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Math Chat Archive: the CSM columns

Math Chat Columns at the
Christian Science Monitor Web Site

1) Math Chat
1996-07-05 [TEXT ONLY]
 
Frank Morgan Math Chat Frank Morgan Professor of Mathematics, Wiliams College If a Ball Goes Up, When Must It Come Down? Thanks for the tremendous response to the Father's Day Math Chat column June 14. The answer appears below, along with a new challenge question. But first, here's a copy of the last question for those who didn't see it: Challenge question From Father's Day Suppose every husband and wife keep having children until they have a girl and then stop. Assuming boys and girls ar... (351 words)
2) Where Does The Sun Rise?
1996-07-19 [TEXT ONLY]
 
Frank Morgan Where Does The Sun Rise? Is it due East? Answer The answer depends on the time of year. On the first day of spring or fall, the sun rises almost due east. In our winter, when the South Pole is tilted toward the sun, the sun rises south of east. In our summer, when the North Pole is tilted toward the sun, the sun rises north of east. It's all a matter of geometry. (Of course this motion of the sun is only apparent: It's actually the Earth moving.) Challenge question (With thanks ... (175 words)
3) Stationary Suns and Kings and Queens
1996-08-16 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Old Planet X challenge question: My friend on Planet X reports that the sun stayed at the same spot on the horizon for a whole 24-hour day. How can this be? Answer: The planet's axis of rotation is horizontal, not vertical. Two days a year, it points directly at the sun, which appears to hang directly over the North or South Pole. If our friend lives on the equator, he sees it stationary on the horizon. Chuck Gahr remarks that Uranus's ax... (305 words)
4) Math Chat
1996-08-30 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Marbles and Change Old marbles challenge question You have 10 marbles, half black and half white, to put in two bags any way you want. Your opponent (without looking) will select a bag at random and choose a marble at random from that bag. You win if he picks a white marble. What are your chances of winning? What if another enemy looks and switches one pair of marbles between bags before your opponent makes his selection? Answer For ... (418 words)
5) Math Chat
1996-09-13 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Prime Number Digits Old kings and queens challenge question A magician takes a stack of eight cards (four kings and four queens, face down, in some order), turns the top one face up, moves the next one to the bottom of the stack, turns the next one face up, moves the next one to the bottom, and so on. To deal out an alternating sequence of kings and queens, how must he arrange the cards beforehand? Suppose instead he has some number of ca... (377 words)
6) Coins, Prime Numbers, And Checker Boards
1996-09-27 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Old coins challenge question (thanks to Alice Loth) In the US, with our five coin denominations of 1«, 5«, 10«, 25«, and 50«, it can take up to eight coins to make change (99«). Which different five denominations would minimize the number of coins ever needed to make change? Answer. With coins of 1«, 3«, 11«, 27«, and 34«, you never need more than five coins to make change. Tenie Remmel reports that this is one of 1,129 solutions, the o... (611 words)
7) Checkerboards and Fences
1996-10-11 [TEXT ONLY]
 
Frank Morgan Old checkerboard challenge How many squares are there on an 8x8 checkerboard, counting not only the 64 1x1 squares, but also the 2x2 squares, 3x3 squares, etc.? What about a 101x101 checkerboard? How many of those squares contain the little square in the center? Answer On an 8x8 checkerboard, there are 64=82 1x1 squares. There are just 72 (overlapping) 2x2 squares, since they cannot start in the last row or column. There are 62 3x3 squares and so on down to 12=1 8x8 square. The t... (379 words)
8) Double Bubbles And Presidential Elections
1996-10-25 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Old Double Pen Challenge What is the most efficient shape for two identical adjacent pens if any shapes are allowed, not just rectangles? Answer Chuck Gahr and Jan Smit found the most efficient shape: two overlapping circular pens, separated by a straight line as in Figure 1. The three fences meet at equal angles of 120 degrees. (This shape is better than one circle divided by a straight line through the middle.) This fact was proved by a... (415 words)
9) Presidential Elections, Challenges of the Years
1996-11-08 [TEXT ONLY]
 
