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Maths olympiad question · P5–P6★★★★ Expert

Number puzzles: olympiad question 9

An expert question on number puzzles for P5–P6, with a step-by-step solution and the answer. Number puzzles and made-up operations appear in the Hua Xia Cup from the P1 and P2 first rounds to the P6 finals.

The question

ABCD is a four-digit number; writing its digits backwards gives the four-digit number DCBA.

Different letters are different digits, and neither number starts with 0.

A + B + C + D = 11, and

ABCD
−DCBA
3357

What is ABCD?

Show the hints (3)
  1. Hint 1: Write both by place value and subtract: ABCD − DCBA = 999 × (A − D) + 90 × (B − C).
  2. Hint 2: 90 × (B − C) is at most 810, less than 999, so A − D is about 3357 ÷ 999. Find A − D and B − C.
  3. Hint 3: A − D = 3 and B − C = 4. Then use A + B + C + D = 11 to find D + C.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.ABCD − DCBA = (1000A + 100B + 10C + D) − (1000D + 100C + 10B + A) = 999 × (A − D) + 90 × (B − C) = 3357.
  2. 2.B − C lies from −9 to 9, so 90 × (B − C) is at most 810 either way. Try A − D: 999 × 3 = 2997, 3357 − 2997 = 360 = 90 × 4. So A − D = 3 and B − C = 4.
  3. 3.A = D + 3 and B = C + 4; put into A + B + C + D = 11: 2 × (D + C) + 7 = 11, so D + C = 2.
  4. 4.Try each D (not 0, every digit 0 to 9): 4511 (repeated digit, no), 5402 ✓.
  5. 5.So ABCD = 5402. Check: 5402 − 2045 = 3357 ✓, 5 + 4 + 0 + 2 = 11 ✓
  6. ⚠️Common mistake: assuming B − C is positive (it can be negative); or forgetting different letters are different digits and giving a number with a repeat.

Answer: 5402

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