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Category Archives: pattern
Hypercube offsets
In my previous posts, each drawing consisted of two offset copies of the previous drawing. For example, here are the drawings for and : You can see how the drawing contains an exact copy of the drawing, plus another copy … Continue reading
A few words about PWW #27
The images in my last post were particular realizations of the famous Sieve of Eratosthenes. The basic idea of the sieve is to repeatedly do the following: Circle the next number bigger than that is not yet crossed out, call … Continue reading
Posted in pattern, pictures, posts without words, primes
Tagged Eratosthenes, prime, sieve
4 Comments
Post without words #27
Posted in pattern, pictures, posts without words, primes
Tagged Eratosthenes, prime, sieve
7 Comments
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 … Continue reading
PIE: proof by algebra
In my previous post I stated a very formal, general form of the Principle of InclusionExclusion, or PIE.1 In this post I am going to outline one proof of PIE. I’m not going to give a completely formal proof, because … Continue reading
Formal PIE
I’ve been talking informally about the Principle of InclusionExclusion but I realized it would be useful to state it more formally before proceeding to some proofs. The only problem is that a fully formal statement of PIE has a lot … Continue reading
Posted in combinatorics, pattern
Tagged equation, exclusion, formal, inclusion, PIE, sets
5 Comments
Have a piece of PIE
Commenter Stuart LoPresti gave a very nice analysis answering the questions at the end of my last post. And indeed, the punchline is that the answers are very similar to the answers to the post before that. PIE If we … Continue reading
Probabilistic PIE
Commenters Sylvain B., xpil, and Christian Luca all gave correct answers to the challenge from my previous post. If the probability that someone likes X is , then the probability they don’t like X is . Therefore the probability that … Continue reading
Finding the repetend length of a decimal expansion
We’re still trying to find the prefix length and repetend length of the decimal expansion of a fraction , that is, the length of the part before it starts repeating, and the length of the repeating part. In my previous … Continue reading
Posted in computation, group theory, modular arithmetic, number theory, pattern
Tagged decimal, expansion, group theory, rational, repeating, repetend, totient
Comments Off on Finding the repetend length of a decimal expansion
Finding the prefix length of a decimal expansion
Remember from my previous post that we’re trying to find the prefix length and repetend length of the decimal expansion of a fraction , that is, the length of the part before it starts repeating, and the length of the … Continue reading