24 Game Strategy: The Factor-Pair Method and Beyond

Random guessing doesn't scale

With 1,362 solvable four-number hands out of 1,820, luck alone will eventually fail you. What separates consistent solvers from lucky ones is a fixed search order — a ladder of shapes you check from easiest to hardest, stopping at the first rung that works. This is that ladder, rung by rung, with real hands from our solution archive.

Rung 1: Aim for a factor pair of 24

Multiplication is the densest operation: if you can split the four cards into two blocks worth 3 and 8, 4 and 6, or 12 and 2, you are done. The question is never "how do these make 24?" but "can these make a 3 and an 8?" (or 4 and 6, or 12 and 2).

Take 2 3 4 6. The pair 4 × 6 stares at you, leaving 2 and 3 — but 2 and 3 don't multiply to 1, so that split needs help. Try instead: (6 − 3) × 2 × 4 = 24, or 6 ÷ (3 − 2) × 4 = 24. This hand is generous — it has 10 distinct solutions, and most runs through a factor shape. Compare yours against the full 2 3 4 6 solution page.

Rung 2: Try the sum shape

When no factor split works, ask whether the cards divide into two pairs that add to a complementary pair of targets: 18 + 6, 16 + 8, 20 + 4, 12 + 12. Sum shapes are how most "easy" hands actually fall — the archive's easy tier is dominated by them.

Rung 3: Push the extremes — big products, small differences

Some hands only yield when you let one operation do violence. 4 4 10 10 looks like a wall: no factor pair, no sum shape. The way through is a big product: 10 × 10 = 100, subtract 4 to get 96, divide by the last 4 — (10 × 10 − 4) ÷ 4 = 24. Noticing that a double-digit product overshoots 24 by exactly a multiple of your remaining card is a pattern worth memorizing; the 4 4 10 10 solution page confirms it is the only whole-number route on this hand.

Rung 4: The fraction escape, a ÷ (b − c ÷ d)

When the first three rungs fail, stop hunting for multiplications and hunt for a small denominator. If three of your cards can make a tiny value like 1/3, 1/4, or 2/3, the fourth card divided by that value is your 24.

3 3 8 8 is the canonical case and the most famous hand in the game: 8 ÷ (3 − 8 ÷ 3) = 24. Every whole-number path fails — the fraction is the only door. Its sibling 5 5 5 1 works the same way from the other side: 1 ÷ 5 = 1/5, 5 − 1/5 = 24/5, and 5 × (5 − 1 ÷ 5) = 24. The shape a ÷ (b − c ÷ d) — or its multiply-through cousin — covers essentially every fraction-dependent hand. Across the entire archive only 16 solvable hands need fractions at all, and they all live in the expert tier.

Rung 5: Multiply by 1, add 0

The final rung is bookkeeping: using a redundant pair without disturbing the rest. A card paired with a 1 gives you free constructions — 1 × anything, anything ÷ 1 — and differences like (a − a) add 0. This is less a rescue technique and more an explanation for why some hands have absurd solution counts: 1 5 7 12 holds the archive record with 22 distinct solutions, and most of them are the "useful core" 5 + 7 + 12 dressed up as 1 × 5 + 7 + 12, 5 ÷ 1 + 7 + 12, 1 − 12 + 5 × 7, and so on. See the full list on the 1 5 7 12 solution page.

Putting the ladder to work

1. Factor pair? (3×8, 4×6, 12×2) — most hands die here.
2. Sum shape? (18+6, 16+8, 20+4) — most of the rest.
3. Extremes? — one big product or one engineered difference.
4. Fraction escape? — a ÷ (b − c ÷ d).
5. Housekeeping — ×1 and +0 constructions.

Work every hand top to bottom and you will solve the overwhelming majority of the hard tier within a minute. When you are stuck, the solver shows every solution for any four numbers — use it to check which rung you missed, not to skip the thinking.

New to the game? Start with how to play and the rules, then come back here once the first ten hands feel slow.