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:...
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John Rhodes
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August 16, 2023
The King’s Banquet
The Knights of the Order of Fibonacci are preparing a grand feast for their king. There are $n$ knights, and each knight is assigned a distinct number...
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John Rhodes
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August 16, 2023
Square Root Smooth Numbers
A positive integer is called square root smooth if all of its prime factors are strictly less than its square root. Including the number $1$, there are...
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John Rhodes
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August 16, 2023
Moving Pentagon
After buying a Gerver Sofa from the Moving Sofa Company, Jack wants to buy a matching cocktail table from the same company. Most important for him is...
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John Rhodes
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August 16, 2023
Polymorphic Bacteria
Members of a species of bacteria occur in two different types: $\alpha$ and $\beta$. Individual bacteria are capable of multiplying and mutating between the types according to...
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John Rhodes
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August 16, 2023
Proportionate Nim
Two players play a game with two piles of stones, alternating turns. On each turn, the corresponding player chooses a positive integer $n$ and does one of...