John Rhodes, Math Teacher at Senioritis.io

John Rhodes
August 16, 2023
Computational Mathematics

Minimum Area of a Convex Grid Polygon

A symmetrical convex grid polygon is a polygon such that: All its vertices have integer coordinates. All its internal angles are strictly smaller than $180^\circ$. It has...

John Rhodes
August 16, 2023
Computational Mathematics

Secret Santa

Secret Santa is a process that allows $n$ people to give each other presents, so that each person gives a single present and receives a single present....

John Rhodes
August 16, 2023
Computational Mathematics

Summation of Summations

Take a sequence of length $n$. Discard the first term then make a sequence of the partial summations. Continue to do this over and over until we...

John Rhodes
August 16, 2023
Computational Mathematics

Counting Ordered Factorisations

Define $d(n,k)$ to be the number of ways to write $n$ as a product of $k$ ordered integers \[ n = x_1\times x_2\times x_3\times \ldots\times x_k\qquad 1\le...

John Rhodes
August 16, 2023
Computational Mathematics

Coin Loops

A game is played with many identical, round coins on a flat table. Consider a line perpendicular to the table. The first coin is placed on the...

John Rhodes
August 16, 2023
Computational Mathematics

Paths to Equality

Define two functions on lattice points: $r(x,y) = (x+1,2y)$ $s(x,y) = (2x,y+1)$ A path to equality of length $n$ for a pair $(a,b)$ is a sequence $\Big((a_1,b_1),(a_2,b_2),\ldots,(a_n,b_n)\Big)$,...

John Rhodes
August 16, 2023
Computational Mathematics

Divisors of $2n^2$

Let $f(n)$ be the number of divisors of $2n^2$ that are no greater than n. For example, $f(15)=8$ because there are 8 such divisors: 1,2,3,5,6,9,10,15. Note that...

John Rhodes
August 16, 2023
Computational Mathematics

A Bit of Prime

The logical-OR of two bits is $0$ if both bits are $0$, otherwise it is $1$. The bitwise-OR of two positive integers performs a logical-OR operation on...