The goal of this course is to teach students how to solve the 100 rooms problem. When you first hear the problem, it sounds impossible and feels unsettling that there's a solution. In order to solve this problem, you only need three concepts from analysis. In this class, students will build the mathematical tools to solve this problem and along the way use these tools to solve simpler problems that lead up to the 100 rooms problem. Some of the topics that will be covered in this course include understanding sizes of infinity, Hilbert hotel problems, black/white hat problems, and the axiom of choice. The crux of the 100 rooms problem is using the axiom of choice which is simply stated but has shocking implications.
Here’s the 100 rooms problem:
Suppose that there are 100 rooms and in every room there are an infinite number of boxes labelled "box 1", "box 2",... etc. There is a real valued number in each box and the set of boxes in each room are identical. Now there are 100 people and each is assigned to a room and must guess what number is in at least one box. They may open as many boxes as they want, even infinitely many but must leave one unopened. Before the people enter their room, they can come up with a strategy on how to correctly guess the number in some box. It is possible that at least 99 people correctly guess the number in some box.
(1) Sizes of infinity
On the first day, I would like to state the 100 rooms problem and emphasize that this is a shocking problem because we're able to devise a strategy to correctly guess real valued numbers. To help the students understand the magnitude of this result, we'll go over what it means for a set to be finite, countable, and uncountable. To understand the concept of a countable set, we'll go over the simple examples such as the integers, even numbers, and rationals. Then we can apply this knowledge to work on our first logic problem which will be the Hilbert hotel problems. When we go over logic problems in class, I plan to state the problem and then let the class try to work together to figure out the solution. The homework assignment corresponding to this topic would include showing certain sets are countable and some variations of the Hilbert hotel problems done in class. In addition, I'll give them some chapters from David Foster Wallace’s book "Beyond Infinity" as a reference to further help them understand these concepts. I think this would be a great reference because its written by a non-mathematician and the author provides great intuition for the concept of countable vs uncountable.
(2) Analysis: Convergence of Sequences, Equivalence Classes, and Axiom of Choice
The next major topic is teach the students the necessary higher level math to solve the 100 rooms problem. First, we’ll begin with sequences and go over what it means for a sequence to converge, then do lots of practice. Second, we’ll define equivalence classes and go over equivalence relations that we can define on sequences. Lastly, we’ll define the axiom of choice and then see applications of this axiom by going over the black/white hat problems. The black/white problems are a stepping stone to solving the 100 rooms problem. Once the students understand a few of the black/white hat problems, then we’ll be able to go through the solution to the 100 rooms problem.
The underlying goal of this course is to introduce students to higher level math in an engaging and interesting manner. This course helps students develop abstract thinking and problem solving skills rather than focusing on problems typically seen in a high school math class where the goal is to make a calculation. I hope this course will make students more interested in math and give them a new perspective of the kinds of problems we can solve using math.
Prerequisites: There are no prerequisites for this course.
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