Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
A. K. Dewdney once wrote [Dewdney, p. 123]: "To bring off a feat of magic three things are sufficient, if not always necessary: audience, props, and presentation. I have seen magic done without props but never without an audience; any audience will do, but I must leave hornswoggling to the aspiring magician." Trivially, the magician is the fourth and necessary, although not always sufficient, component of a magic presentation. In addition, an eye catching assistant -- neither necessary nor sufficient -- often relieves the magician from the drudgery of bamboozlement. It appears (and I'll state that for the record) that a minimum, and therefore necessary, magic setup comprises audience, magician and an act of magic. Presentation is desirable but not necessary, while an act of magic is. The latter is performed by the magician and experienced by the audience. How can we know that one took place?
In other words, what qualifies a performance as an act of magic? The best indication is wonderment. Watch the audience. If they wonder "How did he do that?" the magic is here, otherwise it's a flop. Wonderment is necessary. Not any kind of wonderement, but that of the indicated sort. For example you may be wondering where I am heading without waiting for a magic act. These four must be sufficient: audience, magician, act, and wonderment.
Simple. Let's run a check up as to what we have right now. An audience is present, right? You are reading this, after all. And as Dewdney put it, any audience will do (no offence meant.) An audience is then out there watching the magician.
I am the magician of course and I plan to peform a magic act shortly. But before I do, I'd like to express my sincere gratitude to my loyal assisstant, Alex Bogomolny. True, calling him eyecatching would stretch things. But, between us of course, being at the beginning of my career, I could not afford anybody more attractive.
Now to the act. It's all about numbers. After all, numbers are my strong point. I felt affinity with numbers right out of the box, you know.
N2 numbers are arranged in a square pattern. Select one of the numbers and try answering my queries. There are just two of them. For that's all I need to reveal your number.
Do you have any questions? Simple, hmm? Well, let's try something else.
The numbers from 1 through 27 are now arranged in three rows of nine numbers each. Select one of those numbers and reply truthfully to just three of my queries. Keep your eyes open.
Is this simple too? [Rouse Ball, p. 328-329] mentions that in 1813-1814 M. Gergonne published a generalization that dealt with NN numbers arranged in N rows of NN-1 numbers each. It is always possible to combine rows in such a manner that after N replies the selected number will appear in any desired spot, not necessarily in the middle of the mid row.
Ok, there's another one. Let N be an integer. This is the lone number at the bottom of the applet below. It can be modified by clicking on it. Currently it is 4. For practical reasons, it can change between 3 and 6, inclusive.
The integers from 1 through N·(N+1) are split into pairs. Select one pair and click the Proceed button. The same numbers will now be arranged in a N×(N+1) array. By just pointing to the two rows in which your selected numbers are located (if the numbers are in the same row, select it twice) you give me enough information to guess your numbers. Try it!
That's it for today. I'd like to explain how the thing works, but that's taboo among us magicians.
But if you are curious, [Rouse Ball, p. 326-328] refers the trick to the classical Problèmes plaisans et délectables by a pioneer of recreational mathematics Claude Gaspar Bachet, sieur de Méziriac (the 1st edition published in 1612, the 2nd in 1624.)
Well, so long. Feels like I need more practice.
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. In September 2002 the site has welcomed its 5,000,000th visitor. Alex 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 © 1996-2002 Alexander Bogomolny