Computational Mathematics

John Rhodes
August 15, 2023
Computational Mathematics

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 $u_0 = -1$...

John Rhodes
August 15, 2023
Computational Mathematics

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...

John Rhodes
August 15, 2023
Computational Mathematics

$60$-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...

John Rhodes
August 15, 2023
Computational Mathematics

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...

John Rhodes
August 15, 2023
Computational Mathematics

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...

John Rhodes
August 15, 2023
Computational Mathematics

Best Approximations

Let $x$ be a real number. A best approximation to $x$ for the denominator bound $d$ is a rational number $\frac r s $ in reduced form,...

John Rhodes
August 15, 2023
Computational Mathematics

Prize Strings

A particular school offers cash rewards to children with good attendance and punctuality. If they are absent for three consecutive days or late on more than one...

John Rhodes
August 15, 2023
Computational Mathematics

Maximising a Weighted Product

Let $S_m = (x_1, x_2, \dots , x_m)$ be the $m$-tuple of positive real numbers with $x_1 + x_2 + \cdots + x_m = m$ for which...