Category Archives: pattern

Hypercube offsets

Posted on January 30, 2020 by Brent

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 →

Posted in pattern | Tagged , , , , , , , | 2 Comments

A few words about PWW #27

Posted on October 24, 2019 by Brent

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 , , | 4 Comments

Post without words #27

Posted on October 21, 2019 by Brent
Posted in pattern, pictures, posts without words, primes | Tagged , , | 7 Comments

PIE: proof by counting

Posted on August 13, 2019 by Brent

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 →

Posted in combinatorics, pattern, proof | Tagged , , , , | 1 Comment

PIE: proof by algebra

Posted on August 7, 2019 by Brent

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 proof, because … Continue reading →

Posted in combinatorics, induction, pattern, proof | Tagged , , , , , , | 2 Comments

Formal PIE

Posted on July 27, 2019 by Brent

I’ve been talking informally about the Principle of Inclusion-Exclusion 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 , , , , , | 5 Comments

Have a piece of PIE

Posted on July 21, 2019 by Brent

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 →

Posted in pattern, probability | Tagged , | 6 Comments

Probabilistic PIE

Posted on July 17, 2019 by Brent

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 →

Posted in pattern, probability, solutions | Tagged , | 7 Comments

Finding the repetend length of a decimal expansion

Posted on February 8, 2019 by Brent

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 , , , , , , | Comments Off on Finding the repetend length of a decimal expansion

Finding the prefix length of a decimal expansion

Posted on February 2, 2019 by Brent

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 →

Posted in number theory, pattern | Tagged , , , , | 3 Comments