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# Category Archives: pattern

## 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

Posted in combinatorics, pattern, proof
Tagged counting, exclusion, inclusion, PIE, proof
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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 proof, because … Continue reading

## Formal PIE

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

## 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

## 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