Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

Tribute to Invariance

December 1997

As a competitive activity, the Squares and Circles game is not very challenging in itself. That is to say, the outcome of the game does not depend on how the game proceeds. The challenge is to come up with this realization and establish its veracity. After the secret is revealed, its verification still provides an entertaining and purposeful exercise in counting much removed from the customary rote.

To resolve the enigma, suffice it to note that the parity of the number of circles remains invariant under legitimate moves. Therefore, the outcome of the game is predetermined by the original amount of circles. If the number is even then the last shape will be a square regardless of the sequence of moves. If that number is odd, the last object will be a circle.

The game extends to a 3-shape puzzle. Let there be three kinds of shapes: squares, circles, and triangles. A move consists in selecting two objects of different shapes which are then replaced with an object of the remaining shape. The question is, Is it always possible to achieve a state with a single object? If so, what shape might it be?

The puzzle has a mass-conserving variation. In the latter, the two selected objects of different shapes metamorphose each into the remaining shape. The goal here is to get a situation where only objects of the same kind remain. Is it always possible? Hint: think of quantities that remain invariant under legitimate moves. A thicker hint: in the original 3-shape game, think of invariance modulo 2. In the mass-conserving variation, modulo 3 arithmetic is more useful.

Writes Sherman K.Stein [Ste]:

Of all subjects, mathematics can be the best taught, or the worst. In mathematics, all the cards can be put on the table: Nothing has to be taken on faith or on the say-so of some authority figure. Everything should make sense. Guided by a well-prepared teacher, students can conduct their own experiments, make their own discoveries, and uncover many of the basic principles without being told. These experiments require no fancy equipment. Pencil and paper, a calculator, a ruler, a piece of string, dice, and pennies will do.

(I would, of course, include simple Java applets into the list of widely available and useful instruction tools.)

As in the above puzzles, invariance of some quantity is concealed in a variety of problems of quite an elementary nature. Many disguise counting exercises in a game or puzzle format. Most allow for very meaningful variations and generalizations and all contribute to the development of abstract imaging. Breaking Chocolate Bar puzzle is another example.

Various invariance principles are of such a fundamental nature in mathematics and other sciences that inclusion of activities similar to the described above into instruction process not only provides students with meaningful and entertaining exercises but also gives a teacher an opportunity to place the study in a very general, abstract framework relating various sciences to each other as well as the science in general to the real world phenomena. In this column I wish to argue in favor of inclusion of games and puzzles based on the invariance principles into an early math curricula.

  • Piaget and Post-Piaget Experiments
      I describe the famous Piaget's experiments that put children's ability for number conservation into a question which are counterbalanced by more recent results that point to the innate ability of children for rudimentary math comprehension
  • Importance and Abstraction of Invariance
      A very superficial account of invariance principles that extends glossary term definitions.
  • Brain's Evolution
      Quotations and an argument that combine to foster the idea that presently children are not given instruction in abstract when they need it most.
  • Invariance: Sample Activities
      Description of several activities based on the invariance principles.
  • Glossary
      A short collection of terms related to the notion of invariance. Might be helpful in drawing a general picture.

Addition and multiplication tables are made to be memorized, a dull and uninspiring experience for most of us. For the next month's column there is an applet that presents addition and multiplication tables in various bases. I am not suggesting to teach preschoolers how to count in various bases. (Though why not?) However, even older children enjoy non-routine tasks. Turning something as dull as a table of addition or multiplication into a research tool will definitely enliven class instruction. What properties of the basic arithmetic operations can be discerned from the observation of these tables?

Are you game?


  1. [Cha] J-P Changeaux, A.Connes, Conversations on Mind, Matter, and Mathematics, Princeton University Press, 1995.
  2. [Deh] S.Dehaene, The Number Sense, Oxford University Press, 1997
  3. [Dev] K.Devlin, MATHEMATICS: The Science of Patterns, Scientific American Library, 1995
  4. [Fom] D.Fomin,S.Genkin,I.Itenberg, Mathematical Circles (Russian Experience), AMS, 1996
  5. [Gar] M.Gardner, Puzzles From Other Worlds, Vintage Books, 1984
  6. [McK] W.D.McKillip,E.J.Davis, Mathematics Instruction: Early Childhood, Silver Burdett Co., 1980
  7. [Po1] G.Polya, Induction and Analogy in Mathematics, Princeton University Press, 1954
  8. [Po2] G.Polya, How To Solve It, Princeton University Press, 2nd ed, 1957
  9. [Ste] S.K.Stein, Strength in Numbers, John Wiley & Sons, 1996
  10. [Stw] I.Stewart, Nature's Numbers, BasicBooks, 1995

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