Expert Solutions | Senioritis

John Rhodes
August 16, 2023
Computational Mathematics

Expressing an Integer as the Sum of Triangular Numbers

Gauss famously proved that every positive integer can be expressed as the sum of three triangular numbers (including $0$ as the lowest triangular number). In fact most...

John Rhodes
August 16, 2023
Computational Mathematics

Planetary Gears

A circle $C$ of circumference $c$ centimetres has a smaller circle $S$ of circumference $s$ centimetres lying off-centre within it. Four other distinct circles, which we call...

John Rhodes
August 16, 2023
Computational Mathematics

Square Subsets

For a set of positive integers $\{a, a+1, a+2, \dots , b\}$, let $C(a,b)$ be the number of non-empty subsets in which the product of all elements...

John Rhodes
August 16, 2023
Computational Mathematics

Numbers with a Given Prime Factor Sum

Consider the numbers $15$, $16$ and $18$: $15=3\times 5$ and $3+5=8$. $16 = 2\times 2\times 2\times 2$ and $2+2+2+2=8$. $18 = 2\times 3\times 3$ and $2+3+3=8$. $15$,...

John Rhodes
August 16, 2023
Computational Mathematics

Mirror Power Sequence

For two integers $n,e \gt 1$, we define an $(n,e)$-MPS (Mirror Power Sequence) to be an infinite sequence of integers $(a_i)_{i\ge 0}$ such that for all $i\ge...

John Rhodes
August 16, 2023
Computational Mathematics

Creative Numbers

Alice plays the following game, she starts with a list of integers $L$ and on each step she can either: remove two elements $a$ and $b$ from...

John Rhodes
August 16, 2023
Computational Mathematics

The Millionth Number with at Least One Million Prime Factors

Consider the natural numbers having at least $5$ prime factors, which don’t have to be distinct. Sorting these numbers by size gives a list which starts with:...

John Rhodes
August 16, 2023
Computational Mathematics

Special Partitions 2

An integer partition of a number $n$ is a way of writing $n$ as a sum of positive integers. Partitions that differ only by the order of...