Olympiad topic · P1–P6
Number puzzles
Number puzzles and made-up operations appear in the Hua Xia Cup from the P1 and P2 first rounds to the P6 finals.
Try one
★★ MediumMing 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:
What is the smallest AB can be?
- 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.
- 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.
Answer: 12
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsOne or two missing digits, following a rule.
- ★★MediumFree3 question types · unlimited questionsA number minus its reverse, a multiplication with gaps.
- ★★★HardFree2 question types · unlimited questionsA rule with unknown numbers in it, letter sums.
- ★★★★Expert (competition level)VIP2 question types · unlimited questionsA rule about runs of numbers, a four-digit number and its reverse.
Number puzzles: what it is and how to solve it
Sums with hidden digits or letters for digits: work each digit out. Also made-up operations with their own rules.
For example: 5□ + □7 = 70. What do the two □s add to? Ones: □ + 7 ends in 0, so the first □ is 3, carry 1; tens: 5 + □ + 1 = 7, so the second is 1. 3 + 1 = 4.
The key idea: Work from the ones digit, minding carries and borrows; start with the surest places, like the leading digit or a 0. For a made-up operation, put the numbers into its rule.
Practise this topic free →Easy to Hard (★ to ★★★) all free, unlimited questions, no download