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The chromatic number of the plane, part 4: an upper bound
In my previous posts I explained lower bounds for the Hadwiger-Nelson problem: we know that the chromatic number of the plane is at least 5 because there exist unit distance graphs which we know need...
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Efficiency of repeated squaring: proof
My last post proposed a claim: The binary algorithm is the most efficient way to build using only doubling and incrementing steps. That is, any other way to build by doubling and incrementing uses an...
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Efficiency of repeated squaring: another proof
In my previous post I proved that the “binary algorithm” (corresponding to the binary expansion of a number ) is the most efficient way to build using only doubling and incrementing steps. Today I want...
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Chinese Remainder Theorem proof
In my previous post I stated the Chinese Remainder Theorem, which says that if and are relatively prime, then the function is a bijection between the set and the set of pairs (remember that the...
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Computing the Euler totient function, part 3: proving phi is multiplicative
We are trying to show that the Euler totient function , which counts how many numbers from to share no common factors with , is multiplicative, that is, whenever and share no common factors. In my...
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PIE: proof by algebra
In my previous post I stated a very formal, general form of the Principle of Inclusion-Exclusion, or PIE.1 In this post I am going to outline one proof of PIE. I’m not going to give a completely formal...
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PIE: proof by counting
Recall the setup: we have a universal set and a collection of subsets , , , and so on, up to . PIE claims that we can compute the number of elements of that are in none of the (that is, ) if we take...
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A simple proof of the quadratic formula
If you’re reading this blog you have probably memorized (or used to have memorized) the quadratic formula, which can be used to solve quadratic equations of the form But do you know how to derive the...
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Ways to prove a bijection
You have a function and want to prove it is a bijection. What can you do? By the book A bijection is defined as a function which is both one-to-one and onto. So prove that is one-to-one, and prove that...
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The Natural Number Game
Hello everyone! It has been quite a while since I have written anything here—my last post was in March 2020, and since then I have been overwhelmed dealing with online and hybrid teaching, plus a...
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