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...
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John Rhodes
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August 16, 2023
Distinct Rows and Columns
The complexity of an $n\times n$ binary matrix is the number of distinct rows and columns. For example, consider the $3\times 3$ matrices $$ \mathbf{A} = \begin{pmatrix}...
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John Rhodes
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August 16, 2023
Feynman Diagrams
Let $F(n)$ be the number of connected graphs with blue edges (directed) and red edges (undirected) containing: two vertices of degree 1, one with a single outgoing...
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John Rhodes
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August 16, 2023
Toriangulations
For positive real numbers $a,b$, an $a\times b$ torus is a rectangle of width $a$ and height $b$, with left and right sides identified, as well as...