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 $B_0$...
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John Rhodes
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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...
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John Rhodes
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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...
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John Rhodes
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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, $g(53)=4$...
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John Rhodes
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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...
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John Rhodes
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August 16, 2023
Iterated Composition
Let $\Bbb R^2$ be the set of pairs of real numbers $(x, y)$. Let $\pi = 3.14159\cdots\ $. Consider the function $f$ from $\Bbb R^2$ to $\Bbb...
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John Rhodes
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August 16, 2023
Hybrid Integers
An integer of the form $p^q q^p$ with prime numbers $p \neq q$ is called a hybrid-integer. For example, $800 = 2^5 5^2$ is a hybrid-integer. We...
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John Rhodes
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August 16, 2023
Pentagonal Puzzle
Pentagonal numbers are generated by the formula: $P_n = \tfrac 12n(3n-1)$ giving the sequence: $$1,5,12,22,35, 51,70,92,\ldots $$ Some pentagonal numbers can be expressed as the sum of...