Expert Solutions | Senioritis

John Rhodes
August 15, 2023
Computational Mathematics

Digital Root Sums of Factorisations

A composite number can be factored many different ways. For instance, not including multiplication by one, $24$ can be factored in $7$ distinct ways: \begin{align} 24 &=...

John Rhodes
August 15, 2023
Computational Mathematics

Lexicographical Neighbours

Taking three different letters from the $26$ letters of the alphabet, character strings of length three can be formed. Examples are ‘abc’, ‘hat’ and ‘zyx’. When we...

John Rhodes
August 15, 2023
Computational Mathematics

Base-10 Diophantine Reciprocal

Consider the diophantine equation $\frac 1 a + \frac 1 b = \frac p {10^n}$ with $a, b, p, n$ positive integers and $a \le b$. For...

John Rhodes
August 15, 2023
Computational Mathematics

Counting Digits

Starting from zero the natural numbers are written down in base $10$ like this: $0\,1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\cdots$ Consider the digit $d=1$. After we write down each number $n$, we...

John Rhodes
August 15, 2023
Computational Mathematics

Counting Capacitor Circuits

An electric circuit uses exclusively identical capacitors of the same value $C$. The capacitors can be connected in series or in parallel to form sub-units, which can...

John Rhodes
August 15, 2023
Computational Mathematics

Exploring Pascal’s Pyramid

A triangular pyramid is constructed using spherical balls so that each ball rests on exactly three balls of the next lower level. Then, we calculate the number...

John Rhodes
August 15, 2023
Computational Mathematics

Investigating Gaussian Integers

As we all know the equation $x^2=-1$ has no solutions for real $x$. If we however introduce the imaginary number $i$ this equation has two solutions: $x=i$...

John Rhodes
August 15, 2023
Computational Mathematics

There are several ways to write the number $\dfrac{1}{2}$ as a sum of square reciprocals using distinct integers. For instance, the numbers $\{2,3,4,5,7,12,15,20,28,35\}$ can be used: $$\begin{align}\dfrac{1}{2}...