Random Rectangles
Three points, $P_1$, $P_2$ and $P_3$, are randomly selected within a unit square. Consider the three rectangles with sides parallel to the sides of the unit square...
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John Rhodes
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August 16, 2023
Cube-full Divisors
A positive integer $n$ is considered cube-full, if for every prime $p$ that divides $n$, so does $p^3$. Note that $1$ is considered cube-full. Let $s(n)$ be...
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John Rhodes
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August 16, 2023
Finite Sequence Generator
Two positive integers $x$ and $y$ ($x > y$) can generate a sequence in the following manner: $a_x = y$ is the first term, $a_{z+1} = a_z^2...
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John Rhodes
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August 16, 2023
Siegbert and Jo
Siegbert and Jo take turns playing a game with a heap of $N$ pebbles: 1. Siegbert is the first to take some pebbles. He can take as...
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John Rhodes
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August 16, 2023
Tom and Jerry
Tom (the cat) and Jerry (the mouse) are playing on a simple graph $G$. Every vertex of $G$ is a mousehole, and every edge of $G$ is...
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John Rhodes
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August 16, 2023
Binary Series
For $0 \le x \lt 1$, define $d_i(x)$ to be the $i$th digit after the binary point of the binary representation of $x$. For example $d_2(0.25) =...
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John Rhodes
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August 16, 2023
Piles of Plates
We stack $n$ plates into $k$ non-empty piles where each pile is a different size. Define $f(n,k)$ to be the maximum number of plates possible in the...
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John Rhodes
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August 16, 2023
Shuffling Cards
A standard deck of $52$ playing cards, which consists of thirteen ranks (Ace, Two, …, Ten, King, Queen and Jack) each in four suits (Clubs, Diamonds, Hearts...