Computational Mathematics

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}...

John Rhodes
August 15, 2023
Computational Mathematics

A Preference for A5

A printing shop runs 16 batches (jobs) every week and each batch requires a sheet of special colour-proofing paper of size A5. Every Monday morning, the supervisor...

John Rhodes
August 15, 2023
Computational Mathematics

Sub-triangle Sums

In a triangular array of positive and negative integers, we wish to find a sub-triangle such that the sum of the numbers it contains is the smallest...

John Rhodes
August 15, 2023
Computational Mathematics

Maximum-sum Subsequence

Looking at the table below, it is easy to verify that the maximum possible sum of adjacent numbers in any direction (horizontal, vertical, diagonal or anti-diagonal) is...