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Cassini’s Identity for Fibonacci Numbers

https://www.youtube.com/watch?v=WED3oFewP-4 The Fibonacci numbers are among the most famous of number sequences. They satisfy a plethora of identities, including Cassini’s identity. Cassini is not easy to prove by elementary means, but an ingenious trick via matrix multiplication and the mutliplicativity of the determinant does the job. We provide this proof in this video.

Classifying the Platonic Solids

https://www.youtube.com/watch?v=sgXXcwdn5p8 The Platonic solids are regular polyhedra, meaning each vertex has the same number of edges incident on it, and each face has the same number of edges on its border. It is known that there exist five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. By using Euler’s formula for planar graphs, and …

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A Criterion for Planar Graphs

https://www.youtube.com/watch?v=be8AKUAAuZs It is difficult to know in general when a graph has a planar embedding without actually seeing a planar embedding. However, we can derive a necessary criterion for planarity using Euler’s characteristic formula, depending only on the number of vertices and edges of the graph (making it independent of any particular planar embedding). By …

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Balls & Urns, Sticks & Stones, Stars & Bars

https://www.youtube.com/watch?v=lcjNoFJ3Ick In how many ways can $n$ indistinguishable balls be placed into $k$ distinguishable boxes? What if each box must receive at least one ball? It turns out that the answers can be expressed as binomial coefficients. However, the standard journey to finding them requires an ingenious trick. We discuss and apply the technique in …

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