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...
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John Rhodes
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August 16, 2023
Amoebas in a 3D Grid
Consider a three dimensional grid of cubes. An amoeba in cube $(x, y, z)$ can divide itself into three amoebas to occupy the cubes $(x + 1,...
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John Rhodes
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August 16, 2023
Amoebas in a 2D Grid
Consider a two dimensional grid of squares. The grid has 4 rows but infinitely many columns. An amoeba in square $(x, y)$ can divide itself into two...
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John Rhodes
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August 16, 2023
Runner and Swimmer
Two friends, a runner and a swimmer, are playing a sporting game: The swimmer is swimming within a circular pool while the runner moves along the pool...