Fibonacci Words
For any two strings of digits, $A$ and $B$, we define $F_{A, B}$ to be the sequence $(A,B,AB,BAB,ABBAB,\dots)$ in which each term is the concatenation of the...
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John Rhodes
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August 15, 2023
Four Representations Using Squares
Consider the number $3600$. It is very special, because \begin{alignat}{2} 3600 &= 48^2 + &&36^2\\ 3600 &= 20^2 + 2 \times &&40^2\\ 3600 &= 30^2 + 3...
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John Rhodes
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August 15, 2023
Minkowski Sums
Let $S_n$ be the regular $n$-sided polygon – or shape – whose vertices $v_k$ ($k = 1, 2, \dots, n$) have coordinates: \begin{align} x_k &= \cos((2k –...
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John Rhodes
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August 15, 2023
The Chase
The Chase is a game played with two dice and an even number of players. The players sit around a table and the game begins with two...
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John Rhodes
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August 15, 2023
Tribonacci Non-divisors
The sequence $1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, 1201, \dots$ is defined by $T_1 = T_2 = T_3 = 1$...
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John Rhodes
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August 15, 2023
Almost Right-angled Triangles II
Let us call an integer sided triangle with sides $a \le b \le c$ barely obtuse if the sides satisfy $a^2 + b^2 = c^2 – 1$....
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John Rhodes
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August 15, 2023
Almost Right-angled Triangles I
Let us call an integer sided triangle with sides $a \le b \le c$ barely acute if the sides satisfy $a^2 + b^2 = c^2 + 1$....
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John Rhodes
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August 15, 2023
Sphere Packing
What is the length of the shortest pipe, of internal radius $\pu{50 mm}$, that can fully contain $21$ balls of radii $\pu{30 mm}, \pu{31 mm}, \dots, \pu{50...