An Infinite Game
Peter is playing a solitaire game on an infinite checkerboard, each square of which can hold an unlimited number of tokens. Each move of the game consists...
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John Rhodes
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August 16, 2023
Sums of Subarrays
Let $t_k$ be the tribonacci numbers defined as: $\quad t_0 = t_1 = 0$; $\quad t_2 = 1$; $\quad t_k = t_{k-1} + t_{k-2} + t_{k-3} \quad...
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John Rhodes
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August 16, 2023
Fibonacci Paths
Alice walks on a lattice grid. She can step from one lattice point $A (a,b)$ to another $B (a+x,b+y)$ providing distance $AB = \sqrt{x^2+y^2}$ is a Fibonacci...
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John Rhodes
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August 16, 2023
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...