What's with the wad?

E4
Want a clue on how to solve the problem downstairs of this para? Take a look at the two people who've given solutions to the crumpled wad of paper problem coming up below that. You'll find that if you think laterally, there's a tactical similarity operating between each and hopefully you'll be able to squeeze your respective craniums betwixt them.   
A monk rises at sunrise and climbs a tall hill to meditate (as if he couldn't have meditated in his kitchen). The path is a narrow spiralling one to the summit and he walks at varying speeds, even stopping frequently to rest. After meditating for some days, the monk comes down - starting again at sunrise - with varying speed along the same path. Can it be proven (and please, without going into quantum chromodynamics, Hamiltonian graphs or Hilbert spaces) that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day?
 
DEAR MS
(Problem #1. If you were to rip out a page of a book, crumple it and then lay the wadded ball back on the book, at least one point on the crumpled page will always be directly over its original position. Why is that? - MS)
I've got a huge amount of answers that deal with Brouwer's fixed point theorem in topology which states that for any continuous function 'f' mapping a compact convex set into itself - and so on and so forth and so forth and so on, etc etc, etc - that a commerce graduate, history teacher, sociology major or law student would repeatedly retch on. Therefore here are the simplest two answers instead.
 
Wad-To-Do-Dept:
The wad covers a small portion of the original area and the remaining part is irrelevant. Remove from the wad the paper corresponding to the irrelevant area. Now the area covered becomes smaller, giving rise to a fresh irrelevant portion. Repeat these removals. It will end with the wad being reduced to a point above its corresponding point.
- Altaf Ahmed, ctrlaltaf@yahoo.in  

Simply put, if you take a map of a country and lay it out on a table inside that country, there will always be a point on the map which will be on top of the same point of the country. A sort of 'You Are Here' indicator!
- Saifuddin S F Khomosi, saif_sfk@hotmail.com
 
(Second problem: "A cube is made of white material but its exterior is painted black. If the cube is cut into 64 smaller cubes of exactly the same size, how many of the cubes will have at least two of their sides painted black?" - MS)
 
High-I-Cube-Dept:
The number of small cubes unpainted, one side painted and two sides painted, have standard relationships with the original cube's size. Three sides painted will always be 8 pieces, irrespective of the cube's size. For a cube of side n units (n = 2 or more), the number of cubes painted on two sides will be 12*(n - 2), which, in this case is 24. Hence the total number of small cubes with at least two sides painted will be 8 + 24 = 32.
- Sheikh Sintha Mathar, sheikhsm7@gmail.com
(Among the first five who also got it right are: Niraj Nandish, nirajnandish@icloud.com; Jaelyne Tauro, jaelynetauro@gmail.com; Siddharth Patkar, siddhpatkar@gmail.com; Ayan Joshi, ayanjoshi@yahoo.com; Priyanka Awatramani, priyankag1010@gmail.com.)
 
(The third problem was: Can a ball roll on ice in - ideally - zero friction conditions? - MS)
 
Nice-Ice-Dept:
A ball on a frictionless surface, when rolled, keeps on rolling or rotating at the same place, as there is no friction to push it forward or act on it.
- Ramakrishna Bhogadi, rambhogadi@gmail.com
 
ENDGAME(S)
1. Take some detergent powder in your palm and place it inside a bucket of cool water. As the powder rises, the hand feels hot. Does it make a difference if done in warm water?
2. You're given a large bucket filled with water, a wooden ruler and a soccer ball. Determine (approx.) the diameter of the ball using only these items.
(Mukul can be reached at mukul.mindsport@gmail.com)

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