Make 24: how to play the 24 game and win

How to play

1. You get four numbers. Use every one of them exactly once, with addition, subtraction, multiplication and division, to make exactly 24.
2. Tap a number, then an operation, then a second number. The two rune stones fuse into one stone holding the result, so four stones become three, then two, then one.
3. The order matters for − and ÷: the first number you tap comes first. Tapping 8, ÷, 2 makes 4; tapping 2, ÷, 8 makes 1/4.
4. Results can be negative or fractions, and they stay exact. 8 ÷ 3 is kept as 8/3, not 2.67, so 8 ÷ (3 − 8 ÷ 3) really makes 24.
5. When one stone is left, the hand is over: 24 wins. Anything else means a step went the wrong way; Undo steps back one fusion and Reset returns to the four numbers.
6. Hint lights up a first step from the stones you have now. Give up shows one full answer and opens every answer in the 24 Game Solver.

Controls

Mouse

Click a number, an operation and a second number to fuse them. Click a selected number again to drop it.

Touch

Tap a stone, tap + − × or ÷, then tap a second stone. Every button is large enough for a thumb.

Keyboard

Keys 1 to 4 pick the stones in reading order, + - * (or x) and / pick the operation, and Backspace undoes a step. Tab and Enter reach every button.

Score and progress

The clock starts with your first tap and pauses when you choose Pause or leave the tab. Hands solved in a row without Hint or Give up form your streak. Your fastest unaided time for 1–9 and for 1–13, your best run and your daily results are saved in this browser when local storage is available.

Tips

• Look for 3 × 8 and 4 × 6 first; they cover more hands than any other pattern.
• Hold on to a big number. Fusing 9 with something small early often throws away the factor you need.
• When a hand has two 1s, use them together: 1 + 1 = 2 is often more useful than wasting each one.
• A hint costs your streak, not your time. Use it once, then finish the hand yourself.

Want every answer for a hand, or have four cards from a real game? Type them into the 24 Game Solver, which lists every different way to make 24, with the steps.

How to make 24: a method that works

Most hands fall to a few patterns. Look for a way to reach a factor pair of 24 first, then fall back on the rarer tricks.

1. Know the targets: 24 is 1 × 24, 2 × 12, 3 × 8 and 4 × 6, and also 12 + 12, 20 + 4 or 30 − 6. Pick one number in the hand and ask which partner it needs.
2. Build the partner from the other three. With 4 7 8 8, keep 8 and make 3 from 4, 7 and 8: 4 + 7 − 8 = 3, and 8 × 3 = 24.
3. Spend extra numbers neutrally. A 1 can multiply or divide without changing anything, and two equal numbers can make 1 (n ÷ n) or 0 (n − n).
4. Try making the bigger part first. With 1 2 7 7, 7 × 7 = 49, 49 − 1 = 48, and 48 ÷ 2 = 24.
5. If whole numbers fail, think in fractions. With 1 5 5 5, 1 ÷ 5 = 1/5, 5 − 1/5 = 24/5, and 5 × 24/5 = 24. With 3 3 8 8, 8 ÷ (3 − 8 ÷ 3) = 24.

How many hands can be solved?

With four numbers from 1 to 9 there are 495 different hands, ignoring order, and 404 of them can make 24; the other 91, such as 1 1 1 1 or 1 1 1 2, cannot. With numbers from 1 to 13, as with playing cards, 1,362 of the 1,820 hands can. Make 24 only deals hands that can be solved, and both counts come from the same exact-fraction search the game uses for its hints.