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Spinning a point

E4
Here's a fun thing. 111,111,111 × 111,111,111 = 12,345,678,987,654,321. Look at it carefully, and wonder in shock and awe, before figuring it out with an instant 'Aha!' solution. However, the next one's going to require a little more of a (M)aha! working out solution.  
Two numbers - not necessarily different - are chosen from the range of positive integers greater than 1 and not greater than 20. Only the sum of the two numbers is given to person A and only the product to person B. Here's their subsequent cell phone conversation: A: "I see no way you can determine my sum." B (one hour later): "I know your sum." A (some time later): "Now I know your product." What are the two numbers?
 
DEAR MS
(The problem concerned an 80m dia sphere dropped down a 100m dia tunnel through the Earth, with a smaller stone simultaneously dropped down a smaller tunnel through the sphere. What would be the joint configuration when they reached the end? - MS)
 
A-Stone's-Throw-Dept:
The stone's position would be at the centre. The stone dropped in it would freefall toward the centre of the Earth until it flew past it. When it reaches the centre, it keeps going because of inertia. When it reaches the other side, gravity pulls it back toward the centre again, slowing its ascent toward the surface, until it began again to fall toward the centre. It would repeat this yo-yo-ing freefall, with air friction slowing it down a bit more each time, until the stone came to rest, floating in the centre of the hole. It's simple harmonic motion!
- Priyanka Awatramani,
priyankag1010@gmail.com
 
(Regarding the analogies problem, 28 people replied - with only three (yes, just three) getting all 12 correct. Fourteen got only one or two wrong but the heaviest hassle happened with #11, with almost everybody thinking the answer was Maurice Ravel. Alas! - MS)
 
An-Alogy-Dept:
(1) Strip is to Mobiüs as Bottle is to...? KLEIN; (2) Thought is to Action as Obsessive is to...? COMPULSIVE; (3) Mice is to Men as Cabbages is to...? KINGS; (4) Sword is to Damocles as Bed is to...? PROCRUSTES; (5) Jekyll is to Hyde as Eloi is to...? MORLOCKS; (6) 4 is to Hand as 9 is to...? SPAN; (7) Hollow Victory is to Pyrrhic as Hollow Village is to...? POTEMKIN; (8) Easy Job is to Sinecure as Guiding Light is to...? CYNOSURE; (9) Leg is to Ambulate as Arm is to...?  BRACHIATE; (10) Astronomy and Physics is to Astrophysics as History and Statistics is to...? CLIOMETRICS; (11) Language Games is to Ludwig as Piano Concerto for the Left Hand is to...? PAUL; (12) Set of Sets Not Member of Themselves is to Russell as Darkness of the Night Sky in an Infinite Universe is to...? OLBERS.
- Surya Narayanan Krishna Moorthy, surya661666@yahoo.com
 
(Yes, Jaelyne Tauro, jaelynetauro@gmail.com and Saifuddin Khomosi, saif_sfk@
hotmail.com, you also got it right. - MS)
 
(The third problem was: "If a year consists of 365 days, how many times does the Earth rotate during this year? The answer is 366. But the question is why?" - MS)  
 
Day-Of-Reckoning-Dept:
If you want a precise answer, you need to specify which definition of 'year' you're referring to. Our calendars are based on the 'tropical year'. The length of the tropical year is about 365.24 'mean solar days' of exactly 24 hours. Divide that year length by the Earth's rotation period relative to the fixed stars, called a 'sidereal day'. (That rotation period is about 23 hours 56 minutes and 4 seconds.) The answer in that case is approximately 366 spins.
- Abirami Sundari Ramalingam, abirami262002@gmail.com
 
ENDGAME(S)
1. To get a peeled hard-boiled egg into a bottle, just drop a burning taper in and the partial vacuum created by the loss of oxygen sucks the egg in. But the O2 loss is compensated by the production of CO2 and water vapour. So what sucks the egg in?
2. What colour is hidden in the following sentence: "One dancer I see is out of step."

(Mukul can be reached at mukul.mindsport@gmail.com)