A Long Chess Match
Two friends $A$ and $B$ are great fans of Chess. They both enjoy playing the game, but after each game the player who lost the game would...
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John Rhodes
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August 16, 2023
Largest Prime
Consider the sequence $n^2+3$ with $n \ge 1$. If we write down the first terms of this sequence we get: $4, 7, 12, 19, 28, 39, 52,...
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John Rhodes
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August 16, 2023
Incomplete Words II
In the context of formal languages, any finite sequence of letters of a given alphabet $\Sigma$ is called a word over $\Sigma$. We call a word incomplete...
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John Rhodes
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August 16, 2023
Incomplete Words
In the context of formal languages, any finite sequence of letters of a given alphabet $\Sigma$ is called a word over $\Sigma$. We call a word incomplete...
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John Rhodes
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August 16, 2023
Palindromic Sequences
Given an irrational number $\alpha$, let $S_\alpha(n)$ be the sequence $S_\alpha(n)=\lfloor {\alpha \cdot n} \rfloor – \lfloor {\alpha \cdot (n-1)} \rfloor$ for $n \ge 1$. ($\lfloor \cdots...
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John Rhodes
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August 16, 2023
Divisible Palindromes
The numbers $545$, $5\,995$ and $15\,151$ are the three smallest palindromes divisible by $109$. There are nine palindromes less than $100\,000$ which are divisible by $109$. How...
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John Rhodes
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August 16, 2023
Neighbourly Constraints
Let $T(n, m)$ be the number of $m$-tuples of positive integers such that the sum of any two neighbouring elements of the tuple is $\le n$. For...
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John Rhodes
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August 16, 2023
Frictionless Tube
Consider a horizontal frictionless tube with length $L$ millimetres, and a diameter of 20 millimetres. The east end of the tube is open, while the west end...