The article points out that since Al Gore and Joe Lieberman ran together in the 2000 presidential election, Gore has gone to the left, and Lieberman to the right; would that have happened had they been elected in 2000?
The article ends:
Ron Klain, who was Mr. Gore’s vice presidential chief of staff, said the Gore-Lieberman schism reminded him of Algebra I.
"It’s like one of those old math problems,” he said. “Two trains leave Chicago at the same time traveling in opposite directions. How far apart are they in three hours? Very far apart."
I don't remember having to do problems like that. This may be why I still like mathematics. (And why is it always "trains" in these problems? I suspect they may actually have been more common in a time when trains were common; if the probems were written now they would be about cars or airplanes.)
The article's not bad re: politics; this is just yet another mathification, i. e. gratuitous insertion of math where it doesn't really make sense.
Trains have well-defines tracks. Planes don't.
I vaguely remember word-problems involving trains. Stereotypically they involve two trains leaving two different cities at two different times heading towards each other at two different speeds. The question is usually when, and where, they will pass each other.
The variant I always liked best was: Two trains are on a straight east-west track 100 km apart. The eastern one is heading west at 10km/h, and the western one is heading east at 10km/h. A bird gets its kicks flying at 50km/h back and forth between the trains. It can sustain very high accelleratons since when it reaches a train it immediately turns around and flies in the opposite direction without losing any time. As the trains get closer, the bird's flights between trains get shorter and shorter, and overall the bird will make a infinite number of flights. How far did the bird fly in those infinite number of flights before the NTSB investigators find it mangled between the crashed engines?
I like that one too, although more for the story involving John von Neumann first hearing it than for the problem itself.
Try this for size (to be solved without using algebra):
Two old women started at sunrise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on that day?
Vladimir Arnold said in his interview
(the first paragraph on page 433) that solving this problem was his first experience in mathematical research.
Here is a similar puzzle from Calculus Lite by Frank Morgan (section 29.6, calculus and algebra allowed):
Snow has been falling steadily for a little while when a snowplow starts out at noon. It plows a constant wolume of snow per unit time. It covers 2 miles in the first 2 hours, but only 1 mile in the second 2 hours. When did the snow start?
I remember the snow plow problem from very long ago. Probably the trains are traditional, from a time when people had far more experience riding trains than airplanes, if any.
Whilst making love, a necklace broke / A row of pearls mislaid. One sixth fell on the floor / One fifth fell on the bed / The woman saved a third / One tenth were caught by her lover. If six pearls remained on the string / How many pearls were there altogether?
Georges Ifrah, The Universal History of Numbers, from Sanskrit.
Carl, I bet some parents will complain if you offer such a problem in primary school (where it properly belongs, mathematically).
The answer to your first puzzle should be 3am. Let's say the two old women met at point X. It takes the second woman (walking from B to A) 9 hrs to talk from X to A. This means the first woman must also have taken exactly 9 hrs to walk from A to X! So, sunrise is at 1200 hrs - 900hrs = 300 hrs, or 3AM.
This is exactly the type of problem that I immensely enjoyed reading/solving from those popular Russian math/science books, especially the ones by Y.I. Perelman. Here is a similar kind of problem (on my blog) that should be solved using only basic arithmetic (and no algebra!)
Oops! "talk" should be "walk".
Vishal, the women don't walk at the same speed. Check the other leg of the journey and you'll see the problem.
The time they walked before they met is the geometric mean of the times they walked after they met: six hours.
Dr Armstrong: Thanks for that clarification! I knew something was wrong about 3am. I mean how many places (even in Russia) really have sunrise at 3am!
But... awww... you killed the problem for me by giving away the answer.
Here is my solution, anyway. Let's say they meet at point X. Also, suppose the first woman took 't' hrs to walk from A to X. Then, it took the second woman also 't' hrs to walk from B to X. Now, we note that speed is inversely proportional to time (for a fixed distance). Also, the ratio of the speeds of the two women is always constant, and this ratio for the first leg of the journey is just 9/t. Similarly the ratio for the second leg of the journey is t/4. But, we have 9/t = t/4, which gives t = 6. And so the sunrise was at (12-6)am = 6am.
Hope the above solution doesn't count as "algebraic"!
Glad you had fun with the problem. By the way, there are plenty of places in Russia where the sun is up at 3 a.m. in the summer, St. Petersburg is one of them.
The problem suggested by topologicalmusings is really nice, the answer and the reasoning behind it are rather surprising. I don't want to give it away. Offering more puzzles like this in elementary school would be very good for mathematical education, but how many teachers can handle it? Y.I. Perelman's books are still very popular in Russia, they are real treasure troves. Unfortunately not many of them are translated into English. Kordemsky collection is pretty good too.
Misha: I vividly remember spending hours and hours on whatever Russian popular science books I could lay my hands on during my middle-school years. (They were very inexpensive! American popular science books are way too expensive.) And through many of those books, I learned about the names of many of the famous Russian scientists/mathematicians who made significant contributions to science in the erstwhile Soviet Union: Tsiolkovsky , Sobolev, Igor Tamm, Fock, Fomenko, Lomonosov, Lobachevsky, Landau, Lysenko, Sergei Korolev, Mandelshtam, Mendeleyev, Markov, Ostrogradsky, Sakharov, Ioffe, Alferov, Popov. The names of all of these people, I remember quite well! Ah, the very thought of those awesome science books sends a thrill down my spine even now! Haha.
And, thanks for letting me know about Kordemsky! And, yeah there is a nice arithmetic solution to the problem I posed on my blog.
(Isabel: My apologies for hijacking the thread/post!)
Lysenko was a crook, and Fomenko is.
My bad! I should have been careful about mentioning Lysenko and Fomenko.
By the way, almost all the Perelman's books are downloadable
No English translations? :( I can't read Russian!
Very few, and not easy to find, mostly out of print.
The train problem, in print, again, today, in the NYT:
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