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...
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John Rhodes
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August 16, 2023
Sum over Bitwise Operators
Define $$\displaystyle g(m,n) = (m\oplus n)+(m\vee n)+(m\wedge n)$$ where $\oplus, \vee, \wedge$ are the bitwise XOR, OR and AND operator respectively. Also set $$\displaystyle G(N) = \sum_{n=0}^N\sum_{k=0}^n...
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John Rhodes
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August 16, 2023
A Squared Recurrence Relation
The function $f$ is defined for all positive integers as follows: \begin{align*} f(1) &= 1\\ f(2n) &= 2f(n)\\ f(2n+1) &= 2n+1 + 2f(n)+\tfrac 1n f(n) \end{align*} It...
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John Rhodes
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August 16, 2023
Buckets of Water
There are 3 buckets labelled $S$ (small) of 3 litres, $M$ (medium) of 5 litres and $L$ (large) of 8 litres. Initially $S$ and $M$ are full...
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John Rhodes
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August 16, 2023
Stealthy Numbers
A positive integer $N$ is stealthy, if there exist positive integers $a$, $b$, $c$, $d$ such that $ab = cd = N$ and $a+b = c+d+1$. For...
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John Rhodes
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August 16, 2023
Not Zeckendorf
Consider the Fibonacci sequence $\{1,2,3,5,8,13,21,\ldots\}$. We let $f(n)$ be the number of ways of representing an integer $n\ge 0$ as the sum of different Fibonacci numbers. For...