Make 24 Strategy: How to Solve Four Numbers to 24 (2026)

1. The library: 36 frozen puzzles, 577 solution paths

Make 24 gives you four whole numbers and one target: leave a single tile reading exactly 24. The library is frozen at 36 puzzles, twelve in each tier and numbered 1–36.

A solution path is one complete sequence of three merges that ends at 24; the counts are exact, not estimates. All twelve easy puzzles have at least 39 paths, which is why they feel forgiving. The hard tier has only 34 paths in total across its twelve puzzles, and three puzzles in the whole library have exactly one path: #19 (medium, 3 5 7 13), #25 (hard, 1 3 4 6) and #26 (hard, 2 3 5 12). On those, any merge that is not on the path makes the puzzle unwinnable — Undo is not a convenience there, it is the difference between finishing and restarting.

Nothing is generated at runtime. Every puzzle was proved solvable offline with the same exact move set you play, and the shipped solution count for each one was re-derived in this session.

2. One move is a merge: the three facts that decide everything

1. A move takes two tiles, applies one operator and leaves one result. So four numbers always take exactly three merges. There is no way to use a number twice, and no way to finish in two moves.
2. The first tile you pick is the left operand. Picking 8 and then 3 and pressing minus means 8 − 3 = 5; picking 3 first gives 3 − 8 = −5. Addition and multiplication do not care; subtraction and division do. Section 8 shows why this decides more puzzles than it looks like it should.
3. Only division by zero is illegal. Negative results and fractions are legal and are kept exactly: 8 ÷ 3 stays 8/3, never 2.666…, and the solver reasons with the same exact value. There is no rounding to make a nearly-right answer pass.

Because a merge can express any bracketing — the three merges are a binary tree over the four numbers — you never need a parentheses button. The interesting question is never "how do I write it"; it is "which pair do I merge, in which order, and with which operator".

The board shows the merge log above the tiles. Read it before you press Undo: it tells you which value came from which pair, and that is usually enough to see where the position went wrong.

3. The tier is a fact about the solutions, not a score

The three tiers are not a difficulty slider. Each one is a statement about what the solutions of that puzzle are allowed to contain, and the shipped library satisfies the statement on every single puzzle:

• Easy — some solution uses only plus and minus. All 12/12 easy puzzles have at least one purely additive path, and 9 of the 12 have exactly 39 paths, all of them additive.
• Medium — no additive solution exists, but some solution never leaves the integers. All 12/12 medium puzzles have an integer-only path and none has an additive one.
• Hard — every solution needs a fractional intermediate. All 12/12 hard puzzles have zero integer-only paths; all 34 hard solution paths pass through a fraction.

That last line is the one players disbelieve. The table below shows the smallest fraction each hard puzzle forces, measured over all of its paths:

A hard puzzle is not "a medium puzzle with bigger numbers". It is a puzzle whose shortest way to 24 runs through a value that is not a whole number, and often through a denominator of 5, 7 or 11. If you refuse to accept a fraction because it looks unfinished, you cannot solve a single hard puzzle. The hard tier is not a random sample: of all four-number multisets drawn from 1–13, only 16 can be solved at all while having no integer-only path. The library uses twelve of those sixteen.

4. What a win actually looks like in each tier

I grouped all 577 solution paths by the operator of the last merge. The pattern is strong enough to plan around:

Easy finishes by addition or subtraction — 95.9% of its paths. The last step is usually two medium-sized values such as 14 + 10, 15 + 9, 12 + 12 or 25 − 1. So on easy, do not hunt for a product; build two numbers whose sum or difference is 24.

Medium finishes by multiplication — 72.7% of its paths end in ×, and the common shapes are factor pairs of 24: 6 × 4 (13 paths), 2 × 12 (9), 8 × 3 (8), 12 × 2 (4), 3 × 8 (3), 4 × 6 (1). Plan backwards: which two of your four numbers can become 6 and 4, or 8 and 3, while the other two become the second factor? Medium is also where negative intermediates show up: #16 (1 3 5 7) wins with 3 − 1 = 2, 5 + 7 = 12, 2 × 12 = 24, and #19 uses 5 × 13 = 65, 65 + 7 = 72, 72 ÷ 3 = 24.

