 above, and its odd-n L₁ maximum to (n² − 1)/2; the other figures we recomputed reproduce exactly.

K. Selkirk, “Re-designing the dartboard,” Mathematical Gazette 60 (1976) 171–178 — the L₂ maximum ⅓n³ − 4n/3 + 2. Everson & Bassom (Math. Spectrum, 1994/5) and Eiselt & Laporte (JORS, 1991) give the L₁ result.

Scoring & checkout counts: OEIS A242718, A242681, A242678, A241746 (the first three by Colin Barker, the fourth by John Bibby) and A338551 (Carmen Bruni). All recomputed here, not copied.

On the layout's history: the Gamlin (1896) attribution is folklore, well repeated and unevidenced; darts historian Patrick Chaplin calls Gamlin a 1970s invention. The board's arithmetic is real; its authorship is not settled, and this page does not pretend otherwise.

Two models, stated plainly. “Ways to score” counts distinct number values (a 20 is a 20 however you hit it); “ways to check out” counts distinct physical areas thrown in order (the darts rules care where the last one lands). The two OEIS families make exactly these two choices; we follow each.

Corrected 2026-09-28 (found by a scan of the site's pages for verdicts printed whatever the page computed): the chip "170 = T20 T20 Bull ✓" beside the impossible finishes was fixed text, not a result of the checkout finder. The claim was true, but the page did not check it; the chip's route and tick now come from the same checkout functions as the rest of the section.

Corrected 2026-10-13 (claims pass): This page said everything else in Cohen & Tonkes reproduces exactly; their L₁ maximum for odd n, 2⌊n/2⌋², is also wrong (it is (n² − 1)/2; at n = 3 every board scores 4), though nothing about the real 20-number board changes. It credited the scoring sequences in OEIS to T. D. Noe; A242718, A242681 and A242678 are Colin Barker's and A241746 is John Bibby's. Its headline said the board was arranged this way on purpose, which its own history section calls an inference; it now says the board behaves as if arranged so. Smaller fixes: Patrick Chaplin, whom it cites, doubts Gamlin existed and favours Thomas Buckle (1913), and the page now says so; the three-dart peak of 191 ways falls at both 68 and 70; the page said the search re-proved the formulas for the maxima, when it confirms them at n = 4, 6, 8 and the proofs are the paper's; and the verifier, one of whose checks had an escape clause, now runs 42 checks, brute-forcing the L3 and L4 formulas too.