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How to Solve the 24 Game: Strategies and Examples
By HumanCalc · Published · 7 min read
The 24 game gives you four numbers and the four operations, and you must use every number exactly once to reach 24. The fastest way in is to stop trying combinations at random and aim for a pair that makes 24: 3 × 8, 4 × 6 or 2 × 12. Make one half from some of your numbers and the other half from the rest.
We searched all 1,820 possible sets of four numbers from 1 to 13, so the figures below are exact counts, not estimates. They show how often each approach works, how many sets cannot be solved, and which 16 sets need a fraction.
How many sets can be solved?
About three in four. Of the 1,820 sets of four numbers from 1 to 13, 1,362 can make 24 and 458 cannot. Order does not matter in this count, so 3 8 3 8 and 3 3 8 8 are the same set. Sets like 1 1 1 1, 7 7 7 7 and 9 9 9 9 have no solution. HumanCalc never deals a set without one.
The number of ways to solve a set varies a lot. Counting a + b and b + a as the same solution, 429 sets have 12 or more solutions and 170 have exactly one.
| Numbers from | Possible sets | Can make 24 | Need a fraction |
|---|---|---|---|
| 1–9 | 495 | 404 (82%) | 7 |
| 1–10 | 715 | 566 (79%) | 10 |
| 1–13 | 1,820 | 1,362 (75%) | 16 |
Strategy 1: aim for 3 × 8, 4 × 6 or 2 × 12
Multiplication is the most common way to finish. Of the 1,362 solvable sets, 956 (70%) have a whole-number solution that ends in a multiplication, and 902 (66%) can end in exactly 3 × 8, 4 × 6 or 2 × 12. Check those three first.
There are two ways to set this up. Split the numbers two and two and make each half from a pair, or keep one number as it is and build its partner from the other three.
| Set | Solution | Last step |
|---|---|---|
| 3 4 5 7 | (7 − 4) × (3 + 5) | 3 × 8 |
| 2 4 5 9 | (9 − 5) × (2 + 4) | 4 × 6 |
| 4 5 6 7 | (6 − 4) × (5 + 7) | 2 × 12 |
| 2 3 4 5 | 4 × (5 + 3 − 2) | 4 × 6 |
| 2 5 6 8 | 8 × (6 − 5 + 2) | 8 × 3 |
Strategy 2: start from one number and work backward
Pick the number that looks most useful and ask what it needs. A 6 needs a 4 from the other three. An 8 needs a 3. A 9 is not a factor of 24, so try adding 15 to it or subtracting it from 33. If you have a number left over, multiply by it if it is a 1, or cancel it against an equal number: 9 ÷ 9 = 1 and 9 − 9 = 0 both leave your total alone, which is how 3 8 9 9 becomes 3 × 8 × (9 ÷ 9).
Set: 1 3 6 7
- You hold a 6, so you need a 4 from 1, 3 and 7.
- 7 − 3 = 4.
- The 1 is spare. Multiplying by it changes nothing.
6 × (7 − 3) × 1 = 24
Strategy 3: land near 24, then add or subtract
A whole-number finish by addition exists for 725 solvable sets (53%) and by subtraction for 708 (52%). A set can have both. This approach suits numbers that multiply past 24: make 20, 25, 27 or 30, then correct with what is left. The last three sets below have no other solution, so factor pairs will not help you there.
| Set | Solution | Last step |
|---|---|---|
| 1 4 4 5 | 4 × 5 + 4 × 1 | 20 + 4 |
| 5 5 5 5 | 5 × 5 − 5 ÷ 5 | 25 − 1 |
| 3 3 3 3 | 3 × 3 × 3 − 3 | 27 − 3 |
| 5 6 6 9 | 6 × 9 − 5 × 6 | 54 − 30 |
Strategy 4: build a big number, then divide
This one is easy to miss, because it means going well past 24 on purpose. 493 solvable sets (36%) have a whole-number solution that finishes with a division, such as 48 ÷ 2, 72 ÷ 3 or 96 ÷ 4. For 1 2 7 7 and 4 4 10 10 it is the only way: (7 × 7 − 1) ÷ 2 and the example below.
