Expert Solutions | Senioritis

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,...

John Rhodes
August 16, 2023
Computational Mathematics

Squares on the Line

Sam and Tom are trying a game of (partially) covering a given line segment of length $L$ by taking turns in placing unit squares onto the line...

John Rhodes
August 16, 2023
Computational Mathematics

$2$-Friendly

Two positive integers $a$ and $b$ are $2$-friendly when $\gcd(a,b) = 2^t, t \gt 0$. For example, $24$ and $40$ are $2$-friendly because $\gcd(24,40) = 8 =...

John Rhodes
August 16, 2023
Computational Mathematics

A Long Row of Dice

Consider a row of $n$ dice all showing 1. First turn every second die,$ (2,4,6,\ldots)$, so that the number showing is increased by 1. Then turn every...

John Rhodes
August 16, 2023
Computational Mathematics

Shut the Box

Bob plays a single-player game of chance using two standard 6-sided dice and twelve cards numbered 1 to 12. When the game starts, all cards are placed...

John Rhodes
August 16, 2023
Computational Mathematics

Summing a Multiplicative Function

A multiplicative function $f(x)$ is a function over positive integers satisfying $f(1)=1$ and $f(a b)=f(a) f(b)$ for any two coprime positive integers $a$ and $b$. For integer...