24 Game Solver,
every way to make 24.

Field guide

How the 24 game solver works.

The 24 game is a mental-arithmetic puzzle. You are dealt four numbers, often the values of four playing cards, and you must use each one exactly once, combining them with addition, subtraction, multiplication, division and parentheses, until the answer is 24. You cannot skip a number, reuse one, or join two digits into a bigger number. That is all there is to the rules, yet some hands take a long time by hand and a few have exactly one answer.

Entering a hand

Type your numbers into the boxes in any order; the same value can appear more than once. Choose three, four or five numbers, and leave the target at 24 or change it to any whole number from 0 to 9,999. The answers appear as soon as every box is filled. The Random hand button deals card values from 1 to 13 that the solver has already confirmed can be solved, and hides the answers, so you can attempt the hand first and reveal them afterwards.

True fractions, never rounded decimals

Many hands only work through a fraction. The classic example is 3, 3, 8, 8, where the route is to turn 8 ÷ 3 into a fraction, subtract it from 3, and divide 8 by the result.

A calculator that rounds 8 ÷ 3 to 2.666667 gets 24.000024 and might miss the answer or accept a wrong one. This solver stores every intermediate value as a fraction made of two whole numbers, a numerator and a denominator, so the final comparison with the target is a comparison of whole numbers, with no rounding anywhere. Answers that pass through a non-whole step are labelled, and a checkbox hides them if you only want whole-number routes.

Which answers count as the same

The search tries every order and grouping, which produces many expressions that are really one answer. The list merges them. The order and grouping of a sum do not matter, so 2 + (3 + 4) and 4 + 3 + 2 are one answer. Subtraction and division fold into the surrounding sum or product, so 8 − (3 − 1) is shown as 8 + 1 − 3. A division by 1 is shown as a multiplication by 1, and two equal numbers are interchangeable. Answers with different structure, such as (a + b) × (c + d) and a × b + c × d, remain separate entries.

Worked example

Take 4, 7, 8, 8. The two eights divide to 1, 7 − 1 gives 6, and 4 × 6 makes 24, written 4 × (7 − 8 ÷ 8). The solver lists that and every other distinct route for the hand, with whole-number answers first and the fewest brackets next, so the simplest ideas are at the top.

Hands that cannot reach 24

Not every hand has a solution. Among the 1,820 distinct hands of four cards from ace to king, 1,362 can make 24 and 458 cannot, according to the count published at 4nums.nfshost.com, and this solver agrees with both totals. When a hand has no answer, the result panel says so in plain words after checking every order, grouping and fractional step, rather than leaving you wondering whether you missed one.

Tips for solving by hand

• Aim for factor pairs. 24 is 3 × 8, 4 × 6, 2 × 12 and 1 × 24, so look for two numbers that make one factor and two that make its partner.
• Create a 1 or a 0. Dividing two equal numbers gives 1, and subtracting them gives 0, which are useful for adjusting a total without changing it.
• Try a fraction when stuck. If nothing works in whole numbers, test whether a quotient such as 5 ÷ 5 or 1 ÷ 4 can sit inside a subtraction before a final division.

Disclaimer: This tool is an independent fan creation and is not affiliated with, authorized, or endorsed by Suntex International Inc. “24 Game” is a trademark of its respective owner, and the words “24 game” are used here only to describe the arithmetic puzzle. Provided for practice and entertainment only.