Want a clue on how to solve the problem rising out downstairs from under this para? Take a look at the two people who’ve given solutions to the kawphy cup surface and 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?
(The leftover problem was: “If you stir a cup of coffee and then let it come to rest. At least one point on the surface will be back in its original place. Why? Or, 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. Again, why?”)
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, geology major or law student would repeatedly retch on. Therefore here are the simplest two answers instead.
One important observation is that the wad should not overhang the area originally occupied by the page. Now the wad covers a small portion of the original area and the remaining part is irrelevant. Next 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, email@example.com
Take a map of a country and place it on top of a table in the same country. Won’t at least one point on the map correspond to the same geographical point in the country? – Rajnath V J, firstname.lastname@example.org
(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?”)
Thirty two pieces will have at least two black faces (of these, eight will have three black faces, while 24 will have exactly two faces black). -- Kishore Rao, email@example.com
Eight of the smaller cubes will have paint on three sides and 24 smaller cubes will have paint on two sides (total 32). 24 smaller cubes will have paint on only one side and eight smaller cubes will not have paint on any of their sides. -- Dr P Gnanaseharan, firstname.lastname@example.org
(Among the first five to get it correct also are: Advaith Ram. email@example.com; Rajesh Patil, firstname.lastname@example.org; Saishankar Swaminathan, email@example.com; Gopunatarajan Natarajan, firstname.lastname@example.org; Animesh Jena, email@example.com)
(The third problem was: “Can a ball roll on ice in -- ideally -- zero friction conditions?”)
A ball on an ideally 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, firstname.lastname@example.org
The ball will not move on ice since the ball with any weight will lower the melting point and form a small hollow and thus will settle at each point. -- Gadepalli Subrahmanyam, email@example.com
BUT GOOGLE THIS NOW
1. Take some detergent powder in your palm and place it inside a bucket of cool water. As the powder rises up, the hand feels hot. Does it make any difference if it’s done in warm water?
2. You’re given a large bucket filled with water, a wooden ruler and a soccer ball. Determine (approximately) the diameter of the ball using only these items.
— Sharma is a scriptwriter and former editor of Science Today magazine.(firstname.lastname@example.org)