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Number Names

Think of a positive integer (e.g. 23).

Count the number of letters in its name (e.g. "twenty-three" has 11 letters).

Repeat this process using the new number (e.g. 11 produces the number 6).

What happens as the process continues?

Can you prove why this happens?

Also, what happens if you replicate the problem using numbers in other languages such as German and Italian?


Hint: What is true if the initial number is 4 or less?

Why did Dr. Matrix (i.e. Martin Gardner's alter ego) call 4 an "honest" number?

What is true about the number of letters of the name for n if n an integer greater than 4...and what happens as the process is repeated?


Solution Commentary: Using the hint sequence, you should first discover by exhaustion that applying the process to the numbers 1, 2, 3, and 4 all converge to the the number 4 (i.e. a "fatal attractor."

Next, for numbers n > 4, the number of letters for n is less than n itself.

Finally, applying to process to numbers n > 4 will lead to a decreasing sequence of numbers (again, why?), which eventually must converge to the honest number 4 (again, why?).

Note: Problem solvers who want to explore this problem further with other variations (e.g. Sisyphus String, Kaprekar's Constant, Trigg's Constant) or other tangential connections to chaos theory, consider Monte Zerger's "Fatal Attraction" (Mathematics and Computer Science, Spring 1993, pp. 116-124).