Two Creepers Climbing a Tree

Two creepers, one jasmin and other rose, are both climbing up and round a cylindrical tree trunk. jasmine twists clockwise and rose anticlockwise, both start at the same point on the ground. before they reach the first branch of the tree the jasmine had made 5 complete twists and the rose 3 twists. not counting the bottom and the top, how many times do they cross?

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 Coincidence

We 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.
jasmine and rose creepers puzzle
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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 Coincidence

An 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)
8

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.

Source : puzzles.nigelcoldwell.co.uk/thirtytwo.htm

Wires, Batteries and Lightbulbs

You are standing in a house in the middle of the countryside. There is a small hole in one of the interior walls of the house, through which 100 identical wires are protruding.

From this hole, the wires run underground all the way to a small shed exactly 1 mile away from the house, and are protruding from one of the shed's walls so that they are accessible from inside the shed.

The ends of the wires coming out of the house wall each have a small tag on them, labeled with each number from 1 to 100 (so one of the wires is labeled "1", one is labeled "2", and so on, all the way through "100"). Your task is to label the ends of the wires protruding from the shed wall with the same number as the other end of the wire from the house (so, for example, the wire with its end labeled "47" in the house should have its other end in the shed labeled "47" as well).

To help you label the ends of the wires in the shed, there are an unlimited supply of batteries in the house, and a single lightbulb in the shed. The way it works is that in the house, you can take any two wires and attach them to a single battery. If you then go to the shed and touch those two wires to the lightbulb, it will light up. The lightbulb will only light up if you touch it to two wires that are attached to the same battery. You can use as many of the batteries as you want, but you cannot attach any given wire to more than one battery at a time. Also, you cannot attach more than two wires to a given battery at one time. (Basically, each battery you use will have exactly two wires attached to it). Note that you don't have to attach all of the wires to batteries if you don't want to.

Your goal, starting in the house, is to travel as little distance as possible in order to label all of the wires in the shed.

You tell a few friends about the task at hand.

"That will require you to travel 15 miles!" of of them exclaims.

"Pish posh," yells another. "You'll only have to travel 5 miles!"

"That's nonsense," a third replies. "You can do it in 3 miles!"

Which of your friends is correct? And what strategy would you use to travel that number of miles to label all of the wires in the shed?

Two Threads Are Bruning.?????

You have two lengths of rope. Each rope has the property that if you light it on fire at one end, it will take exactly 60 minutes to burn to the other end. Note that the ropes will not burn at a consistent speed the entire time (for example, it's possible that the first 90% of a rope will burn in 1 minute, and the last 10% will take the additional 59 minutes to burn).

Given these two ropes and a matchbook, can you find a way to measure out exactly 45 minutes?