What Sports Goats?

E4
Imagine that you're on a television game show and the host presents you with three closed doors. Behind one of them sits a sparkling, brand-new sports car; behind the other two are smelly, old goats. The host asks you to pick a door, and you select door #1. Then, the host, who is well aware of what's going on behind the scenes, opens door #3, revealing one of the goats. "Now," he says, turning toward you, "do you want to keep door #1, or do you want to switch to door #2?"
Suppose we pause at this point, and suddenly a flying saucer settles down onto the stage. A little green woman emerges, and the host asks her to point to one of the two unopened doors. The chances that she'll randomly choose the one with the prize are 1/2, all right. But that's because she lacks the advantage the original contestant had - the help of the host. In other words, which choice gets you the car - keeping your original door, or switching? (HINT: If, like most people, you posit that your odds are 50-50, think again - unless, of course, you like winning goats as much as you like new cars, in which case you'll win 100 per cent of the time.)
 
DEAR MS
(The problem was: "How can you tell if an X drawn inside a closed curve is really inside it or actually outside but giving the appearance of being inside due to the convolutions of the closed curve?" - MS)
 
Border-Crossings Dept:
Draw a ray from the point X. Count the number of times it crosses the closed curve. If the number is even, then X lies outside the curve. If it is odd, then X lies inside the curve. In fact, it doesn't have to be a ray; just as long as it is a curve that starts at X and goes off to infinity.
- Saifuddin S F Khomosi,
saif_sfk@hotmail.com
 
(The other problem was: "An insect crawls along a rubber band which is attached to a wall at one end and a car moving away at the other end. The car moves at 1m/s while the insect is only moving at 1cm/s. If the band is uniformly and infinitely elastic, will the insect ever reach the car?" - MS)
 
No-Zeno-Paradox Dept:
Three important data are missing in the question - what is the length of the rubber band, what is the initial position of the insect, and does it crawl from the wall to the car or in the opposite direction. Unless its initial position is at the car-side-end of the band, it can never reach the car, as the car is moving away at 100 times the speed of the insect.
- Sheikh Sintha Mathar, sheikhsm7@gmail.com (Sorry Double-S M,
wrong! - MS)
 
We think the insect will never reach the car. Actually, it will, after an enormous amount of time has elapsed. Given such a colossal amount of time, whatever may be the length of the stretched rubber band and the relative speed of the insect, if it is subject to steady stretching and speed, the insect will reach the car. In case the insect has covered 1/5000 of the rubber band after a second, then in the next second the insect would have moved the same distance, but it may be lesser than the size of the rubber band at 1/10000.  For a long time, this will continue to happen and the fraction will keep decreasing. Summation of the fractions proves that success will result.
- Suryanarayanan Krishnamoorthy, surya661666@yahoo.com
 
ENDGAME
If you've seen police action serials like CSI, Law & Order etc, you know that when the alleged perps are taken into cop custody for the old third degree, there's often this huge one-way mirror on one wall which lets the other cops watch the fun and games from the outside, while all that the poor sweating sap sees from the inside  is a moving mugshot of his own making. Meaning, if you were in his or her position and wanted to know if it was, in fact, a real mirror or one of those sneaky one-way tricks, what's the easiest way to find out?

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

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