Each issue Jerry, O4R's puzzle guy, brings you a set of puzzles. As always, the answers will be published in the next issue. They are now posted here as well! Questions and discussion may be directed to Jerry.
PROBLEM 1 A bear walks a mile South, a mile West, a mile North and returns to her starting point. Identify the places from which she could have started. (Due to Paul Halmos, mathematician extraordinaire.)
PROBLEM 2 Why don't they build 200 story buildings?
Problem 1. This is a fun problem because almost always some, or all, of the recipients will insist they know the answer-the North Pole. On a percentage (or measure) basis you don't get much, in fact you get zero for that correct answer. Now, hie thee-self South till you are near the South Pole. Stop when circumnavigation of the Pole is exactly one mile, and you have another starting place. Go a bit closer, till you can make two circles around the pole before returning to your starting place. Obviously, you can continue this process for 3, 4, ......n. So, here we have as many answers for starting places as there are integers. But even with all of these, you still have zero percent of the answers. For along each circle, you could have started at any point whatsoever, and it turns out that all the integers can't hold a candle to all the points on a line (or curve) segment. (c.f. any elementary text on set theory and check on orders of infinity).
Problem 2. Answer: People will only wait so long for an elevator, or walk so far to an alternate bank, before they take their business elsewhere when possible. This means that approximately every ten floors requires another elevator. (Actually, you probably don't know of a ten-story building with only one elevator.) Now, of course, elevators that only go from floor A to floor B speed things up for those passengers; but alas, we then need roughly the same number of elevators for each equivalent set of deleted floors. For 200 floors (with a footprint small enough so one bank of elevators suffices), you'd need more than 20 elevators and, if you count the areas of the elevators and approaches, there will be precious little (or zero) space remaining for whatever purposes people were taking the elevator in the first place. Incidentally, when Peter Cooper built the five-story Cooper Union in New York (in 1855) he was sure someone would invent a means of vertical assent and included a cylindrical shaft five stories tall. It is still used today.
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