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The Four Fours

This year I'm a volunteer teacher at JLS in Palo Alto to help guide a small number of curious children into the wonders of advanced mathematics over 12 classes in 6 weeks. The program is called "The Number Devil". It uses chapters from the similarly named book by Hans Magnus Enzensberger as anchor points for discussion during each class.

Just before the holidays started, the students were to do a warm-up exercise in preparation for diving into the deeper wonders of Number Theory in upcoming classes. Here is the exercise, called the Four Fours. Using exactly 4 digits, all of which are fours, and any number of the arithmetic operators Plus, Minus, Times, Divide, Exponentiation, Factorial, Square Root, and Parenthesis, derive each of the numbers from 1 to 50. E.g. One way you can derive 1 is obviously 4/4 *4/4. No doubt there are a number of different ways to derive each number and the goal is to get students to think about this, and to come up with expressions that are different from those of their colleagues.

We, however, decided to turn this little exericse into a game, which as a matter of fact, also turned out to be a good game to occupy young minds during the holidays! Here is how you would play it:

The game has 15 rounds. Each round lasts 1 minute. We tried to cover 15 numbers in the game.

Each round consists of:

  1. Host (me): Calling out a different random integer between 1 and 50.
  2. Players: Each of them has 30 seconds to write an expression with exactly 4 4s and the arithmetic operators to get this number.
  3. We compare answers and points are scored for this round as follows:
    • Zero points - No expression or invalid expression (meaning it doesn't evaluate to the number called)
    • M-N points - Otherwise. M is the number of players and N is the number of other players who have the same expression as you. If your expression is unique, you get the full M points for the round.

A great deal of fun though this game was, I was pleasantly surprised by the number of related meta-issues that we could discuss that laid the stage for all kinds of advanced concepts one would learn in grad-school and beyond. Here is just a sampling of them:

  • Just how many legal expressions can you ever make (A legal expression is some combination of the 4 fours and the operators that will compute correctly, i.e. is syntactically correct)? What if you couldn't use Square Roots, Factorials and Parentheses? Can you still make an infinite number of expressions? If not exactly how many?

  • What is the largest number you can calculate? What if you didn't have Factorial?

  • How can you tell if two expressions are the same? E.g. If John came up with 4*4 * 4/4 = 16 and Mary came up with 4/4 * 4*4 = 16, how can you tell they were the same? Notice that it's not always this trivial. This discussion leads to concepts of Normal Forms, how to reduce to Normal forms, whether even such reduction is always possible, whether unique normal forms exist, and how one can algorithmically (and efficiently) compare Normal forms.

  • This point, especially, I love because it goes to the root of Science, Math, Non-Science and their distinctions! But I've completely forgotten the discussion chain that led to this point in class! Suppose John and Mary were to form one expression each as above, but instead used a variable for one of the fours. By substituting a value for this variable, each expression would evaluate to a certain number. However, it happens to be the case that we are unable to reduce either expression to Normal form and thus compare them directly. How could we now tell whether the two expressions are in fact the same? This brings up all kinds of interesting discussion points - Conjectures and Theorems Versus Hypotheses and Theories, Deductive and Inductive reasoning, Mathematical Induction, Empirical testing and indeed, gives students a real taste of that tantalizing field of exploration - Epistemology!


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