Frank Morgan US Presidential Election Challenge Question Assume that there are just two candidates and say half the population in each state votes. What is the fewest number of votes with which one could be elected president? What if there are three candidates? Answer Basically you need to win about half the votes in states with about half the electoral votes, or about 25 percent of the popular vote. Actually, since smaller states get more electoral votes per resident, you can win with 39 sma... (357 words)
10) Math Chat
1996-11-22 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Hours, Days, and Years Old junior challenge If you live for 6,000 years, how many days will that be? Answer A good start is (6,000 years) x (365 days per year) = 2,190,000 days. An extra day in a leap year every four years adds another 1,500 days to make 2,191,500. But every hundred years we skip a leap year (1900 was not a leap year), so we subtract 60 days to get 2,191,440. But every 400 years we put the leap year back in (2000 will b... (394 words)
11) Math Chat: Daylight Saving Time And Water Levels
1996-12-06 [TEXT ONLY]
 
Frank Morgan Daylight Saving Time And Water Levels Old time challenge. John lives in an Atlantic coast state of the United States, and Mary lives in a Pacific coast state. When talking on the telephone from home recently, they realized it was the same time in both locations. How could this have been possible? Solution Florida and Oregon extend into the Central and Mountain time zones. Winners Timothy Clark, Mike Soskis, Chuck Gahr, and Eric Brahinsky explain that if John lives in Pensacola... (362 words)
12) Math Chat: Water Levels And Calendars
1996-12-20 [TEXT ONLY]
 
Frank Morgan Old water level challenge (Thanks to Jeff Bradford.) When you throw the anchor over the side of your boat, does the water level in the pond rise or fall? When ice melts in a glass of warm water, does the water level rise or fall? Answers Imagine someone on the dock first removing the anchor from the boat and then throwing it in the water. When it is lifted, the boat rises and the water level falls an amount depending on the anchor's weight. When it is thrown into the water, the le... (375 words)
13) Making a Better Calendar
1997-01-03 [TEXT ONLY]
 
Frank Morgan Old calendar challenge Our current calendar divides the 365-day year into 12 months, or 52 weeks plus one extra day. Find a better calendar system. For example, are 7-day weeks best? Is there any way to avoid needing a new calendar every year? Answers You can use the same calendar every year (except leap years, which have two extra days) if you make the extra day a special holiday, perhaps at the beginning or end of the year, not part of any week. C. Gahr suggests calling it Janua... (384 words)
14) Airplane Weight And World Population
1997-01-17 [TEXT ONLY]
 
Frank Morgan Last week, American mathematicians gathered in San Diego for our annual meeting. The Cole Prize went to Andrew Wiles, who proved Fermat's Last Theorem, which had eluded us for 350 years. The Morgan Prize for research by an undergraduate went to recent Harvard graduate Manjul Bhargava. Largest prime number A new largest-known prime was found in November by Joel Armengaud of Paris. Unlike the previous supercomputer discovery of September, this one came on a personal computer, with h... (343 words)
15) Math Chat: World Population and Four Dice
1997-01-31 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Old challenge Estimate what fraction of the people who have ever lived on Earth are alive on Earth today. Answer Aubrey Dunne assumes that each generation is twice as large as the one before. So the previous generation was just 1/2 as large as the current generation, the one before that just 1/4 as large as the current generation, the one before that just 1/8 as large as the current generation, and so on. Since 1/2 + 1/4 + 1/8 + 1/16 + =... (566 words)
16) Math Chat: Dice and Valentine's Day Symmetries
1997-02-14 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Old dice challenge (S. H. Logue) The six faces of four dice have these numbers on them: A 1 1 5 5 5 5 B 4 4 4 4 4 4 C 3 3 3 3 7 7 D 2 2 2 6 6 6 You and your opponent each choose one die to use throughout the game. Each round you both roll, and the high number wins. Which is the best die to have? Answer Timothy Dawn observes that "This game has the intriguing property that while D beats A, A beats B, and B beats C, C b... (671 words)
17) Math Chat: Symmetries and The Millennium
1997-02-28 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Old Symmetries Challenge What are the symmetries of a cube in space? (What about a hypercube in 4-dimensional space?) Answer Jan Smit counted up all 24 symmetries. You can rotate the cube 90 degrees (a quarter turn), 180 degrees, and 270 degrees about a vertical axis, and similarly about two horizontal axes, for a total of 9 symmetries. You can rotate the cube 180 degrees about a horizontal diagonal axis and five other similar axes for a t... (525 words)
18) Where Will the New Millennium Begin?
1997-03-14 [TEXT ONLY]
 
