Ruff Numbers
Let $S_k$ be the set containing $2$ and $5$ and the first $k$ primes that end in $7$. For example, $S_3 = \{2,5,7,17,37\}$. Define a $k$-Ruff number...
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John Rhodes
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August 16, 2023
Balanceable $k$-bounded Partitions
A $k$-bounded partition of a positive integer $N$ is a way of writing $N$ as a sum of positive integers not exceeding $k$. A balanceable partition is...
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John Rhodes
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August 16, 2023
Pseudo Geometric Sequences
We define a pseudo-geometric sequence to be a finite sequence $a_0, a_1, \dotsc, a_n$ of positive integers, satisfying the following conditions: $n \geq 4$, i.e. the sequence...
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John Rhodes
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August 16, 2023
Delphi Flip
A and B play a game. A has originally $1$ gram of gold and B has an unlimited amount. Each round goes as follows: A chooses and...
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John Rhodes
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August 16, 2023
Binary Quadratic Form II
Consider the following binary quadratic form: $$ \begin{align} f(x,y)=x^2+5xy+3y^2 \end{align} $$ A positive integer $q$ has a primitive representation if there exist positive integers $x$ and $y$...
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John Rhodes
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August 16, 2023
Chandelier
A certain type of chandelier contains a circular ring of $n$ evenly spaced candleholders. If only one candle is fitted, then the chandelier will be imbalanced. However,...
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John Rhodes
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August 16, 2023
Window into a Matrix II
A window into a matrix is a contiguous sub matrix. Consider a $16\times n$ matrix where every entry is either 0 or 1. Let $B(k,n)$ be the...
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John Rhodes
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August 16, 2023
Trillionaire
Starting with 1 gram of gold you play a game. Each round you bet a certain amount of your gold: if you have $x$ grams you can...