Tonight I introduced the sad tale of Daedalus and Icarus to my son. In case you are not familiar with the story, Daedalus and Icarus are stuck in the tower in the middle of the maze with a horrible minotaur blocking their escape. Daedalus builds wings and they try and fly away.
In the mathematical version of this story, created by Gordon Hamilton, Daedalus “flies” by choosing a number. If the number is odd, he multiplies it by 3 and then subtracts 1. If it is even, he divides it by 2. For Icarus if the number is odd, he multiplies it by 3 and adds 1. If it is even, he also divides it by 2. The objective is to pick a number that does not eventually end up as a 1 because this means that Daedalus or Icarus just crashed into the ground.
My son and I played this game (based on the Collatz Conjecture) and quickly found a value of Daedalus that allows him to fly away and escape the maze. We then started trying to find values that allow Icarus to escape (we haven’t succeeded yet).
At one point my noted the following:
You know, Dad. There are an infinite number of solutions to the Daedalus problem. If you have a solution, to find another starting number that works, just double it, and so on. We can keep finding new solutions by doubling whatever number we have. Since we have one answer that works, we know there are an infinite number of ones that do.
This is essentially inductive reasoning. He’s reasoned that if we know case n works, we can find a larger case by doubling it. Since we know at least one solution for Daedalus works, and that we can double as many times as we like, then we must have an infinite number of solutions.