burger fun / 24 Game Solver

The hardest 24 game puzzles

Every hand below was measured the same way: an exhaustive solver found every genuinely different way to make 24 (rearrangements like 3 × 8 vs 8 × 3 count once), with exact fraction arithmetic. The hardest hands are the ones with exactly one solution — all 94 of them are here, nothing cherry-picked. Each links to its own page where you can try the hand before revealing the answer.

The 94 single-solution hands

Exactly one distinct way to make 24 — miss it and there is nothing else to find. This is the complete list, hardest tier first, not a selection.

For contrast: the 4 most generous hands

The other end of the scale — hands with 12 or more distinct solutions. Good confidence builders, terrible tiebreakers.

The classics

Hands with a reputation rather than a record: the ones every classroom deals first.

And the famous lost cause

Not hard — impossible, provably. Its page explains why.

Of the 495 possible hands of digits 1 to 9, 404 can make 24 and 91 cannot. 94 solvable hands have exactly one distinct solution — every one of them is listed above. The 24 game solver takes any four digits, and practice mode deals brutal hands on demand.

FAQ

What is the hardest 24 game puzzle of all?

By reputation, 3, 3, 8, 8 — its only solution is 8 ÷ (3 − 8 ÷ 3), which forces you through 1/3 as an intermediate value. Its rival is 1, 3, 4, 6, solved only by 6 ÷ (1 − 3 ÷ 4). By the numbers, they are two of 94 single-solution hands; what makes those two famous is that their lone solutions need fractions.

How many 24 game hands have exactly one solution?

Exactly 94 of the 495 possible hands of digits 1 to 9. Another 94 have exactly two solutions, 146 have three to five, and 70 have six or more — while 91 hands cannot make 24 at all. All of it comes from the same exhaustive solve.

How were these hands found?

An exhaustive solver evaluated every possible expression over every hand with exact fraction arithmetic, then deduplicated solutions structurally so trivial rearrangements do not inflate the count. The dataset is verified by an independent test — a separately written search and parser must agree with every claim on these pages.

Does one solution always mean hard for a human?

Mostly, with funny exceptions. Some single-solution hands fall instantly — 1, 5, 9, 9 is just 1 + 5 + 9 + 9. The reliably cruel ones are those whose only solution needs a fractional step, like 3, 3, 8, 8. The count measures how few roads exist, not how well-lit they are — which is exactly why single-solution hands make good tiebreakers in class.

Can every 24 game hand be solved?

No — 91 of the 495 hands of digits 1 to 9 cannot make 24, and that is a proof by exhaustion, not a shrug. The most famous lost cause is 1, 1, 1, 1, which has its own page here explaining just how short it falls.

The Number Lab

Same idea, different constraints: four 9s, four 4s, or any four digits you like.

Or browse more free math games.