Olympiad topic · P1–P6
Most and least
Finding the most and the least appears in the Hua Xia Cup from the P1 and P2 first rounds to the P6 papers.
Try one
★★ MediumThere are 4 digit cards:
0, 5, 7, 8.
3 of them make a three-digit number, each card used once, and 0 can't go first.
Of the three-digit numbers that can be made, how much bigger is the biggest odd one than the smallest even one?
- Hint 1: Which place decides whether a number is odd or even? First think which cards can go in the units place.
- Hint 2: Once the units digit is fixed: for the biggest, put the rest in decreasing order; for the smallest, lead with the smallest non-zero digit, then increasing. Try every card that can go in the units place.
- 1.Biggest odd: the units digit can be 5 or 7, the other cards in decreasing order. units 5 → 875; units 7 → 857. The biggest is 875.
- 2.Smallest even: the units digit can be 0 or 8; lead with the smallest non-zero digit, then increasing. units 0 → 570; units 8 → 508. The smallest is 508.
- 3.The difference: 875 − 508 = 367.
- ⚠️Common mistake: trying only one possible units digit, or putting 0 first (057 isn't a three-digit number).
Answer: 367
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsLargest and smallest numbers, the fewest for all-different amounts.
- ★★MediumFree3 question types · unlimited questionsSharing evenly, river crossings.
- ★★★HardFree3 question types · unlimited questionsThe shortest time to fry toast, splitting a number for the largest product.
- ★★★★Expert (competition level)VIP2 question types · unlimited questionsSplitting into different numbers, the fastest bridge crossing.
Most and least: what it is and how to solve it
The most or least possible: the largest number, the fewest trips, the shortest time.
For example: From cards 7, 5, 9, 1, pick two to make the smallest two-digit number: 1 in the tens, 5 in the ones: 15.
The key idea: Picture the extreme case, then check it can really happen. To make the largest share as small as possible, share as evenly as you can; for the largest product, split into 3s (with 2s for what's left).
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