Card Stacking Game
Two players play a game with a deck of cards which contains $s$ suits with each suit containing $n$ cards numbered from $1$ to $n$. Before the...
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John Rhodes
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August 16, 2023
Cyclogenic Polynomials
A monic polynomial is a single-variable polynomial in which the coefficient of highest degree is equal to $1$. Define $\mathcal{F}$ to be the set of all monic...
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John Rhodes
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August 16, 2023
A Grand Shuffle
A standard $52$ card deck comprises thirteen ranks in four suits. However, modern decks have two additional Jokers, which neither have a suit nor a rank, for...
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John Rhodes
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August 16, 2023
Alternating GCD Sum
For a positive integer $n$, the function $g(n)$ is defined as $$\displaystyle g(n)=\sum_{i=1}^{n} (-1)^i \gcd \left(n,i^2\right)$$ For example, $g(4) = -\gcd \left(4,1^2\right) + \gcd \left(4,2^2\right) – \gcd...
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John Rhodes
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August 16, 2023
Seventeen Points
This problem uses half open interval notation where $[a,b)$ represents $a \le x < b$. A real number, $x_1$, is chosen in the interval $[0,1)$. A second...
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John Rhodes
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August 16, 2023
Too Many Twos
We define $\nu_2(n)$ to be the largest integer $r$ such that $2^r$ divides $n$. For example, $\nu_2(24) = 3$. Define $\displaystyle S(n) = \sum_{k = 1}^n (-2)^k\binom{2k}k$...
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John Rhodes
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August 16, 2023
Average and Variance
Denote the average of $k$ numbers $x_1, …, x_k$ by $\bar{x} = \frac{1}{k} \sum_i x_i$. Their variance is defined as $\frac{1}{k} \sum_i \left( x_i – \bar{x} \right)...
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John Rhodes
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August 16, 2023
Clock Grid
There is a grid of length and width 50515093 points. A clock is placed on each grid point. The clocks are all analogue showing a single hour...