Category Archives: Uncategorized

Beware the “natural” quaternion

Introduction Rotation math can be confusing. But it didn’t need to be this confusing. I think the reason that 3D rotations can be tricky to work with is that there are so many choices of convention– in interpretation, notation, and … Continue reading

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Analysis of Evil Hangman

Introduction This is a follow-up to the previous post that briefly mentioned the game of Hangman, in the context of developing a dictionary of words sorted based on their frequency of occurrence in the recently updated Google Books Ngram dataset. … Continue reading

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An updated Google Books word frequency list

Introduction I think I nerd-sniped myself. This started with the objective of writing a simple program to play Hangman, as a demonstration of a potential programming exercise for students. (Don’t assign the problem if you don’t know the solution.) But … Continue reading

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Balanced clock hands

Given an analog clock with sweeping hour, minute, and second hands, at what times are the hands most evenly “balanced”– that is, most equally separated in angle? It’s a standard high school algebra problem to compute all of the times … Continue reading

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What is the probability of a tie vote?

This article was discussed on Reddit. It is time to vote for a new mathematics department chair. All voting faculty members are on the ballot as eligible candidates… but no one really cares about the job, so each faculty member … Continue reading

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Counting edge-matching puzzles

I recently re-discovered a puzzle that I had mostly forgotten from when I was a kid. The problem is simple to state: rearrange and rotate the seven hexagonal pieces shown below so that each of the twelve pairs of facing … Continue reading

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The Fisher-Yates shuffle is backward

Given a list of elements, such as cards in a deck, what is the right way to shuffle the list? That is, what is the appropriate algorithm to (pseudo)randomly permute the elements in the list, so that each of the … Continue reading

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Among Us: Morse code puzzle

In the online game Among Us, players who visit the Comms room hear a fuzzy audio recording of a series of high-pitched beeps that sound like Morse code. I first heard the recording here, but this more recent video also … Continue reading

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Counting Hotel Key Cards

Introduction Suppose that you are the owner of a new hotel chain, and that you want to implement a mechanical key card locking system on all of the hotel room doors. Each key card will have a unique pattern of … Continue reading

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Exploiting advantage from too few shuffles

Introduction A few days ago a friend of mine referred me to an interesting podcast discussing card shuffling, framed as a friendly argument-turned-wager between a couple about how many times you should shuffle a deck of cards. A woman claims … Continue reading

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