Frank Morgan Millennium challenge Assuming the third millennium arrives on January 1, 2001, where on Earth should the celebration begin? Answer Because of different time zones, the third millennium will arrive earlier in Europe than in North America, still earlier in Japan (that is why it is called "Land of the Rising Sun"), and so on until you hit the International Date Line. The South Pacific island of Tonga, situated in an eastward bulge of the date line, has claimed the millennium will b... (520 words)
19) Math Chat: The Family Photo and Counting on One Hand
1997-03-28 [TEXT ONLY]
 
Frank Morgan Old challenge (thanks to Norman Goodwin) A photographer wants to line up the 11 members of the Goodwin family from shortest to tallest (left to right), starting with anyone, adding someone else on either side, and continuing by always adding someone next to those already assembled. How many different ways are there of doing this? Answer Eric Brahinksy figures that there is 1 way starting with the shortest, plus 10 ways starting with the second shortest (according to which of the ... (433 words)
20) Math Chat: Computers, Free Will, And Marching Ants
1997-04-11 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Old free will challenge Is it logically possible for a computer to have free will? Answer Seth Rogers argues that "since a computer must follow its program it does not have choice. Many contend that the brain works the same way by firing neurons, but they do not distinguish the mind from the brain. Since a computer and a brain perform similar functions, the study of AI [artificial intelligence] can reveal the boundary between the brain and... (512 words)
21) Math Chat: Accurate Census Taking And Orderly Ants
1997-04-25 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Old marching ants challenge (Luke Somers) A column of ants marches in single file, until one ant falls away and they then march two by two, until another ant falls away and they then march three by three, and so on until finally they are marching 10 by 10. How many ants were there to start with? Answer Charles Medler and Andrew Sornborger used computers to find the first possible answer: 2,519 ants. It works because 2,518 is divisible by 2... (515 words)
22) Census Challenges
1997-05-09 [TEXT ONLY]
 
Frank Morgan Old census challenge Gordon Squire notes that planners for the 2000 US Census wonder what to do about missing certain groups of people such as the homeless in New York. Suggestions include statistical random sampling. What approach would you suggest? Answer Two interesting suggestions came from students calling the MathChat cable-TV show in Williamstown, Mass. Second-grader Nicholas de Veaux proposed basing the census on the phone book. Of course, there is usually just one name p... (420 words)
23) Math Chat: On Four 4s And Speed Limits
1997-05-23 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Old 'four 4s' challenge (Howard Sheldon) Can you get past 30 in forming numbers 0, 1, 2, 3, ... from four 4s and the standard mathematical operations +, - , x, /, decimal point, square root, powers, and factorial(!)? Answer Eric Brahinsky, Dick Feren, and Paul Goodrich got 31 = 4! + (4!+4)/4 and 32 = (4+4) x 4. Roger Bliss, Robert Lewis, Mike Soskis, and Richard Thorne also got 33 = 4!+(4- .4)/.4 and made it to 36. William Foster, William ... (401 words)
24) Math Chat: 65-m.p.h. Speed Limits, Large Primes, and Japanese Exams
1997-06-13 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Today's column (No. 26) marks the first full year of Math Chat. Many thanks to readers, in communication or silent, for their willingness to have some fun thinking about mathematics. The winning answers mentioned today are just part of the large reader response. Old speed limit challenge (Marc Abel) When a state increases a highway speed limit from 55 to 65 miles per hour, by what percent does the road capacity (in cars per hour) change? ... (900 words)
25) Math Chat: Powers of 5 and Traveling To the North Pole
1997-06-27 [TEXT ONLY]
 
