John Rhodes, Math Teacher at Senioritis.io

John Rhodes
August 16, 2023
Computational Mathematics

Distinct Values of a Proto-logarithmic Function

Consider the values of $\log_2(8)$, $\log_4(64)$ and $\log_3(27)$. All three are equal to $3$. Generally, the function $f(m,n)=\log_m(n)$ over integers $m,n \ge 2$ has the property that...

John Rhodes
August 16, 2023
Computational Mathematics

Patterned Cylinders

An infinitely long cylinder has its curved surface fully covered with different coloured but otherwise identical rectangular stickers, without overlapping. The stickers are aligned with the cylinder,...

John Rhodes
August 16, 2023
Computational Mathematics

Divisors of Binomial Product

Let $B(n) = \displaystyle \prod_{k=0}^n {n \choose k}$, a product of binomial coefficients. For example, $B(5) = {5 \choose 0} \times {5 \choose 1} \times {5 \choose...

John Rhodes
August 16, 2023
Computational Mathematics

Low-Prime Chessboard Nim

Alice and Bob are taking turns playing a game consisting of $c$ different coins on a chessboard of size $n$ by $n$. The game may start with...

John Rhodes
August 16, 2023
Computational Mathematics

Skipping Squares

For some fixed $\rho \in [0, 1]$, we begin a sum $s$ at $0$ and repeatedly apply a process: With probability $\rho$, we add $1$ to $s$,...

John Rhodes
August 16, 2023
Computational Mathematics

Linear Transformations of Polygonal Numbers

It is possible to find positive integers $A$ and $B$ such that given any triangular number, $T_n$, then $AT_n +B$ is always a triangular number. We define...

John Rhodes
August 16, 2023
Computational Mathematics

Bounded Divisors

Let $n$ be a natural number and $p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ its prime factorisation. Define the Liouville function $\lambda(n)$ as $\lambda(n) = (-1)^{\sum\limits_{i=1}^{k}\alpha_i}$. (i.e. $-1$ if the sum...

John Rhodes
August 16, 2023
Computational Mathematics

Every Day Is a Holiday

On planet J, a year lasts for $D$ days. Holidays are defined by the two following rules. At the beginning of the reign of the current Emperor,...