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1 3 5 12 = 24: 10 solutions

Difficulty: HardOpen in Solver

12 × (5 − 1 × 3) = 24

  1. 1 × 3 = 3
  2. 5 − 3 = 2
  3. 12 × 2 = 24

10 solutions

  • 12 − 3 × (1 − 5) = 24Steps
    1. 1 − 5 = -4
    2. 3 × -4 = -12
    3. 12 − -12 = 24
  • 3 × (12 + 1 − 5) = 24Steps
    1. 1 − 5 = -4
    2. 12 + -4 = 8
    3. 3 × 8 = 24
  • 12 × (1 × 5 − 3) = 24Steps
    1. 1 × 5 = 5
    2. 5 − 3 = 2
    3. 12 × 2 = 24
  • 1 × 12 × (5 − 3) = 24Steps
    1. 1 × 12 = 12
    2. 5 − 3 = 2
    3. 12 × 2 = 24
  • 12 × (5 − 3 ÷ 1) = 24Steps
    1. 3 ÷ 1 = 3
    2. 5 − 3 = 2
    3. 12 × 2 = 24
  • 12 + 3 × (5 − 1) = 24Steps
    1. 5 − 1 = 4
    2. 3 × 4 = 12
    3. 12 + 12 = 24
  • 12 × (5 ÷ 1 − 3) = 24Steps
    1. 5 ÷ 1 = 5
    2. 5 − 3 = 2
    3. 12 × 2 = 24
  • (1 + 5) ÷ (3 ÷ 12) = 24Steps
    1. 1 + 5 = 6
    2. 3 ÷ 12 = 1/4
    3. 6 ÷ 1/4 = 24
  • (5 − 3) ÷ (1 ÷ 12) = 24Steps
    1. 5 − 3 = 2
    2. 1 ÷ 12 = 1/12
    3. 2 ÷ 1/12 = 24

Solving hints

  • Build a factor pair: split the four numbers into two groups and try to make a factor pair of 24 — 3 × 8, 4 × 6, or 12 × 2 — then multiply the two groups.

  • No factor pair divides cleanly? Switch to sums: make two numbers add up to 24, or try shapes like (a ± b) × (c ± d) and (a + b + c) × d.

Related hands

FAQ

Can 1 3 5 12 make 24?

Yes. Every one of them is listed above with step-by-step working.

How many solutions does 1 3 5 12 have?

10 — the list above is exhaustive, so every mathematically distinct way to combine the numbers is shown.

How difficult is 1 3 5 12?

It is rated “Hard” on our four-tier scale, based on whether whole-number intermediate steps suffice, how many distinct solutions exist, and whether the hand includes a two-digit card (10–13).