Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 A schedule repeats every 5 days. Which day of the cycle does day 6 match?
Show worked solution
6 − 5 = 1, so day 6 matches day 1 — the cycle has just restarted.
2 A 7-day cycle runs for 10 days. Write the cycle day that day 8 matches, and the cycle day that day 10 matches.
Show worked solution
8 − 7 = 1 and 10 − 7 = 3.
3 A pattern repeats every 6 symbols. Which position in the cycle does position 13 match?
Show worked solution
13 − 6 − 6 = 1.
Level 2 · Core
4 A display repeats every 4 days. Which day of the cycle does day 30 match?
Show worked solution
30 − 1 = 29, and 29 ÷ 4 = 7 remainder 1, which is one step past day 1, so day 2.
5 A 7-day cycle is followed for 45 days. Write how many complete cycles have finished by day 45, and the cycle day that day 45 matches.
Show worked solution
6 complete cycles take 42 days, and 45 − 42 = 3.
6 A 12-day trip follows an 8-day cycle. How many of its stops are visited twice?
Show worked solution
12 − 8 = 4 days repeat, so 4 stops are visited twice.
Level 3 · Challenge
7 A 10-day tour runs a 7-day cycle.1 Day 9 matches day 2 of the cycle.
2 Three cities are visited twice.
3 Every city is visited at least twice.
Which statements are correct?
2 Three cities are visited twice.
3 Every city is visited at least twice.
Which statements are correct?
Show worked solution
9 − 7 = 2 (1 ✓), and days 8, 9, 10 repeat days 1, 2, 3, so three cities are visited twice (2 ✓). The other four are visited once, so not every city repeats (3 ✗). 1 and 2.
8 Six tours all follow the same 7-city cycle, each starting one step further along. Write the cycle position that day 10 corresponds to, and how many of the seven cities are never visited on day 10 by any tour.
Show worked solution
Day 10 is cycle position 3. Six tours put six different cities there, and with seven cities in the ring exactly 1 is left out.
9 A light flashes in a cycle of 9 colours. Which position in the cycle is flash number 100?
Show worked solution
100 − 1 = 99, and 99 ÷ 9 = 11 remainder 0, so it is position 1 — exactly 11 whole cycles have finished.