Product of Gauss Factorials
The Gauss Factorial of a number $n$ is defined as the product of all positive numbers $\leq n$ that are relatively prime to $n$. For example $g(10)=1\times...
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John Rhodes
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August 16, 2023
Fermat Equation
Fermat’s Last Theorem states that no three positive integers $a$, $b$, $c$ satisfy the equation $$a^n+b^n=c^n$$ for any integer value of $n$ greater than 2. For this...
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John Rhodes
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August 16, 2023
Powers of $1+\sqrt 7$
When $(1+\sqrt 7)$ is raised to an integral power, $n$, we always get a number of the form $(a+b\sqrt 7)$. We write $(1+\sqrt 7)^n = \alpha(n) +...
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John Rhodes
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August 16, 2023
Concatenation Coincidence
A non-decreasing sequence of integers $a_n$ can be generated from any positive real value $\theta$ by the following procedure: \begin{align} \begin{split} b_1 &= \theta \\ b_n &=...
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John Rhodes
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August 16, 2023
Optimal Card Stacking
Card Stacking is a game on a computer starting with an array of $N$ cards labelled $1,2,\ldots,N$. A stack of cards can be moved by dragging horizontally...
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John Rhodes
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August 16, 2023
Near Power Sums
A positive integer, $n$, is a near power sum if there exists a positive integer, $k$, such that the sum of the $k$th powers of the digits...
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John Rhodes
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August 16, 2023
Triangular Pizza
Mamma Triangolo baked a triangular pizza. She wants to cut the pizza into $n$ pieces. She first chooses a point $P$ in the interior (not boundary) of...
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John Rhodes
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August 16, 2023
A Messy Dinner
$n$ families, each with four members, a father, a mother, a son and a daughter, were invited to a restaurant. They were all seated at a large...