John Rhodes, Math Teacher at Senioritis.io

John Rhodes
August 16, 2023
Computational Mathematics

Sum over Bitwise Operators

Define $$\displaystyle g(m,n) = (m\oplus n)+(m\vee n)+(m\wedge n)$$ where $\oplus, \vee, \wedge$ are the bitwise XOR, OR and AND operator respectively. Also set $$\displaystyle G(N) = \sum_{n=0}^N\sum_{k=0}^n...

John Rhodes
August 16, 2023
Computational Mathematics

A Squared Recurrence Relation

The function $f$ is defined for all positive integers as follows: \begin{align*} f(1) &= 1\\ f(2n) &= 2f(n)\\ f(2n+1) &= 2n+1 + 2f(n)+\tfrac 1n f(n) \end{align*} It...

John Rhodes
August 16, 2023
Computational Mathematics

Buckets of Water

There are 3 buckets labelled $S$ (small) of 3 litres, $M$ (medium) of 5 litres and $L$ (large) of 8 litres. Initially $S$ and $M$ are full...

John Rhodes
August 16, 2023
Computational Mathematics

Stealthy Numbers

A positive integer $N$ is stealthy, if there exist positive integers $a$, $b$, $c$, $d$ such that $ab = cd = N$ and $a+b = c+d+1$. For...

John Rhodes
August 16, 2023
Computational Mathematics

Not Zeckendorf

Consider the Fibonacci sequence $\{1,2,3,5,8,13,21,\ldots\}$. We let $f(n)$ be the number of ways of representing an integer $n\ge 0$ as the sum of different Fibonacci numbers. For...

John Rhodes
August 16, 2023
Computational Mathematics

Product of Gauss Factorials

The Gauss Factorial of a number $n$ is defined as the product of all positive numbers $\leq n$ that are relatively prime to $n$. For example $g(10)=1\times...

John Rhodes
August 16, 2023
Computational Mathematics

Fermat Equation

Fermat’s Last Theorem states that no three positive integers $a$, $b$, $c$ satisfy the equation $$a^n+b^n=c^n$$ for any integer value of $n$ greater than 2. For this...

John Rhodes
August 16, 2023
Computational Mathematics

Powers of $1+\sqrt 7$

When $(1+\sqrt 7)$ is raised to an integral power, $n$, we always get a number of the form $(a+b\sqrt 7)$. We write $(1+\sqrt 7)^n = \alpha(n) +...