We Enumerated Every 24-Game Hand: 1,820 Combos, 458 Impossible

The short answer

Exactly 1,362 of the 1,820 possible four-number hands can make 24. The other 458 cannot. Those two numbers are not estimates and not the output of a sampling run — they come from exhaustively enumerating every distinct hand and every possible expression on it. The full enumeration is published as our open dataset, free to download and to cite.

How many hands are there, anyway?

With the classic rule (four numbers, each from 1 to 13, repetition allowed, order irrelevant), the count is a plain combinations-with-repetition problem: C(16, 4) = 1,820 distinct hands. That is small enough to brute-force completely, which is exactly what we did — there is no excuse for approximating a space this size.

How the enumeration works

Three properties make the result trustworthy rather than merely plausible:

1. Exact fraction arithmetic. Every intermediate result is an exact rational number — a pair of integers, never a floating-point value. 8 ÷ 3 stays 8/3, not 2.6666…, so a hand is never misclassified because of rounding.
2. Canonical multisets. Hands are deduplicated as sorted multisets, so 3 8 8 3 and 8 3 3 8 count once, not twice. The enumeration runs over all 1,820 multisets exactly once.
3. Exhaustive binary-tree search with deduplication. The solver tries every way to combine the numbers with + − × ÷, prunes commutative and associative duplicates via a canonical form, and counts only mathematically distinct solutions. Two expressions that differ only by reordering terms count as one.

If you want to verify this yourself, the methodology is reproducible from the dataset's documentation, and the solver exposes the same engine interactively.

Surprises the data turned up

Only 16 hands truly need fractions. The famous fraction hands — 3 3 8 8 with 8 ÷ (3 − 8 ÷ 3), 5 5 5 1 with 5 × (5 − 1 ÷ 5) — feel like the tip of an iceberg. They are nearly the whole iceberg. Of 1,362 solvable hands, exactly 16 have no solution with whole-number intermediates. The other 1,346 all yield to classroom rules.

The most-solved hand is not exotic. 1 5 7 12 holds the record with 22 distinct solutions — and most of them are the same core dressed up with ×1 and ÷1 bookkeeping. High solution counts come from a 1 in the hand, not from cleverness.

4 4 10 10 is secretly easy. It looks like an expert wall and intimidates players who meet it cold. But its only solution, (10 × 10 − 4) ÷ 4, uses nothing but whole numbers and a single big product. Difficulty is about how hard a hand is to find, not how exotic its solution is — which is why our difficulty tiers score solvability patterns, not expression shapes.

The unsolvable hands get honest pages too. Every one of the 458 impossible hands — 1 1 1 1 being the canonical example — has its own page stating plainly that no expression reaches 24, alongside the proof-by-exhaustion that backs the claim. A directory that only celebrates solvable hands would be quietly lying by omission.

Using the data

The dataset page offers the complete enumeration as downloadable files, covering every hand, its solutions, and its classification. If you build on these numbers in research, a classroom resource, or an article, cite the dataset — and if you find an error, we genuinely want to know: with 1,820 hands, a reproducible discrepancy is a bug in our enumeration, not a rounding artifact.

Not sure where to start? Browse the full solution directory, pick a tier, and try to beat the archive to an answer.