Perfection Quotients
For a positive integer $n$, let $\sigma(n)$ be the sum of all divisors of $n$. For example, $\sigma(6) = 1 + 2 + 3 + 6 =...
-
John Rhodes
-
August 15, 2023
Top Dice
There are $1111$ ways in which five $6$-sided dice (sides numbered $1$ to $6$) can be rolled so that the top three sum to $15$. Some examples...
-
John Rhodes
-
August 15, 2023
Twenty-two Foolish Primes
A set of disks numbered $1$ through $100$ are placed in a line in random order. What is the probability that we have a partial derangement such...
-
John Rhodes
-
August 15, 2023
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
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
Luxury Hampers
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
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
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...