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### 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.