Answer to Puzzle #32: Two Creepers Climbing a Tree
This puzzle was emailed to me by Pramod S and solved by Pratik Poddar who runs a maths blog with many interesting puzzles that are slightly more mathematical than mine, his page on the same puzzle is here, there are a couple of different approaches there.
The first thing to realise is the vertical aspect of the puzzle is something of an illusion. For convenience we will proceed as if they both travel vertically at the same speed and thus complete the journey in the same unspecified time Ttotal, but this is for convenience regardless of the times the path described is the same.
From here we may now consider their motion as being constrained to the outer edge of a circle in the horizontal plane. The problem now becomes not dissimilar to a puzzle involving the overlap of the hands of a clock, though of course the hands are moving in opposite directions.
Each of the creepers will have speeds 5w and 3w. Where 'w' is some unspecified angular velocity in degrees per second, although the units are arbitrary. Using speed is distance over time or time = distance / speed, knowing that the total distance travelled 360 degrees times 5 revolutions or 3 revolutions we can easily show Ttotal as a function of w. Ttotal = 360 * 5 / (5w) or Ttotal = 360 * 3 / (3w) hence Ttotal = 360 / w. We will need this later.
The First CoincidenceWe now have two motions around a circle starting at one point, travelling in opposite directions, with speeds 3w and 5w. We need to solve this for time.
We know a total distance of 360 degrees is travelled by two objects with speeds 5w and 3w. Our equations of motion give us:
360=5wt + 3wt = 8wt
t = 360 / 8w
So our first overlap is at t = 360 / 8w. In terms of subsequent overlaps the puzzle is in some ways reset. They are co-existent at this time and travelling in opposite directions. if we were to calculate the next time of overlap it would be the same. We now know an overlap occurs every 360 / 8w
Total CoincidenceAn overlap occurs every 360 / 8w for a time of Ttotal = 360 / w hence the number of coincidence is
:Ttotal / time between coincidence
:(360 / w) / (360 / 8w)
since the division is exact we know that the last coincidence is at t = Ttotal, hence there are 7 overlaps not including the top and bottom.
Simpler ModelSince they are moving in the opposite direction we can actually just add the two. 5 + 3 (minus the top and bottom.) This is a reference frame argument, for the justification we will use it again when we look at overlapping hands of a clock.
Bard first refused to answer. Then got it wrong.
If you're curious what Bard made of this puzzle...
The least common multiple of 5 and 3 is 15.
Therefore, the creepers cross 15 times before reaching the first branch.
So the answer is 15
If you're curious what ChatGPT made of this puzzle...
Let's call the height of the cylindrical tree trunk "h" and the distance between two consecutive twists "p" (pitch). Since the jasmine and the rose start at the same point on the ground, they will both complete their first twist at the same height.
The jasmine makes 5 complete twists before reaching the first branch, which means that it travels a distance of 5p in height. Similarly, the rose makes 3 complete twists before reaching the first branch, which means that it travels a distance of 3p in height.
Since the jasmine and the rose twist in opposite directions, they will cross each other every time the distance traveled by one is equal to the distance traveled by the other. Therefore, the number of crossings will be equal to the number of times the jasmine and the rose have traveled the same distance in height.
We can find the height at which the jasmine and the rose will first cross each other by finding the least common multiple of 5p and 3p, which is 15p. This means that the jasmine and the rose will first cross each other at a height of 15p from the ground.
Similarly, we can find the height at which they will cross each other for the second, third, and fourth times. The jasmine will have completed a total of 10, 15, and 20 twists respectively, while the rose will have completed a total of 6, 9, and 12 twists respectively when they cross each other for the second, third, and fourth times.
The height at which they will cross each other for the second time is 30p (2 x 15p), the height at which they will cross each other for the third time is 45p (3 x 15p), and the height at which they will cross each other for the fourth time is 60p (4 x 15p).
Therefore, the jasmine and the rose will cross each other 4 times (excluding the bottom and top) before reaching the first branch of the tree.
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