Integer Ladders
In the classic “Crossing Ladders” problem, we are given the lengths $x$ and $y$ of two ladders resting on the opposite walls of a narrow, level street....
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John Rhodes
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August 15, 2023
An Amazing Prime-generating Automaton
A program written in the programming language Fractran consists of a list of fractions. The internal state of the Fractran Virtual Machine is a positive integer, which...
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John Rhodes
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August 15, 2023
Chip Defects
$k$ defects are randomly distributed amongst $n$ integrated-circuit chips produced by a factory (any number of defects may be found on a chip and each defect is...
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John Rhodes
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August 15, 2023
Paper-strip Game
The following game is a classic example of Combinatorial Game Theory: Two players start with a strip of $n$ white squares and they take alternate turns. On...
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John Rhodes
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August 15, 2023
Reflexive Position
Let’s call $S$ the (infinite) string that is made by concatenating the consecutive positive integers (starting from $1$) written down in base $10$. Thus, $S = 1234567891011121314151617181920212223242\cdots$...
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John Rhodes
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August 15, 2023
Primonacci
For any positive integer $n$ the function $\operatorname{next\_prime}(n)$ returns the smallest prime $p$ such that $p \gt n$. The sequence $a(n)$ is defined by: $a(1)=\operatorname{next\_prime}(10^{14})$ and $a(n)=\operatorname{next\_prime}(a(n-1))$...
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John Rhodes
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August 15, 2023
Multiples with Small Digits
For a positive integer $n$, define $f(n)$ as the least positive multiple of $n$ that, written in base $10$, uses only digits $\le 2$. Thus $f(2)=2$, $f(3)=12$,...
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John Rhodes
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August 15, 2023
Strong Achilles Numbers
A positive integer $n$ is powerful if $p^2$ is a divisor of $n$ for every prime factor $p$ in $n$. A positive integer $n$ is a perfect...