Hard never finishes with + or −. Its 34 paths split into two shapes:

• 24/k × k (24 paths): build 24 divided by one of the numbers, then multiply by that number. #33 (3 3 7 7): 3 ÷ 7 = 3/7, 3 + 3/7 = 24/7, 24/7 × 7 = 24.
• x ÷ (1/k) (10 paths): build 1 divided by one of the numbers, then divide by it. #34 (3 3 8 8): 8 ÷ 3 = 8/3, 3 − 8/3 = 1/3, 8 ÷ 1/3 = 24.

That is the whole hard tier in two sentences. The fraction is not a detour; it is the mechanism. Once you see the shape, "impossible" puzzles become arithmetic.

5. The opening is the decision

Every puzzle has 36 legal opening merges (6 unordered pairs × 6 operator orders; no pair can divide by zero in this library). I tried all 1,296 of them and asked whether the remaining three tiles could still reach 24.

A random first merge is a one-in-four shot even on easy, and a one-in-fourteen shot on hard. This is why "just try something" feels bad: most of the time you are not exploring, you are ending the puzzle.

The operator matters even more than the tier:

Multiplying first is the single worst habit in the game. It makes a large number, and a large number is hard to bring back to 24 with two merges; only 4 of the 216 opening products in the whole library leave a winnable position. Subtraction and addition keep the values in a range you can still steer. Division is the exception, and on hard it is the only thing that works.

6. Hard mode: the first merge must be a fraction

Of the 340 opening merges on hard puzzles that keep both operands and the result as whole numbers, none survives. All 30 surviving openings are divisions whose result is not an integer. This is a proof, not a heuristic: adding, subtracting or multiplying two integers cannot create a fraction, and every integer-only position on a hard puzzle is dead, so the first merge has to be a division that leaves a fraction.

The twelve hard puzzles, with the openings that actually keep them alive:

Look at the shape of the surviving opening: it is always the smaller number divided by the larger one, so the result is below 1. Dividing the other way — 10 ÷ 2 = 5 on #30, 4 ÷ 3 = 4/3 on #25 — leaves a value too large to fold back into 24 with two merges. On hard, "divide small by large" is the first thing to try.

7. "I can see 24 already" is usually a trap

Twelve of the 36 puzzles contain two starting numbers that merge to exactly 24: nine pairs multiplying to 24 and three pairs adding to 24. It is almost always bait. Merging that pair first leaves the other two numbers to combine with 24 into 24, which requires them to make 1 (for × or ÷) or 0 (for + or −). They usually cannot.

The measurement: in 11 of those 12 puzzles, the merge that already reads 24 makes the position unwinnable. The single exception is #3 (3 8 9 10): 3 × 8 = 24, 10 − 9 = 1, 24 × 1 = 24. Every other case is a dead end, including all four 3 × 8 pairs on #34 (3 3 8 8) — the puzzle whose actual answer divides 8 by 3 and never multiplies the 3s and 8s directly.

The practical rule: when you spot a pair that makes 24, do not merge it. Ask what the other two numbers can make, and whether 24 can absorb that value. If the answer is not 1 or 0, the pair is a decoy. On #34 the winning move is the one that looks least like progress: turn 8 ÷ 3 into a fraction.

8. Order matters: the first tile is the left number

Swapping the two tiles changes the result of − and ÷, and that changes whether the puzzle survives. I checked both directions for every pair on every puzzle: 41 ordered cases on 20 of the 36 puzzles flip from winning to dead when the operands are swapped.

Two concrete ones:

• #3 (3 8 9 10): 10 − 9 = 1 keeps the puzzle alive; 9 − 10 = −1 kills it.
• #34 (3 3 8 8): 8 ÷ 3 = 8/3 is the only kind of opening that wins; 3 ÷ 8 = 3/8 is dead.

So the habit is not "pick the bigger number first" or "pick the smaller one first". It is: say the subtraction or division out loud in the order you intend — "ten minus nine", "eight divided by three" — and tap the left operand first. On addition and multiplication the order is free, which is exactly why those are the safe operators to open with.

