24 Game Solver
Enter your numbers and see every way to make 24.
Allow fraction steps like 8 ÷ (3 − 8 ÷ 3)
Solutions
Enter your numbers and press Solve to see the results here.
How the 24 game works
You get a hand of numbers — four in the classic card game, each between 1 (ace) and 13 (king). Use every number exactly once, combine them with addition, subtraction, multiplication and division, and group with parentheses to control the order. Reach exactly 24 and the hand is solved.
Some hands only work through a fraction step — 3, 3, 8, 8 famously needs 8 ÷ (3 − 8 ÷ 3). Switch "Fraction steps" on above to allow non-integer intermediate results, or leave it off for classic whole-number rules. With three or five numbers the same rule applies: use each number once and land on 24.
Where the difficulty comes from is no mystery either. Hands containing a two-digit card (10 through 13) drive most of the challenge — they multiply into big numbers fast and crowd out the friendly factor pairs. Hands built from 1 to 9 stay gentler, and the handful that force a fraction step are the famous ones. The solver treats them all the same: every arrangement, every operation, checked. The solution pages carry the tier for every hand, so you can steer by difficulty without trusting anyone's intuition — including your own.
The strategies behind every solution
The solver never guesses — it tries every arrangement of your numbers, every operation, every way to place the parentheses, which is why its answers (and its "No solution") can be trusted. Fast humans get there first with a handful of repeatable patterns. Learn the six below and you can read any solution the solver hands you — and usually beat it to the punch. Each pattern is short enough to teach in one minute and general enough to carry across every hand you meet.
Start from the factor pairs of 24
24 = 3 × 8 = 4 × 6 = 2 × 12. When a solution ends in multiplication, that final step was the whole plan: the two factors were built first. Spot one factor among your four numbers — an 8, say — and the problem shrinks to a smaller one: can the other three cards make its partner, the 3? It opens more hands than any other habit, which is why it comes first.
Check 24's neighbors
24 is also 18 + 6, 21 + 3, 20 + 4 and 30 − 6. When no clean multiplication is visible, good players aim three cards at a near-24 and let the fourth card close the gap. A solution like 5 × 4 + 6 − 2 is really "20 + 4" in disguise — the solver finds it either way, but a human who checks the neighbors finds it faster. Four checks, a few seconds, and a whole class of "multiplication-proof" hands opens up.
Split into two pairs
Many solutions pair the cards off: (a op b) op (c op d). Aim each pair at friendly values — a 6 and a 4, a 12 and a 2 — then finish with a single operation. When the solver's answer has exactly one operator outside the parentheses, that's the pair split it found. It is also the natural division of labor when two students share one screen: each owns a pair and calls out its value.
Take the fraction road
Only 16 of the 1,362 solvable hands force a fraction into the middle steps — but the famous hands live there. In 8 ÷ (3 − 8 ÷ 3), the inner 8 ÷ 3 is not a mistake: it manufactures the 1/3 that the outer division needs. When every whole-number route fails on a hand the solver insists is solvable, hunt for a division whose remainder is worth keeping. The "Fraction steps" switch above controls exactly this — turn it off and those hands will honestly report no solution.
Work backwards from 24
Pin one card aside and ask what the other three must produce. Holding an 8, the rest needs a 3 for multiplication or a 16 for addition; holding a 6, a 4 or an 18. Each choice shrinks the search — and it is exactly how a printed solution reads in reverse, last operation first. Reading answers this way is also the fastest way to check one: verify the last step, then the two halves that fed it.
When all else fails: exhaust, don't guess
Humans give up early; the solver never does. It walks every permutation of the four numbers, every choice of operation, every parenthesization — that is why its "No solution" is a proof rather than a shrug. Exhaustion is also how it finds the odd routes no pattern suggests: sometimes the answer can only be met by trying everything. And the full search of a four-number hand finishes before you notice it started.
Classic hands, answered
The most-searched 24 game hands, solved. Tap one in the example chips above to load it into the solver and browse every alternative — each entry below links to the hand's full solution page, where all of its routes sit side by side.
- 4 4 10 10
(10 × 10 − 4) ÷ 4 = 24. The most famous hard hand: 10 × 10 makes 100, minus 4 is 96, and 96 ÷ 4 lands exactly on 24.
- 3 3 8 8
8 ÷ (3 − 8 ÷ 3) = 24. No whole-number solution exists — 8 ÷ 3 leaves 1/3 inside the parentheses, and 8 ÷ 1/3 is 24.
- 1 5 5 5
5 × (5 − 1 ÷ 5) = 24. Another fraction key: 1 ÷ 5 trims the second 5 to 4.8, and 5 × 4.8 is exactly 24.
- 3 3 7 7
7 × (3 + 3 ÷ 7) = 24. The other fraction classic: 3 ÷ 7 leaves 3/7, adding 3 makes 24/7, and multiplying by 7 lands exactly on 24. Like 3 3 8 8, no whole-number route exists — the fraction step is the only way in.
Solver FAQ
Can a hand be unsolvable?
Yes — plenty of hands have no solution. 1, 1, 1, 1 is the extreme case: no arrangement of ones can reach 24. When this solver answers "No solution" it has checked every permutation, every operation order and every parenthesization, so the verdict is final. Deal yourself another hand. The engine behind it is exact — fractions included — so a near-miss never gets reported as a solution.
Which operations are allowed?
Addition, subtraction, multiplication and division, grouped with parentheses. No exponents, square roots, concatenation or factorials, and every number must be used exactly once — those are the classic rules and the ones this solver applies.
Are fraction steps allowed?
With "Fraction steps" on, intermediate results may be fractions — 8 ÷ 3 = 8/3 is fine as long as the final result is exactly 24. Turn it off to enforce whole numbers at every step, the stricter classic variant.
How many solutions can one hand have?
Anywhere from one to 22. The record holder is 1 5 7 12 with 22 distinct solutions; plenty of solvable hands have exactly one. Open any hand in the solution archive to see its full set side by side. The count matters for teaching too: hands with many solutions make forgiving warm-ups, single-solution hands make sharp tiebreakers.
How common are unsolvable hands?
458 of the 1,820 possible four-number hands have no solution — about one in four. The solver has checked every arrangement before it says so, which is what makes the verdict worth trusting rather than disheartening. The remaining 1,362 hands all have at least one route.
Can I solve with three or five numbers instead of four?
Yes — the mode switcher above sets the hand to 3, 4 or 5 numbers. The rule never changes: use each number exactly once with + − × ÷ and parentheses, and land on exactly 24. Three-number hands make a gentle on-ramp for younger players; five-number hands show up in classroom variations of the game.
More 24 game tools
Same engine everywhere: the hands you solve here are the hands that print on a worksheet and deal in a live classroom room. Start from the archive to browse by difficulty, or take today's hand from the daily puzzle.