Prime-proof Squbes
We shall define a sqube to be a number of the form, $p^2 q^3$, where $p$ and $q$ are distinct primes. For example, $200 = 5^2 2^3$...
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John Rhodes
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August 15, 2023
Iterative Circle Packing
Three circles of equal radius are placed inside a larger circle such that each pair of circles is tangent to one another and the inner circles do...
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John Rhodes
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August 15, 2023
Ambiguous Numbers
A best approximation to a real number $x$ for the denominator bound $d$ is a rational number $\frac r s$ (in reduced form) with $s \le d$,...
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John Rhodes
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August 15, 2023
A Recursively Defined Sequence
Given is the function $f(x) = \lfloor 2^{30.403243784 – x^2}\rfloor \times 10^{-9}$ ($\lfloor \, \rfloor$ is the floor-function), the sequence $u_n$ is defined by $u_0 = -1$...
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John Rhodes
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August 15, 2023
Prime Triplets
Build a triangle from all positive integers in the following way: 1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 20 21...
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John Rhodes
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August 15, 2023
$60$-degree Triangle Inscribed Circles
Let’s call an integer sided triangle with exactly one angle of $60$ degrees a $60$-degree triangle. Let $r$ be the radius of the inscribed circle of such...
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John Rhodes
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August 15, 2023
Coloured Configurations
Consider graphs built with the units $A$: and $B$: , where the units are glued along the vertical edges as in the graph . A configuration of...
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John Rhodes
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August 15, 2023
Squarefree Numbers
A positive integer $n$ is called squarefree, if no square of a prime divides $n$, thus $1, 2, 3, 5, 6, 7, 10, 11$ are squarefree, but...