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. That is not an accident of tradition. 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.
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.
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; not really, 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: no patent has been found, and darts historians note the attribution is repeated far more confidently than it is documented (a rival claim names Thomas William Buckle). 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.
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, is the only escape from a common trap: 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 in the high sixties (191 ways, at 68) and then falls away toward the 180 maximum. The shape below is exact — it is OEIS sequence A242678, recomputed here in your browser.
Hover or drag across the chart.
Show the check — 32/32 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 Dp follow Cohen & Tonkes (2001). The closed forms for the maxima are re-proved by exhaustive search on small boards. 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).
- 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 Dp(A) = Σ|ij − ij+1|p and proves the maximisers. The printed value D₂(Ad) = 2374 is corrected to 2478 above; everything else in the paper reproduces 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 (T. D. Noe / contributors) and A338551. All recomputed here, not copied.
On the layout's history: the Gamlin (1896) attribution is folklore — well repeated, poorly evidenced (see darts historian Patrick Chaplin). 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.