The analogy my professor used to teach us induction is as follows:
You are making a train of dominoes. The goal is to have the whole chain fall over when you push down the first domino. The base case is checking to make sure that the first domino will actually fall over. The induction step is checking to make sure that each domino will knock over the next domino in line. Thus, you know that the whole chain will fall over, because you know the first domino will fall over and you know that it will knock down the second domino, which will knock down each subsequent domino, etc.
I really liked that analogy.
When I read this chapter it was intriguing to me to see a different approach to induction. It is still the same concept, but the way it was explained and some of the formatting differed from what I learned. It's interesting to see how different people approach the same idea.
The pigeonhole-principle is pretty cool. I remember learning about it indirectly in sixth grade. I like how math builds on itself. E.g. the principles that I learned when I was younger prepared me to build upon them to learn more complex principles now.
This principle comes in very handy for solving problems that I wouldn't know how else to approach. We used it a lot at my summer math camp.
Batman
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