A Piece of the Pi: mathematics explained

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Continued fractions
A continued fraction is an expression obtained by representing a number as the sum of its integer part and the reciprocal of another number, and then…
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Richard Green
5
Friedman numbers
A Friedman number is a positive integer that can be written (nontrivially) using its own digits, together with parentheses and the operations of…
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Richard Green
105 trillion digits of pi
The number π (3.1415926…) has now been computed to 105 trillion decimal places! The computational background behind this feat is discussed by Jordan…
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Richard Green
The infinite monkey theorem
The infinite monkey theorem is the observation that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will…
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Richard Green
3
Corridor numbers, Fibonacci numbers, and Pascal’s triangle
The corridor numbers count the number of ways one can take n diagonal steps down a corridor of fixed width, starting in the corner. The picture above…
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Richard Green
Generalizations of Euler’s Tonnetz
The picture above by Konstanze Rietsch shows a Tonnetz of type G2. This is an infinite tessellation of 30°–60°–90° triangles whose edges are labelled by…
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Richard Green
Stacks of cards and the harmonic series
This picture shows a stable deck of 52 cards in which the top card overhangs the bottom card by about 1.5 card widths. In theory, it is possible to do…
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Richard Green
The combinatorics of redistricting plans
This new picture by Jamie Tucker-Foltz is a tiling of a 42×42 grid by 4-ominoes, otherwise known as Tetris pieces. The remarkable property of this…
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Richard Green
6
Derangements and the number e
If a teacher randomly hands back 20 homework assignments to the 20 students who wrote them, what is the probability that none of the students receives…
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Richard Green
1
Misleading diagrams
The picture above appears to show a triangle that can be cut into four smaller shapes and rearranged to form a shape with a different area. The red…
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Richard Green
Diophantine m-tuples
A Diophantine quadruple of integers is a set of four integers, such as {1, 3, 8, 120}, with the property that if we multiply any two of them and add 1…
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Richard Green
1
The Proth–Gilbreath conjecture and absolute differences of integers
The Proth–Gilbreath conjecture is an easily stated open problem about prime numbers. Start by writing the first n prime numbers (for example, 2, 3, 5…
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Richard Green
1
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