A medium question on number puzzles for P2–P4, with a step-by-step solution, diagrams 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
Ming thinks of a two-digit number AB, then swaps its digits to get another two-digit number, BA.
A and B are different digits, and neither is 0.
He adds them:
AB
+BA
33
What is the smallest AB can be?
Show the hints (3)
💡Hint 1: AB is A tens and B ones; BA is B tens and A ones. After adding, how many As and Bs are left?
💡Hint 2: AB + BA = 11 × (A + B), so A + B = 33 ÷ 11.
💡Hint 3: To make AB as small as possible, make the tens digit A as small as it can be — B must still be a digit 1 to 9, different from A.
Worked solution
Have a go first, then read the solution step by step.
1.AB + BA = (10 × A + B) + (10 × B + A) = 11 × A + 11 × B = 11 × (A + B).
2.A + B = 33 ÷ 11 = 3.
3.A is at least 1, and then B = 3 − 1 = 2, which works.
4.AB = 12. Check: 12 + 21 = 33.
⚠️Common mistake: answering with the first AB that works, without checking which is the smallest.
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