Computational Mathematics

John Rhodes
August 15, 2023
Computational Mathematics

Infinite String Tour

Create a sequence of numbers using the “Blum Blum Shub” pseudo-random number generator: \begin{align} s_0 &= 14025256\\ s_{n + 1} &= s_n^2 \bmod 20300713 \end{align} Concatenate these...

John Rhodes
August 15, 2023
Computational Mathematics

Tours on a $4 \times N$ Playing Board

Let $T(n)$ be the number of tours over a $4 \times n$ playing board such that: The tour starts in the top left corner. The tour consists...

John Rhodes
August 15, 2023
Computational Mathematics

Suppliers ‘A’ and ‘B’ provided the following numbers of products for the luxury hamper market: Product’A”B’Beluga Caviar5248640Christmas Cake13121888Gammon Joint26243776Vintage Port57603776Champagne Truffles39365664 Although the suppliers try very hard...

John Rhodes
August 15, 2023
Computational Mathematics

An Arithmetic Geometric Sequence

Given is the arithmetic-geometric sequence $u(k) = (900-3k)r^{k – 1}$. Let $s(n) = \sum_{k = 1}^n u(k)$. Find the value of $r$ for which $s(5000) = -600\,000\,000\,000$....

John Rhodes
August 15, 2023
Computational Mathematics

Semidivisible Numbers

For an integer $n \ge 4$, we define the lower prime square root of $n$, denoted by $\operatorname{lps}(n)$, as the largest prime $\le \sqrt n$ and the...

John Rhodes
August 15, 2023
Computational Mathematics

Lattice Points on a Circle

Let $f(N)$ be the number of points with integer coordinates that are on a circle passing through $(0,0)$, $(N,0)$,$(0,N)$, and $(N,N)$. It can be shown that $f(10000)...

John Rhodes
August 15, 2023
Computational Mathematics

The Race

Two players share an unbiased coin and take it in turns to play The Race. On Player 1’s turn, the coin is tossed once. If it comes...

John Rhodes
August 15, 2023
Computational Mathematics

Prime Factorisation of Binomial Coefficients

The binomial coefficient $\displaystyle \binom {10} 3 = 120$. $120 = 2^3 \times 3 \times 5 = 2 \times 2 \times 2 \times 3 \times 5$, and...