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.
- The numbers 5, 9, and 50 in the problem are all arbitrary, and the teacher is free to change them to customize the problem.
- 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.
- 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.
- 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.