XOR-Primes
We use $x\oplus y$ for the bitwise XOR of $x$ and $y$. Define the XOR-product of $x$ and $y$, denoted by $x \otimes y$, similar to a...
-
John Rhodes
-
August 16, 2023
Rational Recurrence Relation
The following is a function defined for all positive rational values of $x$. $$ f(x)=\begin{cases} x &x\text{ is integral}\\ f(\frac 1{1-x}) &x \lt 1\\ f\Big(\frac 1{\lceil x\rceil...
-
John Rhodes
-
August 16, 2023
Reversible Prime Squares
Both $169$ and $961$ are the square of a prime. $169$ is the reverse of $961$. We call a number a reversible prime square if: It is...
-
John Rhodes
-
August 16, 2023
Loops of Ropes
Given a circle $C$ and an integer $n > 1$, we perform the following operations. In step $0$, we choose two uniformly random points $R_0$ and $B_0$...
-
John Rhodes
-
August 16, 2023
Nim on Towers of Hanoi
This problem combines the game of Nim with the Towers of Hanoi. For a brief introduction to the rules of these games, please refer to Problem 301...
-
John Rhodes
-
August 16, 2023
Shifted Multiples
For a positive integer $n$, let $s(n)$ be the integer obtained by shifting the leftmost digit of the decimal representation of $n$ to the rightmost position. For...
-
John Rhodes
-
August 16, 2023
Counting Binary Quadratic Representations
Let $g(n)$ denote the number of ways a positive integer $n$ can be represented in the form: $$x^2+xy+41y^2$$ where $x$ and $y$ are integers. For example, $g(53)=4$...
-
John Rhodes
-
August 16, 2023
Pseudorandom Sequence
Rand48 is a pseudorandom number generator used by some programming languages. It generates a sequence from any given integer $0 \le a_0 < 2^{48}$ using the rule...