Flipping Game
The flipping game is a two player game played on an $N$ by $N$ square board. Each square contains a disk with one side white and one...
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John Rhodes
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August 15, 2023
Permutations of Project
Consider the alphabet $A$ made out of the letters of the word “$\text{project}$”: $A=\{\text c,\text e,\text j,\text o,\text p,\text r,\text t\}$. Let $T(n)$ be the number of...
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John Rhodes
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August 15, 2023
A Polynomial Modulo the Square of a Prime
Let $f(n) = n^2 – 3n – 1$. Let $p$ be a prime. Let $R(p)$ be the smallest positive integer $n$ such that $f(n) \bmod p^2 =...
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John Rhodes
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August 15, 2023
Triangles Containing the Origin II
Define:$x_n = (1248^n \bmod 32323) – 16161$$y_n = (8421^n \bmod 30103) – 15051$ $P_n = \{(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\}$ For example, $P_8 = \{(-14913,...
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John Rhodes
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August 15, 2023
Diophantine Reciprocals III
In the following equation $x$, $y$, and $n$ are positive integers. $$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$ For a limit $L$ we define $F(L)$ as the number of...
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John Rhodes
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August 15, 2023
Lattice Quadrilaterals
A simple quadrilateral is a polygon that has four distinct vertices, has no straight angles and does not self-intersect. Let $Q(m, n)$ be the number of simple...
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John Rhodes
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August 15, 2023
Long Products
Define $F(m,n)$ as the number of $n$-tuples of positive integers for which the product of the elements doesn’t exceed $m$. $F(10, 10) = 571$. $F(10^6, 10^6) \bmod...
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John Rhodes
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August 15, 2023
Modular Inverses
Consider the number $15$. There are eight positive numbers less than $15$ which are coprime to $15$: $1, 2, 4, 7, 8, 11, 13, 14$. The modular...