Frank Morgan, Special to The Christian Science Monitor Old challenge (Howard Sheldon) What is the remainder when you divide 5999,000 by 7? Answer The remainder is 1, as deduced by several readers who recognized that the remainder of powers of 5 after dividing by 7 repeats in cycles of 6. When you ignore multiples of 7 and look only at the remainder, you are doing arithmetic "modulo 7," as if 7 = 0, 25 = 4 (both are 4 more than multiples of 7), and 699 = -1. Modulo 7, 52 = 25 = 4, 53 = 4 x 5 = ... (666 words)
26) Northwest to the Pole And Two-Headed Coins
1997-07-11 [TEXT ONLY]
 
Frank Morgan Old Northwest challenge (Eric Brahinsky) Suppose you start at the earth's equator and travel continuously northwest until you reach the North Pole. What does your path look like? How long is your path? Answer The path is about 9,000 miles long. A tiny right triangle with base 1 and height 1 has hypotenuse ?(12 + 12) = ?(2) ("Pythagorean Theorem"). Therefore as you head northwest, you go ?(2) times as far as your progress northward. Since the distance due north from the equator to ... (303 words)
27) Math Chat: Trick Coins and Ping-Pong Balls
1997-07-25 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan Old challenge (George Murray) You have a fair coin and a trick coin with two heads. You pick one at random, without noticing which one. The first time you toss it, it's heads. What is the probability it will come up heads the second time? (What if you don't actually see the first toss, but three of your friends, each of whom is right 2/3 of the time, all say it was heads?) Answer Gregory Sahagen got both answers right. The chance of choosing the fair coin and getting heads on the ... (557 words)
28) Math Chat: Ping-Pong Balls and Presidents' Names
1997-08-08 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan, Special to The Christian Science Monitor Old challenge (Jonathan Kravis and Dave McMath) A process involving infinitely many ping-pong balls is carried out in one hour as follows. In the first half hour, two balls are placed into a huge empty barrel and then one is removed and discarded. In the next 15 minutes, two more balls are put in and then one of the three inside is removed. At each step, in half of the remaining time, two more balls are added and one is removed. How many ba... (983 words)
29) Math Chat: It's All in the Name Old challenge
1997-08-22 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan, Special to The Christian Science Monitor What is the probability that two people will randomly choose the same letter of the alphabet? What is the probability that the names of two random American presidents begin with the same letter? Estimate the probability that at least two of the names of 10 random Americans will start with the same letter. Answer (Erik Randolph) The likelihood that a second person will choose the same letter as the first is 1 in 26 or about 3.8 percent. As ... (446 words)
30) Math Chat
1997-09-05 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan Zeros, Ping-Pong Balls, And Acceleration Old challenge 5! = 5x4x3x2x1 = 120, which ends in a zero. Determine the number of zeros at the end of 25! (What about 510!) Answer 25! ends in 6 zeros and 510! ends in 2,441,406 zeros. To understand why, start with the easier case of 10! = 10x9x8x7x6x5x4x3x2x1 = 3,628,800. It ends in two zeros, one because of the factor of 10 and one because of the separate factors 5x2. Since twos are much more common than fives, it is the number of factors... (440 words)
31) Stepping on the Gas And Decoding the Bible
1997-09-19 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan Old challenge You are out driving your gas-powered or perhaps new electric battery-powered car. At a freeway entrance, you need to accelerate from 0 to 60 m.p.h. within 20 seconds. What do you think is the best way to do this to use the least amount of energy? Answer Charles Sullivan reports that engines are more efficient at full throttle at high gears, so at least with a stick shift you should give it plenty of gas but shift up as quickly as possible. Marc Abel reports that batt... (393 words)
32) Personal Accounts and Folded Paper
1997-10-03 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan Old challenge Send in a true personal account with an estimate of its mathematical probability of occurring. The least likely will be the winner. Answers In the second-best answer, Steve Gluck reports that, "In a seemingly irresolvable dispute between my son and daughter, I proposed we flip a coin. My son said coin flipping was unfair because his sister always won. [After he chose tails,] 13 consecutive flips were heads. The probability of such an event was 1 in 213 or 8,192." Act... (884 words)
33) Math Chat: Rearranging Letters And Spinning Sprinklers
1997-10-17 [TEXT ONLY] [WITH GRAPHICS]
Frank.Morgan Old challenge A recent National Public Radio Weekend Edition puzzle asked to replace one letter in HOMEGAME by another letter to get the name of a famous philosopher. Bruce Cobi asks how many different letter sequences you can obtain this way. What if you are also allowed to rearrange the letters? (The NPR answer was none of these, but to replace the Greek letter OMEGA with U to get HUME.) Answer Laura Sabel and Michael Stern correctly figure that each of the 8 letters of HOMEGAME... (723 words)
34) Math Chat: Halloween Card Games And Spinning Sprinklers
1997-10-31 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan Old challenge An old-fashioned water sprinkler with S-shaped spiral arms is propelled counterclockwise as the water spurts out. Which way will it turn if you place it upside down under water and have it suck water in? Answer In the best and winning answer, Sirius Fuller explains that the sprinkler "will not turn at all. It may vibrate, but it won't spin. The reason I know this is that I read a book entitled 'Genius' by James Gleick. It's about the life of [famous physicist] Richar... (414 words)
35) Math Chat: Halloween Cards, Sprinklers, & Mirrors
1997-11-14 [TEXT ONLY] [WITH GRAPHICS]
