Puzzle – Two Creepers Climbing a Tree

Question:

Jasmine and rose creepers are two creepers (climbing plants) that are both climbing up and around a cylindrical tree trunk. Both jasmine and rose begin at the same spot on the ground and twist both clockwise and counterclockwise. The rose had done three complete twists and the jasmine had made five before they reached the tree’s first branch. How many times do they cross, excluding the bottom and top?

Solution:

The vertical component of the problem is somewhat of an illusion, which is the first thing to realize. Let us assume that they both travel vertically at the same position and finish the journey at the same undetermined time, but this is for convenience regardless of whether the times are the same.

Let us consider their motion as being restricted to the outside edge of a circle in the horizontal plane. The problem now resembles a riddle involving overlapping clock hands, although the hands are going in opposing directions.

The creepers will move at 5 and 3 wpm, respectively. Where ‘w’ is some arbitrary angular velocity in degrees per second. We may simply illustrate Total as a function of w if we know that the total distance travelled is 360 degrees times 5 revolutions or 3 revolutions (time = distance/speed), i.e., Total = 360 * 5 / (5w) or 360 * 3 / (3w) 

Hence, Total = 360 / w 

The First Coincidence:
Now, we have two movements around a circle that begin at the same position and go in opposing directions with speeds of 3w and 5w, respectively. We want to tackle this for time.

We know that two objects travelling at speeds of 5w and 3w cover a total distance of 360 degrees. Based on our equations of motion, we can:

360 = 5wt + 3wt = 8wt
t = 360 / 8w

As a result, our first overlap occurs at t = 360 / 8w. In terms of successive overlaps, the problem is reset in certain respects. Right now, they are moving in opposing directions while coexisting. Calculating the subsequent overlap would get the same results. We now know that an overlap happens every 360 / 8w.

Total Coincidence
An overlap happens every 360 / 8w for a total duration of 360 / w, resulting in a total number of coincidences of

= Total/time between coincidence
= (360 / w) / (360 / 8w)
= 8

Since the division is accurate, we know that the last coincidence occurs at t = Total, implying that there are 7 overlaps excluding the top and bottom.