24 Game Solver
24 game solver: enter any four digits and see every way to make 24 with + - × ÷ and parentheses, with exact fraction arithmetic. Unsolvable hands get a proof, not a shrug.
Want the game instead of the answers? Play Today's 24 — one deterministic hand a day with a streak — practice mode deals random solvable hands with streaks and four difficulty levels, and the hardest 24 game puzzles lists every single-solution hand — including 3 3 8 8 and 1 3 4 6, each with its own try-it-first page.
A short history of making 24
The 24 Game most people meet in school — a deck of cards, four numbers per card, first player to see the answer wins — was created in 1988 by Robert Sun, an engineer in Easton, Pennsylvania, who wanted arithmetic practice to feel like a sport. His single-digit deck spread through American classrooms in the 1990s, complete with school, city, and national tournaments.
The idea is older than the deck: making 24 from four playing cards is a long-standing Chinese card pastime (èrshísì diǎn, 'twenty-four points'), traditionally played with the cards ace through nine. Sun's contribution was turning it into a curriculum-friendly product with the fractions filed off the hard cards — though the hard cards are the good ones.
Why 24 and not, say, 23? Divisibility. 24 is 1 × 24, 2 × 12, 3 × 8, and 4 × 6, so an unusually large number of roads lead there — which keeps most hands solvable and the game moving. 23 is prime and would make a very quiet classroom.
FAQ
What are the rules of the 24 game?
You get four numbers. Use each one exactly once, combine them with addition, subtraction, multiplication, division, and parentheses, and make exactly 24. No concatenation, no exponents, no square roots — the whole game is + - × ÷ and stubbornness.
How many 24 game hands are solvable?
For digits 1 to 9, there are 495 possible hands (order does not matter), and 404 of them — about 82% — can make 24. This solver checked every hand by exhaustive search with exact fraction arithmetic, so both the solvable and unsolvable verdicts are proofs.
What is the hardest 24 puzzle?
The classic villains are 3, 3, 8, 8 — solved only by 8 ÷ (3 − 8 ÷ 3) — and 1, 3, 4, 6, solved only by 6 ÷ (1 − 3 ÷ 4). Both force you through fractional intermediate values like 1/3, which is exactly the move human brains refuse to try. They are 2 of exactly 94 hands with a single distinct solution — this site lists all 94, each with its own page.
Can I practice the 24 game online?
Yes — practice mode on this site deals you a random solvable hand at your chosen difficulty, from easy to brutal single-solution hands. Your answer is checked with exact arithmetic, and your streak and solved count stay in your browser. Free, no account, no download.
Why does the solver say a hand is impossible?
Because it is. Four numbers and four operations make a small, finite space of expressions; the solver evaluates all of them with exact fractions (no floating-point rounding) and reports 24 only on exact equality. If no expression works, 'unsolvable' is a checked mathematical fact — 1, 1, 1, 1 really cannot make 24, no matter how long the car ride is.
Can I use this with playing cards?
Yes, for the single-digit game: ace counts as 1 and number cards 2 through 9 count as themselves. Editions that include 10s and face cards go beyond what this tool takes — it accepts digits 1 to 9, with repeats allowed.
The Number Lab
Same idea, different constraints: four 9s, four 4s, or any four digits you like.
- Four Nines — the daily puzzle
- Four Nines solver — every target, solved
- Four Fours — the 1881 original
- Four Fours solver — including the impossible five
- Make 24 solver — any four digits (this page)
- Today's 24 — one new hand a day
Or browse more free math games.