Category Archives: Uncategorized

3d Sudoku

Consider this $9\times 9\times 9$ grid, in which some cubes have been shaded. It can be represented in two dimensions by viewing it from one side and using numbers from $1$ to $9$ to give the location of each cube … Continue reading

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Unique Distancing Problem

The most recent of Matt Parker’s Maths Puzzles was the unique distancing problem: For which $n\in\mathbb N$ is it possible to place $n$ counters on an $n\times n$ square grid so that there’s a different distance between each pair of … Continue reading

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A better representation for real numbers

Real numbers can be represented by decimals or by continued fractions, but both of these systems have the problem that the representations are not unique. This nonuniqueness can introduce ugly complications into proofs that use these representations, in particular those … Continue reading

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Dividing by a vector

There are various ways that we can multiply vectors. We can multiply a vector by a scalar to get another vector: $w=\lambda v$. We can multiply a vector by a linear map to get another vector: $w=\alpha v$. We can … Continue reading

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Adjunctions between monoids

In the last post, I talked about Monads on Monoids. We defined these by turning each monoid into a category (its delooping), and then looking at monads on this category. This data could then be reexpressed in terms of the … Continue reading

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Monads on monoids

This post is inspired by a couple of posts on math.stackexchange.com asking what a ‘monad on a monoid’ is. In category theory there are a couple of especially simple kinds of category. For any partial order we can put a … Continue reading

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