The 24-Point Solution Database

Methodology

The dataset is produced by an exact-arithmetic solver: every intermediate result is a reduced fraction, so solvability and solution counts are exact conclusions — never floating-point approximations.

Canonical dedup: hands are sorted multisets, so 3-8-8-3 and 8-3-3-8 count once. The full space is C(16,4) = 1,820 combos — every way to draw 4 numbers from 1–13 with repetition.

Difficulty is a disjoint four-tier partition of the solvable combos: easy = whole-number intermediate steps, all cards 1–9, at least 3 solutions; medium = whole-number steps, all cards 1–9, only 1–2 solutions; hard = whole-number steps with a card in 10–13; expert = requires fraction intermediates (no whole-number solution exists).

Every number is computed offline with the site's own solver — the same engine behind the solver page — and asserted during generation: totals, per-tier counts and per-combo solution counts must match a fresh solve.

Solve API

Send a GET request with a cards parameter — 3 or 4 comma-separated integers from 1 to 13. No API key is required; the target is always 24.

Example request and response:

GET /api/solve?cards=3,3,8,8

{
  "code": 0,
  "message": "ok",
  "data": {
    "cards": [
      3,
      3,
      8,
      8
    ],
    "target": 24,
    "count": 1,
    "solutions": [
      {
        "expression": "8 ÷ (3 − 8 ÷ 3)",
        "steps": [
          "8 ÷ 3 = 8/3",
          "3 − 8/3 = 1/3",
          "8 ÷ 1/3 = 24"
        ]
      }
    ],
    "solution_page": "/solution/3-3-8-8"
  }
}

Five or more numbers are disabled on this endpoint (Cloudflare Workers CPU limits) — download the dataset above for offline analysis. Requests are rate-limited to 30 per minute per IP.