Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

Spreading Memes

October 1997

In his Commencement address delivered to the mathematics graduating class of the University of California at Berkeley, Keith Devlin has infected the world with a meme virus "Mathematics makes the invisible visible" and furthered its spread by placing a little circumscribed copy into the August issue of the Focus newsletter. A short time later, an on-line version of the virus carrier has mysteriously surfaced on MAA Online.

Being in general agreement with the purpose encoded into the Devlinís meme, I aim at offering assistance through a distribution of an immuno-depressant memetic virus "Cut The Knot!" whose lifeís goal is to affect human behavior in certain ways as to weaken their resistance to Devlinís and similar viruses.

A slew of connotations may make "Cut The Knot!" highly contagious.

Its historical roots may appeal to the proponents of the humanist mathematics - according to legend, Alexander the Great cut the Gordian Knot which led to his prompt conquest of a good portion of Asia. Cutting anything even not as complicated as the legendary knot, is a hands-on activity and, as everyone knows, the only way to master math is by doing math. Mentioning knots is suggestive of the fact little known outside the mathematical circles that math is not just about numbers and counting. Back to Alexander the Great, who could, at the time, foresee his solution to so difficult a problem? Thus "Cut The Knot!" reminds us of the need and utility of being creative. Most of all, "Cut The Knot!", in its personal appeal, underscores an individualís responsibility to either (as the case might be) share and contribute or persevere and overcome (parentheses are not needed as the standard precedence rules apply.)

Memes, writes Devlin, are the thoughts and ideas that people produce and make public -- stories, tunes, poems, myths, beliefs, religions, scientific theories, and the like. Richard Dawkins, who coined the term, comments that "memes propagate themselves in the meme pool by leaping from brain to brain via a process which, in the broad sense, can be called imitation. If a scientist hears, or reads about, a good idea, he passes it on to his colleagues and students. He mentions it in his articles and his lectures. If the idea catches on, it can be said to propagate itself, spreading from brain to brain." Now, itís clear that, although memes may "be regarded as living structures, not just metaphorically but technically (Dawkins)", the human element plays a very important role in meme propagation.

R.Smullyan tells an anecdote about Zigmund Freud. Someone asked him, "Would you hold a man responsible for what he dreams?" Freud replied, "Whom else would you hold responsible?"

So, while Devlin wanted "to capture in a single, easily remembered slogan, the very essence of mathematics," "Cut The Knot!" reflects on a more modest goal. People will always disagree on the essence of mathematics (Sylvester thought that mathematics is essentially about seeing "differences in similarity, similarity in difference.") but they may agree that innumeracy is a diminishing condition, that mathematics appreciation enriches oneís life, and they eventually may agree to do something about it. Paraphrasing W.V.O.Quine, "Who are they?" - "Everyone".

Mathematician: You are the prime meme carrier. You know that mathematics is enjoyable. Share it. Spread the right memes. Resist the "Who cares?" infection.
Teacher: You are the prime transmitter. Foster math appreciation. Help students make their own discoveries. Do not shove it down their throats. Utility of mathematics is not the best justification to sweat over it. Thomas Tymoczko explains this the following way:

Mathematics must not necessarily be useful to be applicable. A leaf is symmetric without any derived utility for the human beings. In teaching mathematics, they (math teachers) insist on stressing its utility, even when it has none. As a result, they often hide from their students the excitement and intrinsic interest of mathematics: they hide it behind a facade of utility. It's rather like trying to awaken people to the joy of mountain climbing by trying to convince them that someday they might need to climb a mountain.

Student: You are the target. All kinds of memes lurk out there waiting for the right moment to enter your brain. There are stereotypical memes of ignorance that cause you to develop a hard shell through which you will only see and perceive a small portion of the world. And the part you'll miss is the more exciting. The best protection from the ignorance memes is the curiosity and proactive attitude vaccinations.
Parent: You are the catalyst. You create an environment conducive to some viruses and protective of others. If you are infected with the stereotypical "I never liked math" meme, you do not even have to sneeze to pass it on to your kids. You live in a viral cloud. But consider that you do not have to be a math wiz to foster your kids' interests. You may learn with them. With your critical thinking skills, you may help kids ask better questions. And this is what learning is about - asking the right questions. There is so much available now (books and Web sites) that all it takes is a little curiosity. For your kids it's an innate condition. Just go along.

This column is intended primarily for the last three categories of meme carriers. Each will contain a Java applet in the form of a puzzle or a problem simulation. I'll try to avoid being obvious and will not state the problem directly. The applets are intended to be such that the right answer to as yet unstated problem should be easy to surmise. State your problem, state your answer, try to justify and, perhaps, prove it. In subsequent columns, I'll be giving my problems and my answers.

For starters, here is the situation: you are given a chocolate bar. As chocolate bars go, this too consists of even rows and columns of small chocolate squares. Your task is to break the bar into small squares (which you achieve by clicking on the line where you want the bar broken.) Count the number of splits it takes to completely break it. What's the answer here? What's the problem?

The applet lets you choose the size of the bar. Start with small ones. Do you get a pattern?

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

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