Olympiad topic · P1–P6
Cycles
Cycle questions appear in the Hua Xia Cup from the P1 first round to the P5 papers: which comes in a given place, which day of the week, which last digit.
Try one
★★ MediumMay threads beads for a bracelet, repeating the pattern "1 red, 1 white" over and over.
She threads 23 beads in all.
How many of the beads are white?
- Hint 1: How many beads make a group, and how many of those are white?
- Hint 2: Beads left over after the whole groups — check one by one whether each is red or white.
- 1.A group is 1 + 1 = 2 beads, 1 of them white. 23 ÷ 2 = 11 remainder 1.
- 2.11 groups hold 11 × 1 = 11 white beads.
- 3.1 bead left over: each group starts with red, so the first 1 is red. White beads: 11 + 0 = 11.
- ⚠️Common mistake: counting all the left-over beads as white, or ignoring them.
Answer: 11 beads
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsThe nth one, how many there are.
- ★★MediumFree2 question types · unlimited questionsDays of the week, bead patterns.
- ★★★HardFree3 question types · unlimited questionsWeekdays across months, the sum of the first so many, the last digit of a huge product.
- ★★★★Expert (competition level)VIP4 question types · unlimited questionsTwo cycles, one counting down; weekdays from a sum of dates; digit sums and digit counts of recurring decimals; remainders of a sequence where each is the sum of the two before.
Cycles: what it is and how to solve it
Things repeating in the same order: what the nth one is, how often something comes up, or what day of the week it is.
For example: Writing 1, 2, 3, 4, 5 over and over, what's the 29th digit? 29 ÷ 5 = 5 r 4: the 4th in a group, so 4.
The key idea: Find how long one cycle is, then divide: the remainder says which place in the cycle; if it divides exactly, it's the last one in the cycle. Days of the week are a cycle of 7.
Practise this topic free →Easy to Hard (★ to ★★★) all free, unlimited questions, no download