Iterative Sampling
Given an $n$-tuple of numbers another $n$-tuple is created where each element of the new $n$-tuple is chosen randomly from the numbers in the previous $n$-tuple. For...
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John Rhodes
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August 16, 2023
SET
The SET® card game is played with a pack of $81$ distinct cards. Each card has four features (Shape, Color, Number, Shading). Each feature has three different...
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John Rhodes
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August 16, 2023
Digits in Squares
Define $m = M(n, d)$ to be the smallest positive integer such that when $m^2$ is written in base $n$ it includes the base $n$ digit $d$....
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John Rhodes
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August 16, 2023
Shortest Distance Among Points
We create an array of points $P_n$ in a two dimensional plane using the following random number generator: $s_0=290797$ $s_{n+1}={s_n}^2 \bmod 50515093$ $P_n=(s_{2n},s_{2n+1})$ Let $d(k)$ be the...
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John Rhodes
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August 16, 2023
Group by Value
A pack of cards contains $4n$ cards with four identical cards of each value. The pack is shuffled and cards are dealt one at a time and...
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John Rhodes
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August 16, 2023
Mezzo-forte
$4n$ people stand in a circle with their heads down. When the bell rings they all raise their heads and either look at the person immediately to...
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John Rhodes
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August 16, 2023
XOR-Powers
We use $x\oplus y$ to be the bitwise XOR of $x$ and $y$. Define the XOR-product of $x$ and $y$, denoted by $x \otimes y$, similar to...
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John Rhodes
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August 16, 2023
Bitwise Recursion
Let $b(n)$ be the largest power of 2 that divides $n$. For example $b(24) = 8$. Define the recursive function: \begin{align*} \begin{split} A(0) &= 1\\ A(2n) &=...