Proof

What is Mathematical Induction?

Last week’s exploration of a problem involving the Fibonacci sequence, has led me to delve into that and related concepts. In order to say much about the Fibonacci numbers, we have to first explore the concept of proof by mathematical induction. We’ll introduce it here, and then dig deeper next time.

Disappearing Area?

We’ve been looking at dissection puzzles, where we cut an object into pieces, and rearrange them. Here we’ll examine a mystery posed by two different puzzles, each of which seems to change the area by rearranging the pieces. The answer combines the marvelous Fibonacci numbers and [spoiler alert!] how easily we misjudge areas.

More Handshake Problems

Last week we looked at problems about counting diagonals in a polygon, and the very similar problem of counting handshakes when everyone in a group shakes with everyone else. In the course of searching for those problems, I also found some very different problems that are also about handshakes. We’ll look at those here, just …

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Proving the Law of Cosines

Last week we looked at several proofs of the Law of Sines. Here we will see a couple proofs of the Law of Cosines; they are more or less equivalent, but take different perspectives – even one from before trigonometry and algebra were invented! Proof using coordinates First, here is a question we looked at …

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