Hexagonal Orchards
A hexagonal orchard of order $n$ is a triangular lattice made up of points within a regular hexagon with side $n$. The following is an example of...
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John Rhodes
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August 15, 2023
Constraining the Least Greatest and the Greatest Least
A list of size $n$ is a sequence of $n$ natural numbers. Examples are $(2,4,6)$, $(2,6,4)$, $(10,6,15,6)$, and $(11)$. The greatest common divisor, or $\gcd$, of a...
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John Rhodes
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August 15, 2023
Langton’s Ant
An ant moves on a regular grid of squares that are coloured either black or white. The ant is always oriented in one of the cardinal directions...
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John Rhodes
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August 15, 2023
Sum of a Square and a Cube
Many numbers can be expressed as the sum of a square and a cube. Some of them in more than one way. Consider the palindromic numbers that...
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John Rhodes
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August 15, 2023
Largest Integer Divisible by Two Primes
The largest integer $\le 100$ that is only divisible by both the primes $2$ and $3$ is $96$, as $96=32\times 3=2^5 \times 3$. For two distinct primes...
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John Rhodes
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August 15, 2023
Strong Repunits
The number $7$ is special, because $7$ is $111$ written in base $2$, and $11$ written in base $6$ (i.e. $7_{10} = 11_6 = 111_2$). In other...
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John Rhodes
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August 15, 2023
Matrix Sum
We define the Matrix Sum of a matrix as the maximum possible sum of matrix elements such that none of the selected elements share the same row...
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John Rhodes
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August 15, 2023
Silver Dollar Game
One variant of N.G. de Bruijn’s silver dollar game can be described as follows: On a strip of squares a number of coins are placed, at most...