Proto-Counting (Week 2, Day 1)

The theme for the week is counting. Let’s start off the week with a real life math problem.

Lets say you are organizing a talk at an auditorium. The auditorium has a whole bunch of seats, so you aren’t worried about running out of seating. However as the time for the talk draws near you learn that the speaker has only yesterday been nominated for a prestigious award, and has become something of a celebrity. A crowd of people arrive to see the talk. What is the simplest way to check if there are enough seats for the attendees?

While you mull that over here is a counting related video starring Count Von Count and some guy named Liam.

Answer the the question after the jump.

The simplest way to check if their is enough chairs for the people is to have the people all try to sit down. If there are people left without a seat then there is more people than chairs. It’s a very simple solution I hope you will agree, and it didn’t involve counting the people or the chairs.
The act of putting people in chairs can be abstracted to what is called a one-to-one correspondence. In our case, for every chair we assign a person until we run out of chairs or people.  This idea of one-to-one correspondence will play an important role later on. For now it is important because it is the concept that we simplify by using counting.
Instead of assigning a person to each chair we can assign words, which we call numbers. The first chair we call “one”, the second “two,” and so on until we know the number of chairs. Then we can do the same with the attendees, giving each a number as they enter, and then once we reach the number of chairs we know the auditorium is full.

What are the most difficult counting problems you have had to deal with in your life?