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A hopping circuit is painted on a school playground. It consists of 25 small circles, with the numbers 0 ( at the 12 o' clock place) to 24, arranged as a big circle. Each student jumps either 3 or 4 spaces clockwise(so a student can end up either in circle 3 or 4 on their first hop). Students must go twice around the circuit and start and end at 0 to complete a game. All students must list in order the numbers they land on and record the total number of hops they take.

a) In one game a student took 13 hops. Write down a possible list of numbers he landed on.

I worked out that to clear 50 numbers with 13 numbers you van have 11 4-hops & 2 3-hops. This makes the numbers 4,8,12,16,20,24,3,7,11,15,19,22,0. But is 11 4-hops and 2 3-hops the only combination? And is there some sort of relativity formula to shorten the answer a bit( I can assume there will be tens and hundreds of combinations)

b) Find all possible combinations of the number of 3-hops and number of 4-hops in a game.

Again, I can assume there will be quite a few. Is there a relativity formula for this too?( You aren't limited to 13 hops for this question)

c)What is the smallest number of different numbers a student can land on in one game? Explain your answer

I'm totally stuck on this one . I don't know how are you supposed to prove that you have the smallest amount of different numbers?

d) Jo and Mike decide to play a longer version of the game. They both play simultaneously and start at 0. When Jo jumps 4 circles, Mike jumps 3 and vice-versa. Jo first 5 hops are 4-hops. After that, she takes 3-hops until she reaches or passes 0. How many laps will each of them take until they next meet at 0?

Again, I'm totally stumped. How are you meant to find that out? I've tried tracking the pattern but to no avail

Thanks!

P.S. I'm only a Year 7, so please don't post answers with calculus in them. I won't understand a single bit. Algebra 1 is fine, though