FRANK MORGAN Old Halloween challenge Ralph Thomas proposes a new card game for you and an opponent for Halloween. Each hand, a goblin deals both players a certain number of cards. The highest card wins (e.g., ace beats king, two aces tie). If you win you get two points; if your opponent wins he gets just one point. (You have an advantage.) Do you prefer to use a regular deck, a double deck, or a pinochle deck (which has two of each card from 9 on up to ace)? How many cards would you like the g... (670 words)
36) Math Chat: Mirrors and Roadways
1997-12-05 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan Old challenge (Fred Wedemeier) Why do mirrors reverse left and right and not up and down? Answer This is one of the most confusing questions I know, and there were some very ingenious wrong answers: "The mirror does reflect from top to bottom, but your brain recognizes that the shape should be right-side-up, and flips it for you." "The image in a mirror is rotated about the vertical axis because a person, when turning to regard another person, rotates himself about the vertical ax... (440 words)
37) Math Chat: Roads, Tennis, and the Sun
1997-12-19 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan Old challenge What is the shortest road network connecting the four corners of a square? What about the six vertices of a regular hexagon? Answer The "double Y" of Figure 1 with length ?3 + 1 or about 2.73 (on a unit square) beats the popular answer of the X of length 2?2 or about 2.83, as well as the U of length 3. It is a general principle that at junctions in shortest networks the roads always meet in threes at angles of exactly 120 degrees. Perhaps our interstate highways shou... (486 words)
38) Math Chat: Today's Late Sunrise
1998-01-02 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan Old challenge Why do we have the latest sunrise now, around the first of the year, and the earliest sunset around Dec. 10, compared with the shortest day around Dec. 21? And how can I remember (from year to year) which comes first? Answer The period of daylight in the Northern Hemisphere is longest on the summer solstice, about June 21, when the North Pole tilts toward the sun, and shortest on the winter solstice, about Dec. 21, when the North Pole tilts away from the sun. So you ... (593 words)
39) Math Chat: Prime Numbers and Adding Hymns
1998-01-16 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan Old sequence challenge (Joe Shipman) What is the next number in this sequence: 1 2 3 4 5 6 8 9 10 11 12 14 15 16 17 18 20 21 22 24 ... ? Answer The next number is 26. The sequence is all differences of prime numbers. For example, 24 = 29 - 5 and 26 = 29 - 3. 25 is not on the list; if it were a difference of primes, one of them would have to be even and hence 2 (the only even prime), and therefore the other one would have to be 27, but 27 is not prime. It looks as if every even nu... (652 words)
40) Math Chat: Milk-Bottle Pressure And Adding Hymns
1998-02-03 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan The old challenge leads to a widely forgotten but handy method for checking addition called "casting out nines." Old challenge (Joe Herman) Why does adding a column of three 3-digit hymn numbers yield the same answer whether you add frontwards, backwards, or even sideways? Answer George Dillard's computer experiments show that it works about 80 percent of the time for three hymns from 1 to 429, and here is why. If there were no carries, you would always just get the sum of all the... (619 words)
41) Math Chat
1998-02-18 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan Old Milk Bottles and Hotel Expenses Falling elevators Suppose cartoon character Wile E. Coyote finds himself in a broken, brakeless elevator plummeting downward. Could he save himself by jumping up just before the elevator hit bottom? It might be hard for him to time his jump perfectly, but Lynne Lawson suggests that if he just keeps jumping up and down he'll probably be in the air when the elevator hits. In a recent "Ask Marilyn" column (Parade Magazine, Sunday, Jan. 18), Marilyn ... (505 words)
42) Math Chat: Alien Mathematics and First Day of Spring
1998-03-03 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan PRINCETON, NJ Number systems Why does our mathematics work so well in explaining the universe? Might alien cultures have different mathematics from ours? These intriguing questions appeared in an article, "Useful invention or absolute truth: What is math?" in The New York Times, Feb. 10. An accompanying graphic gave three highly debatable reasons why an alien culture might not even understand our number ª ("pi," or the ratio of the circumference of a circle to its diameter, about 3... (640 words)
43) Math Chat: That Shifting First Day of Spring
1998-03-19 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan PRINCETON, NJ Old spring challenge (Ilan Vardi) Why is the first day of spring in London sometimes on March 20 and sometimes on March 21? Spring arrives when the earth is tilted at a right angle to the sun, not toward the sun as in the summer or away from the sun as in the winter. Consequently night has the same length as day; hence the term spring equinox meaning "equal night." Answer Mike Bevan, George Dillard, Ron Douglass, Aubrey Dunne, William Hasek, Mary Lou Holmes, Erik Ra... (527 words)
44) Lining Up Your Ducks Is More Difficult Than It Appears
1998-04-16 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan PRINCETON, NJ The Mathematical Diary of 1825 Math Chat is certainly not the first mathematics column, even in the United States. Don Beaver reports that in 1825 R. Adrain launched a periodical called The Mathematical Diary in New York, which included challenge questions and answers. Here is the very first challenge question, submitted by Charles Vyse. "New" challenge (1825) "Ten pounds [sterling] a quarter are allowed to five auditors, A, B, C, D, E, of a Fire Office. They are re... (530 words)
45) Math Chat
1998-04-30 [TEXT ONLY]
 
