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Fun Physics

Posted by trickofmind under Fun Physics (16 Responds)

Take oil and water in a beaker. The oil floats in a separate layer above the water. In which direction should one accelerate the beaker to mix the oil and water? (An emulsion can be taken to mean a mixture for our purposes here.) And hey, we need heavy funda to go with your response.

Water and Vinegar

Posted by trickofmind under Logic, Tom (12 Responds)

The old puzzle was you have one litre of water in one bucket and one litre of vinegar in another bucket. Take a spoonful of water from the first bucket and mix it thoroughly with the vinegar in the second. Now take a spoonful of the vinegar-water mixture in the second bucket and mix it back with the water in the first. Problem: does the first bucket have more vinegar or the second bucket more water? Solve it only if you can get at the most elegant solution with no math involved at all.

Not So General Knowledge Quiz

Posted by Karl Sharman under Tom (37 Responds)

1. Which one of the following doesn’t belong: pitying, favor, unkindness, murder, exaltation?
2. A climber leaves camp at midnight for the top of the mountain and arrives there at noon. He immediately turns around and climbs back down by the same route to arrive at camp at midnight, exactly 24 hours after he left it. Is it necessarily the case that at a point on the route his watch’s hands were in the same position on the ascent as on the descent? (Assume the watch operates normally, and the second hand sweeps in a continuous, non-discrete manner)
3. In 1989 what company claimed the interesting distinction of having the world’s smallest logo?
4. At the other end of the spectrum, what was the largest single man-made object visible with the naked eye to Neil Armstrong when he first stepped onto the moon’s surface?
5. If a metal torus (doughnut) is heated, will the hole in the center get larger or smaller?
6. What do the following have in common, besides their obvious association with formality: limousine, champagne, tuxedo, ascot?
7. What is the next element in this series: “sun”, “new”, “inactive”, “hidden”…?
8. How high up would one need to be to be able to see one-third of the Earth’s surface? Assume that the weather is clear and that the Earth is a sphere.
9. If we are referring to this place today, we call it “Istanbul”. If we are referring to what it was 2000 years ago, what is it called?
10. There is a text in a Yale library, worth hundreds of thousands of dollars. No one knows how old it is. No one has ever read it. What is the text most commonly called?
11. List the planets in order of their distance from the sun (nearest to farthest) on the first of January 1999.
12. We are now entering (just about any second now…) the “Age of Aquarius”. What 26,000 year cycle dictates the ages?
13. How many nibbles are there in a megabyte of RAM?
14. In 1974, from a small town in Puerto Rico, a coded message was transmitted. The message was exactly 1679 bits long. Whom was it sent to?
15. What child of a poet is commonly considered the world’s first computer programmer?

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Running the Trains at Pamplona

Posted by Karl Sharman under Tom (14 Responds)

Walking home one day to your little pied-a-terre outside Pamplona, you take a short cut along the train tracks. The tracks cross a narrow bridge over a deep gorge. At the point you are 3/8 of the way across the bridge, you hear the train whistle somewhere behind you. You charge across the bridge, and jump off the track as the train is about to run you down.
As it happens, if you had gone the other way, you would have reached safety just before being run over as well.

If you can run ten miles per hour, how fast is the train moving?

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Coins in Boxes

Posted by Atran under Tom (11 Responds)
In each of six boxes B1, B2, B3, B4, B5, B6 there is initially one coin. There are two types of operation allowed:

        Type 1: Choose a nonempty box Bj with 1 ≤ j ≤ 5. Remove one coin from Bj and add two coins to Bj+1.

        Type 2: Choose a nonempty box Bk with 1 ≤ k ≤ 4. Remove one coin from  Bk and exchange the contents of (possibly empty) boxes Bk+1 and Bk+2.

Determine whether there is a finite sequence of such operations that results in boxes B1, B2, B3, B4, B5 being empty and box B6 containing exactly 2010^2010^2010 coins. (Note that a^b^c= a^(b^c).) 

Spy Games

Posted by Karl Sharman under Tom (17 Responds)

As your basic, average, everyday, ho-hum, run-of-the-mill super spy, you sometimes need to exchange information with other agents. Your preferred method of communicating is to leave messages in a public strongbox. These strongboxes are made from Unobtanium and are indestructible, and immovable. If you want security on these, you need to provide your own padlock and chain.
You have a suitable lock and chain, but there is a flaw with this plan: The other agent will have no way of removing the lock without the key. If you leave the key in the box first, a cunning thief might copy the key and return it, only to open the box himself before the other agent or his henchmen arrive.
How can you communicate securely with this system?

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Eggs-actly which one?

Posted by Karl Sharman under Tom (20 Responds)

In preparation for an Easter Egg Hunt, Euclid’s mother purchases a dozen eggs. She boils half the eggs, marks each of them with a small “X”, and puts them back in their carton with the rest. Presumably, the remaining eggs will be used for some other less festive purpose like breakfast.
The smallest child of the family, Euclid’s Brother later tells his mother that he finished putting X’s on the remaining eggs. Oops!
What is the easiest way for the mother to figure out which eggs are for Easter, and which are for breakfast

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Airports in a Strange World

Posted by Atran under Tom (14 Responds)

In a strange world there are n airports arranaged around a giant circle, with exactly one airplane at each airport initially. Every day, exactly two of the airplanes fly, each going to one of its adjacent airports. Can the airplanes ever gather at one airport?

Means…all around

Posted by Atran under Mathemagic, MathsChallenge, Tom (6 Responds)

a) Prove that the geometric mean of two positive numbers is equal to the geometric mean of their arithmetic and harmonic means.

b) Prove that the harmonic mean of two positive numbers a and b does not exceed the geometric mean, and that the equality holds only if a=b.

c) Prove that the arithmetic mean of three positive numbers is not less than their geometric mean, that is,

(a+b+c)/3 ≥ ³√abc

3 Positive Numbers ABC

Posted by Atran under Mathemagic, MathsChallenge, Tom (12 Responds)

Prove, given any three positive numbers a, b, and c, the following inequality holds:          (a+b)(b+c)(c+a)≥8abc.

Show that the equality holds only for a=b=c.