Fermat-like Equations
If a triple of positive integers $(a, b, c)$ satisfies $a^2+b^2=c^2$, it is called a Pythagorean triple. No triple $(a, b, c)$ satisfies $a^e+b^e=c^e$ when $e \ge...
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John Rhodes
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August 16, 2023
Coloured Graphs
Let $g(n)$ be the number of undirected graphs with $n$ nodes satisfying the following properties: The graph is connected and has no cycles or multiple edges. Each...
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John Rhodes
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August 16, 2023
Matching Digit Sums
Let $d(i,b)$ be the digit sum of the number $i$ in base $b$. For example $d(9,2)=2$, since $9=1001_2$. When using different bases, the respective digit sums most...
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John Rhodes
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August 16, 2023
$2^{\omega(n)}$
Let $\omega(n)$ denote the number of distinct prime divisors of a positive integer $n$. So $\omega(1) = 0$ and $\omega(360) = \omega(2^{3} \times 3^{2} \times 5) =...
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John Rhodes
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August 16, 2023
Beds and Desks
At Euler University, each of the $n$ students (numbered from 1 to $n$) occupies a bed in the dormitory and uses a desk in the classroom. Some...
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John Rhodes
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August 16, 2023
One More One
Consider the following process that can be applied recursively to any positive integer $n$: if $n = 1$ do nothing and the process stops, if $n$ is...
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John Rhodes
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August 16, 2023
Colouring a Loop
A certain type of flexible tile comes in three different sizes – $1 \times 1$, $1 \times 2$, and $1 \times 3$ – and in $k$ different...
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John Rhodes
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August 16, 2023
Colouring a Strip
A certain type of tile comes in three different sizes – $1 \times 1$, $1 \times 2$, and $1 \times 3$ – and in four different colours:...