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- Stefan Banach (March 30, 1892 – August 8, 1945)
- Diderot Mathematical Forums
- Guessing the Numbers
- What is mathematics for Ehrhard Behrends
- What is mathematics for Krzysztof Ciesielski
- The Three Ducks Trick
- What is mathematics for Franka Brueckler
What remains unchanging? On what can one rely? For centuries, mathematicians have searched for invariants, that is, to put it simply, quantities whose values do not change with respect to some operation under consideration.
By Ehrhard Behrends
As an example, let us consider a deck of cards. When you shuffle the deck, there are certain invariants in play. Certainly, the number of cards is unchanged, as are the numbers of jacks, queens, and so on. The situation is different when one does not allow shuffling, but only repeated cutting of the pack and putting the bottom “half” on top of the top “half.” Then the relative order of the cards remains unchanged. If the ace of spaces was to be found three cards down from the queen of hearts, it will remain so.
Of course, we have to interpret the words “down from”: If the queen of hearts happens to be the bottom card in the deck, then the ace of spades will be three cards down from the top. That is, “down from” is to be understood in the sense that when you reach the bottom of the deck, you continue from the top.
One can make use of this invariant in a little magic trick. Remove the kings and queens from the pack and lay them out in a row, as shown in Figure . The key here is to make the distance between cards of the same suit (king of spades and queen of spades, king of clubs and queen of clubs, and so on) exactly four. If you flash this arranged hand of cards in front of your audience, it appears to be more or less randomly arranged. No one will suspect your skulduggery, and if you cut the pack a few times (Figure ), everyone will believe that the cards have been thoroughly mixed.
But you are operating with the knowledge that the distance between cards is an invariant: four cards below the first card is its partner. It is therefore easy to produce a pair from under a cloth or under the table, of course making it seem that you are struggling mightily. You can repeat this process and produce a second pair (though this time the distance between them is three), and then a third pair (separated by two), and finally the last pair.
This trick relies on the existence of some kind of order amidst seeming chaos. In mathematics, the search for the unchanging has become a sort of leitmotiv in research. Once a set of permissible transformations has been described, a systematic search begins for quantities that remain unchanged under those transformations. This idea has been of particular importance as a unifying principle in many branches of geometry. It was proposed in 1872 by the mathematician Felix Klein and has had great influence over research ever since.
The Back Story: The Distance Modulo the Number of Cards Is Invariant
Using the notion of modular arithmetic introduced in another chapter of this book, we can formulate the principle that underlies the trick somewhat more mathematically:
If n cards are stacked, and two of these cards are in positions a and b (counted from the top), then (b−a) modulo n is an invariant: after cutting the pack any number of times, the difference between the positional values has not changed modulo n.
To see this, one has to use modular calculations for negative numbers as well as positive, but that is really no problem. After all, as everyone knows, the day of the week seven days ago is the same as the day today, and the day 13 days ago was Tuesday if today is Monday. Mathematically, −13 modulo 7 is equal to 1.
One must be aware of this subtlety in order to interpret the invariant relationship presented above correctly. An example: In the example that follows we are going to use the fact that −7 modulo 10 is equal to 3. In a pack of ten cards, the ace of hearts and jack of clubs are in positions 2 and 5. The difference is 3. The pack is cut at position 2. Now the ace of hearts is at position 10, the bottom of the pack, and the jack has moved up to position 3. The difference (position of the second card minus that of the first) is thus 3−10=−7, and modulo 10, this is the same number, 3, as before.
We Draw on an Extensible Surface
Only a few mathematical invariants are suitable for magic tricks. Their great significance is that invariants separate the essential in a theory from the inessential. To show this via a somewhat unconventional example, we require a drawing surface made out of some stretchable material.
On our surface we draw a figure: a triangle, a circle, a collection of rectangles, whatever. Now the surface is distorted; we pull it and compress it in any way that strikes our fancy. Our drawing will change significantly. A small circle can become a large circle; a right angle can become obtuse or acute.
If the original figure had the property that one could connect any two points with a curve contained entirely within the figure, which is the case for a circle or triangle, but not a collection of rectangles, then one can still do so after the figure has been altered: connectivity is an invariant under distortion.