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...
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John Rhodes
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August 16, 2023
Minimal Pairing Modulo $p$
Given an odd prime $p$, put the numbers $1,…,p-1$ into $\frac{p-1}{2}$ pairs such that each number appears exactly once. Each pair $(a,b)$ has a cost of $ab...
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John Rhodes
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August 16, 2023
Dominating Numbers
A dominating number is a positive integer that has more than half of its digits equal. For example, $2022$ is a dominating number because three of its...
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John Rhodes
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August 16, 2023
Bézout’s Game
Two players play a game with two piles of stones. They take alternating turns. If there are currently $a$ stones in the first pile and $b$ stones...
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John Rhodes
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August 16, 2023
Billiard
The following diagram shows a billiard table of a special quadrilateral shape. The four angles $A, B, C, D$ are $120^\circ, 90^\circ, 60^\circ, 90^\circ$ respectively, and the...
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John Rhodes
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August 16, 2023
Reciprocal Pairs
Let’s call a pair of positive integers $p$, $q$ ($p \lt q$) reciprocal, if there is a positive integer $r\lt p$ such that $r$ equals both the...