Four 24-Game Patterns That Cover Most Hands (and the One That Doesn't)

The 24 game has the cleanest rules in elementary arithmetic. Take four cards, combine them with +, −, ×, ÷, and parentheses, hit exactly 24. The Chinese playing-card variant draws from 1–13, generating 1,820 distinct hands (C(16,4) = 1820 multisets with repetition); the original Suntex deck patented by Robert Sun in 1988 uses cards 1–9 and yields 495 distinct multisets (C(12,4) = 495). Both decks contain plenty of hands that even experienced players freeze on.

I have spent six weeks playing 24 most evenings while teaching my niece. What started as her homework turned into my own training. Here are the four patterns I have memorised, and the one hand-family where no shortcut helps — you just have to think.

Pattern 1: Sweep the Factor Triplets First

Every winning expression routes through a multiplication that lands on 24 somewhere in its tree. The productive factorisations to scan for are the three human-friendly ones: 4×6, 3×8, and 2×12. The trivial 1×24 almost never helps; the negative factorisations are too brittle for mental work.

Routine: read your four numbers, then ask in this exact order — can I make 4 and 6? 3 and 8? 2 and 12?

Take 2 3 4 6. The 4 and 6 are already on the table; the remaining 2 3 must collapse into a 1, the multiplicative identity. 3 − 2 = 1. So 4 × 6 × (3 − 2) = 24. Five seconds, no pen needed.

Take 1 2 3 4. Sweep: 4 and 6? 2 × 3 = 6 leaves 1 4, which cannot make 1 cleanly with a single operator. Try 3 and 8? 4 × 2 = 8 leaves 1 3; 3 × 1 = 3. So 8 × 3 × 1 = 24. The pattern primes you to spot both 1 × 2 × 3 × 4 and (4 × 2) × (3 × 1) simultaneously — useful when you want to teach a child why the answer is what it is rather than just announce it.

Roughly two-thirds of solvable hands open up under this single sweep. The remaining third is where Patterns 2–4 earn their keep.

Pattern 2: The Fraction Door

Across 200 random hands I cataloged from a 1–13 deck, about 7% had solutions reachable only through a fractional intermediate value. These are the hands that defeat most adults, because grade-school arithmetic conditioning silently assumes intermediate values stay integer. They don't have to.

The canonical example: 3 3 8 8. The only solution is 8 ÷ (3 − 8÷3) = 24. Trace it by hand: 8/3 = 2.666…, 3 − 2.666… = 0.333…, 8 ÷ 0.333… = 24. The unlock is recognising that the inner subtraction can produce 1/3 — and that 8 ÷ (1/3) = 24 is exactly the answer you need.

Two more in this family to keep in your head, all sharing the template (integer − small_fraction) × integer:

4 4 7 7: (4 − 4÷7) × 7 = (24/7) × 7 = 24
1 5 5 5: (5 − 1÷5) × 5 = (24/5) × 5 = 24

Once you have seen this shape three times, you start to recognise families of hands that match it: two near-equal large cards plus a small/large pair. When you see one, try the template first. It works more often than it has any right to.

Pattern 3: The Diff-of-Twelves

If a hand contains a 12, or you can make a 12 cheaply, frame the problem as 12 + 12, 12 × 2, or 24 + 0 and ask what the other half is.

Take 4 4 3 9. Sweep: 12 anywhere? 4 + 4 + 4 is too many cards. But 4 × 3 = 12, which leaves 4 9. Can I make 12 from 4 9? 9 + 4 − 1 = 12 needs an extra card I don't have. Switch: can the other half make 2 for the 12 × 2 framing? 9 − 4 + ... no card left. Different framing: 9 + 3 + ... no. Eventually: (9 − 3) × 4 + ... we only have one 4 left. Try (4 + 4) × 3 = 24, leaves 9 unused — not allowed; every card must be used. Then (4 + 4 − 3) × 9 = 5 × 9 = 45, no. (9 + 3) × (4 − 4) is 12 × 0 = 0. Eventually: (9 − 3) × (4 + ... ) — only one card left. This hand requires Pattern 4.

The point of Pattern 3 is not that it always wins. It is that it fails fast — five seconds — and you move on with a clear head. Slow failures are what burn time at the table.

Pattern 4: Cancel a Pair with Division, Reach for the Rest

When a hand has two cards of equal or near-equal value, divide them to produce a 1 (or a clean small integer) and use it as a multiplier on the rest of the hand.

6 6 4 ? wants a 1 from the two sixes: 6 ÷ 6 = 1, then 1 × 4 × 6 = 24 — but wait, I have already used both sixes. The expression (6 ÷ 6) × 4 × ... needs a fourth card. So for a real hand like 6 6 4 1: (6 ÷ 6 + 1) × 12? No, that needs a 12. Try (6 + 6) × (4 − 1) × ... no card left. 6 × 4 × (6/6 − ... ) runs out. Actually: 6 × 4 + 6 ÷ 1 = 24 + 6 = 30, no. 6 × 4 × (6 − 5) — no 5. This particular hand may not yield cleanly to Pattern 4 either; it goes back to Pattern 1 (the 4×6 is already on the table, and 6 − 6 + 1 × ... doesn't help). Some hands resist multiple patterns and end up requiring the solver.

For a clean Pattern 4 win, consider 2 2 6 4: (2 ÷ 2) × 6 × 4 = 24. The two 2's collapse to 1, the 6 and 4 multiply to 24, the 1 is the identity. Three operators, one cancellation, done in four seconds.

When All Four Patterns Fail, and What Six Weeks of Practice Looked Like

For the hands that resist every pattern, I open the 24-point solver. It walks all 7,680 candidate expressions per hand in under 50 milliseconds and lists every distinct solution. The point is not to copy the answer for the next hand — it is to study the shape of the solution and add it to my pattern library. For verifying an expression I have already built but am unsure of, the scientific calculator handles parenthesised arithmetic with fractional intermediates faster than I can do it on paper.

When a hand returns no solutions at all — about 5.5% of multisets from a 1–13 deck are unsolvable — the solver's empty result is itself the answer: stop trying, move on. The classic unsolvable hand is 1 1 1 1; the maximum reachable value using all four operators and any parenthesisation is 4.

I dealt 20 random hands a night, set a 60-second per-hand timer, and scored myself. Week 1: I solved 11 of 20 within the timer, average time-to-solve 34 seconds for the ones I cracked. Week 6: 18 of 20 within the timer, average 9 seconds for the ones I cracked. The two I missed each night were almost always fraction-door hands where I could not spot Pattern 2 fast enough.

The pattern approach did not replace the solver — it replaced the flailing. The solver is the last resort for hands that require an insight I have not yet internalised. The instant I see one of those, I write it down, identify which template it belongs to, and try to slot it into one of the four patterns. By week six the unfamiliar shapes had mostly stopped appearing.

The 24 game is small enough to be brute-forced by a script and deep enough to keep teaching a human something every fifth hand. That combination is rarer than it sounds.

Made by Toolora · Updated 2026-05-27