Pythagorean Triple Occurrence
Define $Q(n)$ to be the smallest number that occurs in exactly $n$ Pythagorean triples $(a,b,c)$ where $a \lt b \lt c$. For example, $15$ is the smallest...
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John Rhodes
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August 16, 2023
Birds on a Wire
Consider a wire of length 1 unit between two posts. Every morning $n$ birds land on it randomly with every point on the wire equally likely to...
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John Rhodes
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August 16, 2023
Chasing Game
Two cars are on a circular track of total length $2n$, facing the same direction, initially distance $n$ apart. They move in turn. At each turn, the...
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John Rhodes
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August 16, 2023
Chess Sliders
A Slider is a chess piece that can move one square left or right. This problem uses a cylindrical chess board where the left hand edge of...
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John Rhodes
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August 16, 2023
Factor Shuffle
A list initially contains the numbers $2, 3, \dots, n$. At each round, every number in the list is divided by its smallest prime factor. Then the...
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John Rhodes
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August 16, 2023
Square the Smallest
A list initially contains the numbers $2, 3, \dots, n$. At each round, the smallest number in the list is replaced by its square. If there is...
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John Rhodes
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August 16, 2023
123-Separable
A set, $S$, of integers is called 123-separable if $S$, $2S$ and $3S$ are disjoint. Here $2S$ and $3S$ are obtained by multiplying all the elements in...
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John Rhodes
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August 16, 2023
$N$th Digit of Reciprocals
Let $d_n(x)$ be the $n$th decimal digit of the fractional part of $x$, or $0$ if the fractional part has fewer than $n$ digits. For example: $d_7...