E20. The 5-5-9-9 Problem

Introduction: A problem that has been well publicized on the Internet is the "Four Fours" problem. It asks for a representation of all integers from 1 to N (where N is as large as possible) using the digit 4 exactly four times, in addition to basic arithmetic symbols as needed. See, for example, http://www.wheels.org/math/44s.html. I have found this problem to be excellent for group work for students at the middle school or high school level, because it allows students of many different ability levels to participate. However, when I gave this problem to a group of teachers in an online course, one teacher responded by listing the answers available from an online source, verbatim, even though it was clear she did not even understand some of the symbols used in "her" solution. I thought this was a clear case of cheating, but the student felt that using the Internet was a legitimate way to solve a mathematics problem. Maybe she had a point. In any case, it became clear to me that it was time to modify the problem to make it one whose solutions would not be available on line. See the following.


Problem: Show how to represent each of the integers from 1 to 50 using the digit 5 exactly twice and the digit 9 exactly twice, in addition to basic mathematics symbols as needed. Allowed symbols are plus, minus, times, divides (including fraction bar), parentheses, square root, exponentiation, decimal point, factorial, floor function, and ceiling function.

Comments:

1. The numbers 5, 9, and 50 in the problem are all arbitrary, and the teacher is free to change them to customize the problem.
2. The definition of "basic mathematics symbols" is crucial to the problem. For example, if you allow the successor function (++ suffix in the C programming language), the problem is trivial, and uninteresting. On the other hand, disallowing the decimal point, floor function, and ceiling function would make the problem much more difficult.
3. There are some ambiguities that will come up that must be resolved, such as whether (9-5)(9-5) may be used to represent 44. A question like this would make for good class discussion.
4. The following lemma makes the problem easier to solve: If a number can be represented using a subset of the 4 allowed digits, it can be represented using all 4 allowed digits. Proofs involve showing how to use any number of the allowed digits to produce either a factor with value 1, or a term with value 0. I think most students at the high school level could arrive at this lemma without being told about it. In any case, this lemma might make the idea of using a simplifying lemma, so important in mathematics, something students would remember.