Expert Solutions | Senioritis

John Rhodes
August 15, 2023
Computational Mathematics

Collatz Prefix Families

The Collatz sequence is defined as: $a_{i+1} = \left\{ \large{\frac {a_i} 2 \atop 3 a_i+1} {\text{if }a_i\text{ is even} \atop \text{if }a_i\text{ is odd}} \right.$. The Collatz...

John Rhodes
August 15, 2023
Computational Mathematics

Under the Rainbow

$70$ coloured balls are placed in an urn, $10$ for each of the seven rainbow colours. What is the expected number of distinct colours in $20$ randomly...

John Rhodes
August 15, 2023
Computational Mathematics

Exploding Sequence

Define the sequence $a_1, a_2, a_3, \dots$ as: $a_1 = 1$ $a_{n+1} = 6a_n^2 + 10a_n + 3$ for $n \ge 1$. Examples: $a_3 = 2359$ $a_6...

John Rhodes
August 15, 2023
Computational Mathematics

Double Pandigital Number Divisible by $11$

We call a positive integer double pandigital if it uses all the digits $0$ to $9$ exactly twice (with no leading zero). For example, $40561817703823564929$ is one...

John Rhodes
August 15, 2023
Computational Mathematics

Jumping Frog

There are $n$ stones in a pond, numbered $1$ to $n$. Consecutive stones are spaced one unit apart. A frog sits on stone $1$. He wishes to...

John Rhodes
August 15, 2023
Computational Mathematics

Common Factors Between Two Sequences

Let $G(a, b)$ be the smallest non-negative integer $n$ for which $\operatorname{\mathbf{gcd}}$Greatest common divisor$(n^3 + b, (n + a)^3 + b)$ is maximized. For example, $G(1, 1)...

John Rhodes
August 15, 2023
Computational Mathematics

Unbalanced Nim

Alice and Bob have enjoyed playing Nim every day. However, they finally got bored of playing ordinary three-heap Nim. So, they added an extra rule: – Must...

John Rhodes
August 15, 2023
Computational Mathematics

Sums of Power Sums

Let $f_k(n)$ be the sum of the $k$th powers of the first $n$ positive integers. For example, $f_2(10) = 1^2 + 2^2 + 3^2 + 4^2 +...