123 Numbers
We define 123-numbers as follows: 1 is the smallest 123-number. When written in base 10 the only digits that can be present are “1”, “2” and “3”...
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John Rhodes
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August 16, 2023
Randomly Decaying Sequence
Given a fixed real number $c$, define a random sequence $(X_n)_{n\ge 0}$ by the following random process: $X_0 = c$ (with probability 1). For $n>0$, $X_n =...
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John Rhodes
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August 16, 2023
Mahjong
The game of Mahjong is played with tiles belonging to $s$ suits. Each tile also has a number in the range $1\ldots n$, and for each suit/number...
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John Rhodes
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August 16, 2023
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...