Divisor Game
Two players are playing a game, alternating turns. There are $k$ piles of stones. On each turn, a player has to choose a pile and replace it...
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John Rhodes
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August 15, 2023
Divisibility of Factorials
The smallest number $m$ such that $10$ divides $m!$ is $m=5$. The smallest number $m$ such that $25$ divides $m!$ is $m=10$. Let $s(n)$ be the smallest...
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John Rhodes
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August 15, 2023
Gozinta Chains
A gozinta chainfor $n$ is a sequence $\{1,a,b,\dots,n\}$ where each element properly divides the next. There are eight gozinta chains for $12$: $\{1,12\}$, $\{1,2,12\}$, $\{1,2,4,12\}$, $\{1,2,6,12\}$, $\{1,3,12\}$,...
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John Rhodes
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August 15, 2023
Distance of Random Points Within Hollow Square Laminae
Assuming that two points are chosen randomly (with uniform distribution) within a rectangle, it is possible to determine the expected value of the distance between these two...
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John Rhodes
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August 15, 2023
The Floor’s Revenge
Define $f_k(n) = \sum_{i=0}^n f_k(\lfloor\frac i k \rfloor)$ where $f_k(0) = 1$ and $\lfloor x \rfloor$ denotes the floor function. For example, $f_5(10) = 18$, $f_7(100) =...
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John Rhodes
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August 15, 2023
Faulhaber’s Formulas
The sum of the $k$th powers of the first $n$ positive integers can be expressed as a polynomial of degree $k+1$ with rational coefficients, the Faulhaber’s Formulas:...
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John Rhodes
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August 15, 2023
Chromatic Conundrum
Let $F(r, c, n)$ be the number of ways to colour a rectangular grid with $r$ rows and $c$ columns using at most $n$ colours such that...
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John Rhodes
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August 15, 2023
Prime-Sum Numbers
Define function $P(n, k) = 1$ if $n$ can be written as the sum of $k$ prime numbers (with repetitions allowed), and $P(n, k) = 0$ otherwise....