How Do Springs A, B, and C Reach Equilibrium on the 4th and 5th Intervals?

[Werg22]

This one is about bouncing springs. There are three springs bouncing on the floor, A, B and C. Spring A has a period of 2 seconds, spring B a period of 5 seconds and spring C a period of 9 seconds. If they all start at equilibrium position, eventually the springs reach back the equilibrium position on a 3 consecutive second interval (each reaching it once). What will be the orders in which the springs reach the equilibrium position on the 4th and 5th of these intervals?

 

[Werg22]

I'll give the solution here in case you really can't figure it out...

Since B has a period of 5, at the intervals of interest, a number that either ends with 0 or 5 will be included. For the cases in which it ends in 5, we could have

1. x5, x6, x7
2. x4,x5,x6
3. x3,x4,x5

Out of the above, only 1 and 2 respect the condition that each spring has to bounce only once during the interval. So we are looking for a multiple of 9 that either ends in 7 or 3. The first few five are 27, 63, 117, 153, 207.

Now consider the case in for which the second at which spring B reaches back the equilibrium position is a multiple of 10. It gives as possibilities

x0,x1,x2
x8, x9, y0

Out of the above there's no solution that corresponds to the intervals we are looking for. So all the intervals pertain to case 1, and are, in order,

25, 26, 27
63, 64, 65
115, 116, 117
153, 154, 155
205, 206, 207

The order for 4th interval is C - A - B and the order for the 5th interval is B - A - C.