A Piece of the Pi: mathematics explained

A Piece of the Pi: mathematics explained

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Number Theory 🔢

The golden ratio as a number base
The Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …) are one of the most famous sequences of integers.
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Richard Green
7
Balanced Steinhaus triangles
A familiar fact from the game of pool is that 15 objects can be arranged in an equilateral triangle of side length 5.
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Richard Green
1
Spiral Sudoku
Suppose we have a 5×5 grid with 15 circled positions in a spiral shape as shown above.
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Richard Green
5
Counting dimer tilings
A dimer is a 2×1 domino-shaped rectangle.
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Richard Green
2
The Ramanujan Machine
The mathematician Srinivasa Ramanujan (1887–1920) proved many remarkable identities, such as the formula shown above for the square root of πe/2.
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Richard Green
Repunits and prime numbers
A repunit (“repeated unit”) is a number that only contains the digit 1 in some number base.
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Richard Green
4
Ulam words and the Ulam sequence
The Ulam sequence is a sequence of positive integers xn, where x1=1, x2=2, and where each xn for n > 2 is defined to be the smallest integer that can be…
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Richard Green
3
The comma sequence
The comma sequence (1 , 12 , 35 , 94 , 135 , 186 , 248 , …) is so called because the difference between successive terms is given by the digits on…
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Richard Green
5
Egyptian fractions
The ancient Egyptians had notation to represent the fraction 1/n, but not for every fraction of the form a/b. Because of this, they typically…
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Richard Green
6
Magic squares of powers
A magic square of order n is an n by n grid of integers in which each row, each column, and each of the two main diagonals adds up to the same number.
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Richard Green
Generalizations of Hofstadter’s Q-sequence
The Fibonacci numbers are the famous sequence that begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
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Richard Green
3
Partitions and primes
The picture above shows all the partitions of the numbers 1 up to 8.
 • 
Richard Green
1
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