John Rhodes, Math Teacher at Senioritis.io

John Rhodes
August 16, 2023
Computational Mathematics

Factors of Two in Binomial Coefficients

Define $g(n, m)$ to be the largest integer $k$ such that $2^k$ divides $\binom{n}m$. For example, $\binom{12}5 = 792 = 2^3 \cdot 3^2 \cdot 11$, hence $g(12,...

John Rhodes
August 16, 2023
Computational Mathematics

Circular Logic II

Given an integer $n$, $n \geq 3$, let $B=\{\mathrm{false},\mathrm{true}\}$ and let $B^n$ be the set of sequences of $n$ values from $B$. The function $f$ from $B^n$...

John Rhodes
August 16, 2023
Computational Mathematics

Random Connected Area

Consider a rectangle made up of $W \times H$ square cells each with area $1$. Each cell is independently coloured black with probability $0.5$ otherwise white. Black...

John Rhodes
August 16, 2023
Computational Mathematics

Eulercoin

Leonhard Euler was born on 15 April 1707. Consider the sequence 1504170715041707n mod 4503599627370517. An element of this sequence is defined to be an Eulercoin if it...

John Rhodes
August 16, 2023
Computational Mathematics

Triffle Numbers

Let $\sigma(n)$ be the sum of all the divisors of the positive integer $n$, for example: $\sigma(10) = 1+2+5+10 = 18$. Define $T(N)$ to be the sum...

John Rhodes
August 16, 2023
Computational Mathematics

123 Numbers

We define 123-numbers as follows: 1 is the smallest 123-number. When written in base 10 the only digits that can be present are “1”, “2” and “3”...

John Rhodes
August 16, 2023
Computational Mathematics

Randomly Decaying Sequence

Given a fixed real number $c$, define a random sequence $(X_n)_{n\ge 0}$ by the following random process: $X_0 = c$ (with probability 1). For $n>0$, $X_n =...

John Rhodes
August 16, 2023
Computational Mathematics

Mahjong

The game of Mahjong is played with tiles belonging to $s$ suits. Each tile also has a number in the range $1\ldots n$, and for each suit/number...