Archive for June, 2017

Patterns in Poetry, Music and Morse Code

Suppose we have to ascent a flight of stairs and can take only one or two steps at a time. How many different patterns of ascent are there? We start with the simplest cases. With one step there is only one way; with two, there are two: take two single steps or one double step. With three steps, there are three possibilities. We can now proceed in an inductive manner.

Staircase-01

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The Beer Mat Game

Alice and Bob, are enjoying a drink together. Sitting in a bar-room, they take turns placing beer mats on the table. The only rules of the game are that the mats must not overlap or overhang the edge of the table. The winner is the player who puts down the final mat. Is there a winning strategy for Alice or for Bob?

Beermats-picture

Image from Flickr. 

We start with the simple case of a circular table and circular mats. In this case, there is a winning strategy for the first player. Before reading on, can you see what it is?

* * *

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Fractal Complexity of Finnegans Wake

Tomorrow we celebrate Bloomsday, the day of action in Ulysses. Most of us regard Joyce’s singular book as a masterpiece, even if we have not read it. In contrast, Finnegans Wake is considered by some as a work of exceptional genius, by others as impenetrable bafflegab [See TM117 or search for “thatsmaths” at irishtimes.com].

FW-Fractals-03-Squeezed.jpg

Wavelet transform of sentence length sequence in Ulysses. Note the structural change around sentence number 13,000. Image from Drozdz, et al (2016).

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A Remarkable Pair of Sequences

The terms of the two integer sequences below are equal for all {n} such that {1<n<777{,}451{,}915{,}729{,}368},  but equality is violated for this enormous value and, intermittently, for larger values of {n}.

TicTacToe-2-Sequences

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Beautiful Patterns in Maths and Music

The numerous connections between mathematics and music have long intrigued practitioners of both. For centuries scholars and musicians have used maths to analyze music and also to create it. Many of the great composers had a deep understanding of the mathematical principles underlying music. Johann Sebastian Bach was the grand master of structural innovation and invention in music. While his compositions are the free creations of a genius, they have a fundamentally mathematical basis [See TM116 or search for “thatsmaths” at irishtimes.com].

J-S-Bach

Johann Sebastian Bach, the grand master of structural innovation and invention in music.

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