A related detail: the game allows negative intermediate values, and the hint itself uses them. On #1 (1 6 9 10) the recommended first merge is 1 − 6 = −5, then −5 − 9 = −14, then 10 − (−14) = 24. If a negative number makes you stop and restart, you are throwing away a legal and often necessary route.

9. Hint, Undo, Restart and the star economy

The shipped build has no move limit, no time limit and no penalty for using Undo. The only thing that costs you is the star rating, and the rules are exact:

• 3 stars: the solve used no Hint and no Undo.
• 2 stars: no Hint, but Undo was used at least once.
• 1 star: any Hint was used, no matter how many.

Undo is unlimited and steps back exactly one merge. Restart reloads the puzzle and clears the hint count, the undo flag and the timer, so a clean restart protects a three-star attempt in a way that Undo cannot. Your best record per puzzle — stars first, time second — is replaced only by a better star rating or, at the same rating, a faster time, so a bad run never overwrites a good one. It is stored in this browser only, under pn.make-24.best.v1.

Hint solves the position you are actually in, not the starting position, and shows you one legal merge without playing it. Every successful press adds one to the hint count. If the current position can no longer reach 24, Hint says so and does not add to the count; it tells you to undo a move instead. That makes Hint a position checker as well as a solver: pressing it after a suspicious merge is a cheap way to find out whether you have killed the puzzle. (With one tile left, Hint does nothing at all — the status line already tells you the tile is not 24, and Undo or Restart is the only way forward.)

Two consequences follow. First, if you are chasing stars, restart rather than undo after a mistake: Undo caps you at two stars for the rest of that attempt. Second, use Hint as a last resort, not as a first move — the solver's first suggestion is the first merge its search finds in a fixed operator order (+, ×, −, ÷), not a judgement about which merge is most instructive. Measured from the opening board, the Hint opens with a subtraction on 11 of 12 easy puzzles, with + on 8 of 12 medium puzzles, and with a division on all 12 hard puzzles — the last one is not a stylistic preference, it is the only kind of first merge that survives.

10. A practice loop, with three worked examples

1. Read the tier first. Easy: aim for a final sum or difference. Medium: aim for a final factor pair (6 × 4, 8 × 3, 2 × 12). Hard: accept that the first merge is a small-over-large division.
2. Choose the opening before you touch a tile. Prefer + or − on easy and medium, and only a fraction-producing ÷ on hard. Do not open with × unless you have counted why.
3. Say the direction out loud for − and ÷, then tap the left operand first.
4. Count the last move backwards. Two merges remain after the opening: decide what the final pair must be, then check whether the middle merge can produce it.
5. If the position dies, Undo once. If you are chasing stars, Restart instead. If you are unsure whether it is dead, press Hint — it will tell you without charging you.

Easy, #1 (1 6 9 10): the Hint's own solution is 1 − 6 = −5, then −5 − 9 = −14, then 10 − (−14) = 24. The lesson is not the exact path; it is that the negative value is a tool. A player who refuses to go below zero sees no way in.

Medium, #13 (1 2 3 5): 1 + 2 = 3, 3 + 5 = 8, 8 × 3 = 24. The last move is a multiplication, and the two factors are built from the pairs — exactly the medium shape from section 4. On medium, once you can see a 6 and a 4, or an 8 and a 3, you are usually one merge away.

Hard, #34 (3 3 8 8): 8 ÷ 3 = 8/3, 3 − 8/3 = 1/3, 8 ÷ 1/3 = 24. This is the puzzle people call impossible because 3 × 8 = 24 is sitting right there and merging it kills the position. The working path divides first, subtracts to make a unit fraction, then divides by it. #33 (3 3 7 7) uses the other hard shape: 3 ÷ 7 = 3/7, 3 + 3/7 = 24/7, 24/7 × 7 = 24.

