Look and Say Sequence
The look and say sequence goes 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, … The sequence starts with 1 and all other members are obtained by...
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John Rhodes
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August 15, 2023
Factorisation Triples
Let $n$ be a positive integer. An integer triple $(a, b, c)$ is called a factorisation triple of $n$ if:$1 \leq a \leq b \leq c$$a \cdot...
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John Rhodes
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August 15, 2023
Reciprocal Cycles II
A unit fraction contains $1$ in the numerator. The decimal representation of the unit fractions with denominators $2$ to $10$ are given: \begin{align} 1/2 &= 0.5\\ 1/3...
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John Rhodes
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August 15, 2023
A Frog’s Trip
A row of $n$ squares contains a frog in the leftmost square. By successive jumps the frog goes to the rightmost square and then back to the...
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John Rhodes
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August 15, 2023
Titanic Sets
A set of lattice points $S$ is called a titanic set if there exists a line passing through exactly two points in $S$. An example of a...
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John Rhodes
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August 15, 2023
Kaprekar Constant
$6174$ is a remarkable number; if we sort its digits in increasing order and subtract that number from the number you get when you sort the digits...
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John Rhodes
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August 15, 2023
One-child Numbers
We say that a $d$-digit positive number (no leading zeros) is a one-child number if exactly one of its sub-strings is divisible by $d$. For example, $5671$...
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John Rhodes
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August 15, 2023
Gnomon Numbering
For integers $m, n$ ($0 \leq n \lt m$), let $L(m, n)$ be an $m \times m$ grid with the top-right $n \times n$ grid removed. For...