Bogey Numbers

Video based on blog content, generated by NotebookLM

In a darts match, once you reach a score of between 170 and 2 inclusive, you can defeat your opponent with one, two or three darts provided you score doesn't correspond to one of the so-called "bogey numbers". These numbers are:

• \(169 =13^2 \)
• \(168=2^3 \times 3 \times 7 \)
• \(166=2 \times 83 \)
• \(165=3 \times 5 \times 11\)
• \(163 \text{ is prime} \)
• \(162 = 2 \times 3^4\)
• \(159 = 3 \times 53\)

The maximum of score of 170 can be finished with two consecutive treble 20s followed by 50 (the red of the bullseye equivalent to 2 x 25). With a score of 2, you finish the match off with a single dart directed at double 1.

With my 77th birthday fast approaching, I wanted to consider how many ways there are to finish off a darts match from a score of 77. It turns out that there are 905 different ways (permalink) using two or three darts. Here are the possible two dart finishes:

• 2-Darts: T9, DBull
• 2-Darts: T13, D19
• 2-Darts: T15, D16
• 2-Darts: T17, D13
• 2-Darts: T19, D10

The other 900 ways involve three darts, for example:  S1, D13, DBull. Figure 1 shows a plot of the total numbers of ways to finish for scores between 2 and 170. Click to enlarge. A score of 58 offers the largest number of ways to check out, 1404.

Figure 1: permalink

Of course professional darts players only choose from a very narrow selection out of all these possibilities as Gemini explains.

You have hit on the fundamental difference between the math of darts (which is what the code calculates) and the art of darts (which is what pros play).

While Python might find 50 ways to check out a score, a professional like Michael van Gerwen or Luke Littler might only ever consider one or two of them.

Infographic based on blog content
generated by NotebookLM

Here is the breakdown of the logic top players use to filter those thousands of possibilities down to the "correct" shot.

1. The Principle of "Splitting the Double" (The Safety Net)

This is the single most important rule in professional darts. Pros prioritize doubles that can be halved (split) and still remain a double if they miss into the single segment.

• The "King" Double is D16 (32):

  • If you aim for D16 (32) and hit Single 16, you have 16 left. That is D8.

  • If you aim for D8 and miss into S8, you have 4 left (D2).

  • If you aim for D2 and miss into S2, you have 2 left (D1).

  • Result: You have 4 layers of safety. You can miss your double 3 times and still be on a finish without needing a "setup" dart to get back to an even number.

• The Problem with D20 (Tops):

  • D20 is popular, but less safe. If you miss D20 (40) and hit S20, you have 20 left (D10).

  • If you miss D10 (20) and hit S10, you have 10 left (D5).

  • D5 is an odd number. You cannot finish on 5. You must now waste a dart hitting a Single 1 to leave D2.

The Decision: If a pro has a choice between leaving D20 or D16, they will almost always maneuver to leave D16.

2. "Percentage Darts" (Managing the Miss)

Pros assume they might miss the Triple and hit a Single. They choose routes where a "bad" throw (a single) still leaves a finishable score.

• Example: You have 61 points left with 2 darts.

  • The "Amateur" Math Route: Triple 15 (45) leaves D8 (16).

    • Risk: If you miss T15 and hit S15, you have 46 left. You have 1 dart remaining. You cannot check out 46 with 1 dart. Turn over.

  • The "Pro" Route: Single 25 (Outer Bull) leaves D18 (36).

    • Safety: If you miss the Bull and hit S25, you have 36 left (D18). If you miss widely and hit S11 (a common neighbor to the bull), you have 50 left (Double Bull). You still have a shot at a double.

The Decision: The pro chooses the route where a miss keeps the turn alive.

3. Avoiding the "Bogey" Singles (Odd vs. Even)

The goal is to get to an even number as fast as possible.

• If you are on an odd number (e.g., 17, 33, 45), you must hit an odd segment to get to an even finish.

• If you are on an even number, you must hit an even segment (or a Triple even) to stay even.

Pros hate finishing on odd doubles (D19, D17, D15, etc.) because if they hit the single, they are stuck on an odd number again.

• Bad: Aiming for D19 (38). Hit S19. Left with 19. Now you need S3 to leave D8. It's messy.

• Good: Aiming for D18 (36). Hit S18. Left with 18 (D9). Still messy, but D18 is physically close to D4 and D1, making the recovery easier.

4. Board Management (Staying in Rhythm)

Dart players rely heavily on muscle memory. They prefer routes that require minimal movement around the board.

• Example: 82 left.

  • Option A: Bull (50) + D16 (32). (Requires moving from center to right edge).

  • Option B (Pro Preference): D20 (40) + S10 + D16? No.

  • Option C (The "Top" Route): T14 (42) leaves D20 (40).

  • The "Phil Taylor" Route: Bull (50) leaves D16. (Some pros love the Bull because it fixes "messy" math instantly).

5. The "Matchplay" Factor (Aggressive vs. Conservative)

Finally, the choice depends on what the opponent is doing.

• Scenario A: Opponent is way back (on 300 points).

  • Strategy: Go safe. Don't try a risky T20-T20-D20 finish. Just hit big scores to get down to D16 reliably. Use 3 darts to set up the perfect number.

• Scenario B: Opponent is on a finish (e.g., 32).

  • Strategy: Aggressive. You must check out this turn. If you have 170, you must go T20-T20-Bull. You don't care about safety; you care about possibility.

Summary: The "Pro" Algorithm

If you were to program a "Pro Bot" rather than a "Math Bot," the logic would look like this:

1. Can I finish this turn?

2. If yes: Pick the route where a single (missed triple) leaves a finish for the next dart.

3. If no: Pick the route that leaves 32 (D16) or 40 (D20) for the next turn.

4. Avoid: Any route that leaves an odd number remainder.

Gemini created a program that will calculate the better choices when considering checking out. Figure 2 shows what it suggests for 77 (permalink).

Figure 2