11. If you get stuck: a checklist

1. Did you open with ×? That is the first thing to undo — 98.1% of opening products in this library are dead.
2. On hard, is your first merge a division of a smaller number by a larger one? If not, you have not started yet.
3. Did you merge a pair that already read 24? Check whether the other two can make 1 or 0; if not, undo it.
4. Did you tap the operands in the order you meant for − or ÷? Reversing them is a 41-case trap in this library.
5. Are you refusing a negative or a fraction? Both are legal, and the hard tier requires the fraction.
6. Do you know which finish you are aiming for — a sum/difference on easy, a factor pair on medium, 24/k × k or x ÷ (1/k) on hard?
7. Is the puzzle one of the three with a single solution path (#19, #25, #26)? If so, one wrong merge is fatal; Undo before trying anything else.
8. If you are still lost, press Hint once. It re-solves your current position; if it says there is no way, Undo, and if it shows a merge, the position is still alive.

FAQ

How do you play the 24 game?

Take four numbers, use each one exactly once, and combine them with plus, minus, times and divide until one value remains — it has to be exactly 24. In Make 24 you do it with three merges: tap two tiles, tap an operator, and the pair becomes one result. The first tile is the left operand, so tap the number you mean to subtract or divide first.

Is Make 24 the same as the 24 game?

It is our own build of the public four-number puzzle, not a copy of any commercial version. The four-number puzzle has no single author. 24 Game is a registered mark of Suntex International and 24game.com is a commercial site; Make 24 is not affiliated with, endorsed by or derived from it, and the name is used for description and search only.

Is every Make 24 puzzle solvable?

Yes — all 36 were proved solvable offline with exact fractions, and the shipped solution counts were re-enumerated in this session with zero mismatches. Every puzzle has at least one path to 24. A position can become unsolvable after a wrong merge, though; that is a fact about your board, not about the puzzle. Hint tells you when that has happened.

Do I have to use all four numbers?

Yes. A move consumes two tiles and leaves one, so four numbers always take exactly three merges, and the final tile must be 24. You cannot skip a number or reuse a result, and there is no two-move shortcut.

Are fractions and negative numbers allowed?

Yes, and the hard tier requires a fraction. Values are kept exactly — 8 ÷ 3 stays 8/3, never a rounded decimal — so the game will not reject a position because of floating-point error. Negative intermediates are legal too: on #1 (1 6 9 10) the winning line goes through −5 and −14 and finishes with 10 − (−14) = 24.

Why does multiplying first usually fail?

Because a product is large and you only have two merges left. Across the library, only 4 of 216 opening products leave a winnable position (1.9%), against 25.5% for opening with addition and 22.9% for subtraction. On hard, every opening that stays on whole numbers is dead, so the first merge must be a division that produces a fraction.

What does the tier mean?

It describes the solutions, not a score. Easy has a solution that uses only plus and minus. Medium has no additive solution but has one that never leaves the integers. Hard has no integer-only solution at all: every path passes through a fraction, with forced denominators from 2 to 11 depending on the puzzle.

What do the stars mean?

Three stars for a solve with no Hint and no Undo; two stars if you used Undo but no Hint; one star if you used a Hint at all. Undo is unlimited and only costs you the third star. Restart clears the hint count and the undo flag, so restarting protects a three-star attempt. Your best stars and time per puzzle are kept in this browser.

Why does Hint sometimes say there is no way to 24?

Because the current position really cannot reach 24 — some earlier merge was wrong. Hint re-solves the position you are in, not the starting puzzle. When it reports no path it does not add to your hint count; undo one merge and try a different pair. With one tile left it does nothing at all, because the status line already tells you the tile is not 24.

How many puzzles are there, and can I jump to one?

36, twelve per tier, numbered 1–36. Use ?puzzle= with a number, a puzzle id or a tier: ?puzzle=34, ?puzzle=hard or ?puzzle=medium. An invalid value starts at puzzle 1. The arrows move one puzzle at a time and the keyboard works too — 1–4 pick a tile, + − * / pick an operator, U undoes, H hints, R restarts.

Can I play Make 24 on a phone?

Yes. Tiles are at least 40px in both axes and the buttons are at least 28px tall; tapping works with mouse, touch or pen, and the layout has been checked at 390×844 without overflow. There is no download and no sign-up, and your best times stay in the browser you play in.