Set: 4 4 10 10
- 10 × 10 = 100.
- 100 − 4 = 96.
- 96 ÷ 4 = 24.
(10 × 10 − 4) ÷ 4 = 24
Fraction puzzles: 3 3 8 8 and 15 others
Only 16 of the 1,362 solvable sets (1.2%) need a fraction along the way. If whole numbers go nowhere, you may be holding one of them: 1 3 4 6, 1 4 5 6, 1 5 5 5, 1 6 6 8, 1 8 12 12, 2 2 11 11, 2 2 13 13, 2 3 5 12, 2 4 10 10, 2 5 5 10, 2 7 7 10, 3 3 7 7, 3 3 8 8, 4 4 7 7, 5 5 7 11 and 5 7 7 11. Fifteen have exactly one solution; 1 4 5 6 has two. Thirteen contain a repeated number, which is often the one that cancels at the end.
Six of the 16 finish with a division by a small fraction. The idea is that 8 ÷ 1/3, 6 ÷ 1/4 and 12 ÷ 1/2 are all 24. Even 1 3 4 6, which looks easy, has only this kind of solution: 6 ÷ (1 − 3 ÷ 4).
Set: 3 3 8 8
- 8 ÷ 3 = 8/3.
- 3 − 8/3 = 1/3.
- 8 ÷ 1/3 = 8 × 3 = 24.
8 ÷ (3 − 8 ÷ 3) = 24
The other fraction shape: build 24/n, then multiply by n
The other ten finish with a multiplication. You make 24 divided by one of your numbers, then multiply by that number to clear the fraction. Look for it when the same number appears twice, so one copy can make the fraction and the other can cancel it.
Set: 1 5 5 5
- 1 ÷ 5 = 1/5.
- 5 − 1/5 = 24/5.
- 24/5 × 5 = 24.
5 × (5 − 1 ÷ 5) = 24
| Set | Solution | Why it works |
|---|---|---|
| 3 3 7 7 | 7 × (3 + 3 ÷ 7) | 3 + 3/7 = 24/7, then × 7 |
| 4 4 7 7 | 7 × (4 − 4 ÷ 7) | 4 − 4/7 = 24/7, then × 7 |
How HumanCalc's 20 levels use these sets
Make 24 deals one set per level, and clearing it moves you up. Early levels use small numbers and sets with many solutions. Later levels use larger numbers and sets with only one or two. The last three levels deal only the 16 fraction sets. The game keeps exact fractions, so 8 ÷ 3 becomes a card that shows 8/3. Running out of time or skipping costs one of your three lives and shows one way to solve the set.
| Levels | Numbers from | Ways to solve | Sets it can deal | Time per puzzle |
|---|---|---|---|---|
| 1–3 | 1–9 | 12 or more | 111 | 60–56 s |
| 4–6 | 1–9 | 6–11 | 131 | 54–50 s |
| 7–10 | 1–10 | 3–5 | 112 | 48–42 s |
| 11–14 | 1–12 | 1–2 | 198 | 40 s |
| 15–17 | 1–13 | Exactly 1 | 155 | 40 s |
| 18–20 | 1–13 | Needs a fraction | 16 | 50 s |
How to practise
Holding partial results while you test combinations leans on working memory, as DeStefano and LeFevre's 2004 review of mental arithmetic describes. Practice makes you better at this puzzle. That is not evidence of wider cognitive gains.
- Spend the first few seconds of each set checking 3 × 8, 4 × 6 and 2 × 12 before you try anything else.
- When you solve an easy set, look for a second solution. Early levels have 12 or more, so it is good practice at seeing other routes.
- When a puzzle times out, read the solution HumanCalc shows, then work through it once in your head.
- If you reach the fraction levels, try the two shapes above before anything else.
Research sources
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