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The Locker Problem: Investigations and Extensions with Mathematica

Bruce Torrence
Organization: Randolph-Macon College
Department: Department of Mathematics

2006 Wolfram Technology Conference
Conference location

Champaign IL

The locker problem is well known to many secondary, high school, and university students. It goes like this: Imagine a hallway with 1000 lockers, all closed. 1000 students walk the corridor as follows: The first student opens every locker. The second closes every second locker, beginning with the second. The third changes the state of every third locker, beginning with the third, opening closed lockers and shutting open ones. The fourth does this to every fourth locker, and so on. Which lockers remain open after all 1000 students have marched? Mathematica can be used to model extensions of this classic problem, leading to both interesting mathematics and even more questions. For example, if one wishes to keep only the first locker open, which subset of the students should be dispatched? What questions arise from sending the students in a different order? Iterative schemes can be devised whereby the lockers left open by one cadre of marching students will be precisely the students dispatched in the next iteration. What pattern emerges? We will see an interesting connection to Wolfram’s “Rule 60” for cellular automata. This talk is intended for a general audience; anyone who likes puzzles drawn from the realm of recreational mathematics (and who understands prime factorization of natural numbers) will have the requisite background. Ideas for teaching and undergraduate research will be emphasized, as will the theme of using Mathematica as a research tool to reveal nontrivial patterns.

*Education > College
*Mathematics > Number Theory
*Mathematics > Recreational Mathematics

the locker problem

Torrence-Lockers.zip (1.6 MB) - ZIP archive [for Mathematica 6.0]