Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
What, how, and the Web
Misconceptions are often very robust and not amenable to a logical argument. On occasions when all other resources have been exhausted without bearing fruit, I am tempted by a story from Raymond Smullyan's 5000 B.C. and Other Philosophical Fantasies.
A modern day philosopher once had a dream in which all the famous philosophers in history appeared and outlined their theories one by one. After hearing out Aristotle, the philosopher came up with an objection that completely confounded the famous Greek. The sleeping man then noticed that the same argument worked with Plato and other philosophers. Startled by the universality of his discovery, the philosopher woke up, dragged himself to his desk, jotted down the argument, and then fell fast asleep. The first thing in the morning, he anxiously ran to the desk to consult his note. It was, "That's what you say."
The argument works as advertised, but I only use it as a last resort.
I received a few responses to my June's column, Place Value, that brought up a specter of the New Math to question the wisdom of promoting the place value topic. Here's a sample:
There is an obvious confusion there between two questions, what to teach and how to teach it. Does the failure of the New Math to present a specific topic necessarily imply that teaching this topic has no merit? The New Math has embodied both the what's of mathematics and the how's of instructional methodology (not to mention its philosophical outlook.) Can't these be separated if only for the sake of discussion? Otherwise, how can one explain that more traditional topics that were covered by the New Math had survived its downfall?
Most certainly there were other factors that strengthened the public association of positional systems with the failures of the New Math. Another letter pointed to one of them:
In Japan, it is said
We are thirty years past the trauma of the New Math. Much has changed and more changes are forthcoming. With proliferation of the Web and computers and advances in software technology, education is bound to become more personalized. New means of presenting educational material are becoming commonplace. May it be that new instructional technology will succeed where the old one failed? Sure it may. Who will deny such a possibility out of hand?
Positional systems provide fruitful soil for development of mathematical contents. The puzzle with self-documenting sentences in the June's column is just one example. All that is needed to comprehend (and possibly enjoy) the puzzle is understanding that counting by grouping is most easily recorded in the positional notations, radix being the largest group size. Mastery of conversion techniques from base 10 into another base is not required.
I believe there is just one good criterion for selecting a topic to study, to put on the Web, or to include into curricula: does the topic have a mathematical content? If it does, we have a good candidate for an answer to the what question. According to the object oriented way of thinking, this is the primary question. The question of how is secondary, and a good generic answer to it is "in the most suitable way to present each of the selected responses to the what question." For want of a better argument, I cordially refer to Smullyan's quote any one who thinks that positional numeration is not a good answer to the what question. On the other hand, I would value any advice on how best to present this topic.
And here is another angle to look at the situation. Kindergartens and schools collect and display all kinds of natural objects and artifacts ostensibly to arouse curiosity in children. In good schools, students are encouraged to touch and experiment with those objects. In the best schools
A while back, a young fellow wrote to me asking how multiplication works in base 5. As he claimed to be familiar with both the decimal and the binary systems, he only wanted to have one example worked out, 364 times 23. I remarked that 364 was not a number in base 5, as only digits 0 through 4 are allowed; other than that there is no much difference. The multiplication algorithm is exactly the same as in other bases. As transpired from the return mail, the remark did the job, which also left me wondering. In all those years that the boy learned the decimal system and in all those weeks that he learned the binary, was there no proper time to mention what positional numeration is about?
As the Hebrews are used to say, - veday lehaham - enough for the wise.
The following (unrelated to the above) example is taken from Martin Gardner's Mathematical Games column in Scientific American, v 201, No 6, Dec 1959. (I am grateful to Norman Santora who sent me his unpublished manuscript Fun Maths that mentioned Gardner's article.)
Draw a number of vertical lines and a number of horizontal lines (shuttles) connecting pairs of the vertical ones. For every vertical line, start at the top and trace the line downwards. Wherever an end of a shuttle is encountered, trace this shuttle horizontally till its other end. From there, turn downwards again and continue in this manner until you reach the bottom of one of the vertical lines. The interesting thing about this procedure is that, starting at the top of two different lines, one always ends at different "bottoms." This feature is of practical value in a couple of situations.
An obvious one is that of distributing N tasks (or snacks) between N fellows in an equitable manner. Sometimes, there is just one task assigned to a group, but the task is so small that, to avoid stepping on each other's toes, the group may decide that only one person will carry it out. Which one? More generally, K jobs may be distributed between N persons, where
As a class activity, it goes without saying that all members of the group take part in preparation of the diagram: drawing and labeling lines and putting in shuttles.
(In the applet bellow, to draw a shuttle drag the cursor from one line to another. To verify the assignments, select the "Verify" button, and click in turn on each of the vertical lines.)
A very intuitive explanation of the "no line fusion" phenomenon comes from a physical analogue of the puzzle. Think of every vertical line as a rope with a shuttle indicating the cross-point of two ropes. Since the ropes may only cross but not merge, one can find their number either by counting the upper or the lower ends. The result is the same in both cases.
The puzzle provides a simplified model of a set of braiding ropes. The model is simplified since two ropes may cross in two distinct ways depending on which one goes beneath the other. Shuttles do not differentiate between the two possibilities.
Another way to look at the puzzle is by focusing on the effect of each individual shuttle. As in the applet, label upper ends of the vertical lines. Drawing a shuttle between two lines causes a label exchange at the lower ends. Abstractedly, we are given two (ordered) sets of labels: the top labels and the bottom labels. With no shuttles drawn, there is a 1-1 correspondence between the two sets. Each shuttle affects the label order but not their number, thus establishing another 1-1 correspondence between the sets. We may follow shuttles one by one from the top down. The first shuttle reorders the top labels, the second reorders thus reordered labels, the third changes the label order obtained after the second shuttle, and so on. Since no single shuttle changes the fact of there being a 1-1 correspondence of two sets of labels, one gets a 1-1 correspondence with any number of shuttles in place.
A bottom-up approach leads to an inductive argument: assume n-1 shuttles maintain a 1-1 correspondence between two sets of labels. Since this is also true of the n-th shuttle as well, this is true for any n shuttles.
The mathematical term for any reodering operation is a permutation. A simple permutation that only exchanges two elements is known as a transposition. Executing several transpositions (or, for that matter, permutations) sequentially leads to a new permutation. All permutations of the same set of indices form a group of permutations. The group is mutiplicative and its group operation - sequential execution of two permutations - is called product. Every permutation is a product of transpositions.
So here it is, another example of a good answer to the what question. Easy to understand, entertaining to experiment with, and with a good deal of mathematical contents. It may even serve as an introduction into the theory of permutation groups. I am very satisfied to have put such a nice example up on the Web. To wind up the discussion there is a small question intended for those boys and girls who may stumble onto this page by mere chance.
In the applet, top indices are connected by blue lines with their counterparts at the bottom. Let's call such lines itineraries. Prove that for every shuttle there exist exactly two itineraries that pass through it. One that traverses it from left to right, the other from right to left. Think of at least a couple of proofs.
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at email@example.com
Copyright © 1997-1999 Alexander Bogomolny