Frank Morgan PRINCETON, NJ Which Three States Meet Three Times? Old challenge (Mathematical Diary of 1825) "Ten pounds [sterling] a quarter are allowed to five auditors, A, B, C, D, E, of a Fire Office. They are required to attend seven times in the quarter, and the absentees' shares are to be divided equally among such as attend. Now A and B never fail to attend, C and D are each absent twice, and E is absent only once: What is each auditor's share of the given sum?" Answer The best short an... (486 words)
46) Math Chat: Shifting River Beds And Earthquakes
1998-05-14 [TEXT ONLY]
 
Frank Morgan PRINCETON, N.J. Old challenge (Robert Kimble) Can you name three US states that all meet at three different points? Answer Mike Bevan proves that this cannot happen unless one of the states is disconnected. Mike Tupper writes, "I figured the Mississippi River had to be involved in such a ridiculous thing occurring. And sure enough, that's where I found it. The [north-south] border between Tennessee and Kentucky [follows] a straight line," with Missouri to the west on the other sid... (422 words)
47) Math Chat: A Weight-Loss Formula for Space
1998-05-28 [TEXT ONLY]
 
Frank Morgan Old space station challenge The new space station is only about 6 percent farther from the center of the earth than the earth's surface. Therefore gravity should be just about 12 percent less than on the earth's surface (by Newton's "inverse square" law of gravity, if you like). Then why will the astronauts appear weightless? Answer The orbiting astronauts appear weightless because the whole station is in free fall. If the station continued straight ahead, it would shoot off into ... (438 words)
48) Math Chat: Crossing a Rickety Bridge at Night By Flashlight
1998-06-11 [TEXT ONLY]
 
