Two creepers spiralling up the same tree

Question

A jasmine and a rose both start from the same point at the base of a tree trunk and climb to the same point on the first branch. The jasmine winds clockwise around the trunk exactly 5 times on the way up; the rose winds anticlockwise exactly 3 times. Both climb steadily.

Not counting the starting point at the bottom or the meeting point at the top, how many times do the two creepers cross?

Accepted Answer

Seven times. Set up. Measure height as a fraction h of the climb, from 0 at the base to 1 at the branch, and measure position around the trunk as a fraction of a full turn. The jasmine, going clockwise five times, is at angular position 5h (keeping only the fractional part). The rose, going the other way three times, is at angular position minus 3h, again fractional part. Crossings. The creepers are at the same point of the trunk when 5h and minus 3h differ by a whole number, that is, when 8h is a whole number. Between h = 0 and h = 1 that happens at h = 1/8, 2/8, ..., 7/8, and also at the en…

Two creepers spiralling up the same tree

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