Frank Morgan PRINCETON, NJ Old bridge-crossing challenge (Tiku Majum-der and Conrad Weiser) "There are four people who need to cross a river at night. There is a bridge that can only hold up to two people at a time. There is one flashlight that must be used when crossing. (It is extremely dark, and someone must bring the flashlight back to the others; no throwing anything, no halfway crosses, etc.). "The four people take different amounts of time to cross the river. If two people cross togethe... (383 words)
49) If You Could Design Your Own Numbers
1998-06-25 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan BOSTON Old intelligence challenge (Steve Smale) "What are the limits of intelligence, both artificial and human?" In the best written answer, John Robertson argues that there are indeed limits to artificial and human intelligence, since the physical universe itself is finite, but that these limits are quite high as illustrated for example by mathematics. Then he asks the big question: "Will we solve the important problems that mankind faces?" As examples, he mentions the questions... (358 words)
50) Red Herrings And Young Children
1998-07-23 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan, Professor of mathematics, Williams College WILLIAMSTOWN, MA Old challenge (Aubrey Dunne, Michael Marcotty, and Dave Rossum) Census taker: How old are your three daughters? Mrs. S: The product of their ages is 36, and the sum of their ages is the address on our door here. Census taker: I'm good at math, but I cannot tell. Mrs. S: My eldest daughter has red hair. Census taker: Oh thanks, now I know. Can you figure out how old the three daughters are? Since the census taker cannot t... (307 words)
51) Putting Slippers On for That Morning Commute
1998-08-20 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan, Professor of mathematics, Williams College WILLIAMSTOWN, MA Old Transportation Challenge Starting from scratch, what would be the best transportation system of the future for getting everybody to work? Answer Mark Thompson describes trains that do not waste any time stopping at stations. The last little car on the train separates itself, stops, exchanges passengers, and then accelerates to be joined to the front of the next train through the station. Eric Klieber has an idea for ... (349 words)
52) Cracking an Old Stacking Problem
1998-09-03 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan, Professor of mathematics, William's College NORTH ALLENTOWN, PA Old weather records challenge (Michael Marcotty) This July was the warmest ever recorded. What is the probability of that happening just by chance, as opposed to a long-term weather trend? Answer If records have been kept for 100 years, the random probability that July 1998 would be the hottest is 1 percent. The probability that it would be the hottest or the coldest is 2 percent; the probability that some month of ... (220 words)
53) Cracking an Old Stacking Problem
1998-09-03 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan, Professor of mathematics, William's College NORTH ALLENTOWN, PA Old weather records challenge (Michael Marcotty) This July was the warmest ever recorded. What is the probability of that happening just by chance, as opposed to a long-term weather trend? Answer If records have been kept for 100 years, the random probability that July 1998 would be the hottest is 1 percent. The probability that it would be the hottest or the coldest is 2 percent; the probability that some month of ... (220 words)
54) Taking Mt. Everest Down a Peg
1998-09-17 [TEXT ONLY] [WITH GRAPHICS]
Dr. Frank Morgan ALLENTOWN, PA Old Mt. Everest Challenge Although Mt. Everest, at 29,028 feet above sea level, is the highest mountain in the world, it is not the farthest from the center of the earth. The earth's bulge at the equator pushes Chimborazo in Ecuador, at 20,561 feet above sea level, farther. (How much?) Now suppose you run a water pipe from Everest to Chimborazo. Which way would the water flow? Answer Although it seems hard at first to determine water-level equilibrium on a bulgin... (350 words)
55) The King Was in His Counting House...
1998-10-01 [TEXT ONLY] [WITH GRAPHICS]
Frank Morgan BOSTON Today with its final column, Math Chat thanks all its readers and hopes they continue to find enjoyment in mathematics and in every department of life. Old gold challenge (John Dippel) An ancient king requires that each of his 10 chieftains pay him tribute of 2,000 10-gram gold coins. He learns that one of them plans to substitute 9-gram coins, but he does not know which one. He has an accurate scale that reads out exact weight. How many weighings does he need to identify ... (553 words)

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