$n$-sequences
A sequence of integers $S = \{s_i\}$ is called an $n$-sequence if it has $n$ elements and each element $s_i$ satisfies $1 \leq s_i \leq n$. Thus...
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John Rhodes
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August 15, 2023
Box-Ball System
Consider an infinite row of boxes. Some of the boxes contain a ball. For example, an initial configuration of 2 consecutive occupied boxes followed by 2 empty...
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John Rhodes
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August 15, 2023
Prime Connection
Two positive numbers $A$ and $B$ are said to be connected (denoted by “$A \leftrightarrow B$”) if one of these conditions holds: (1) $A$ and $B$ have...
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John Rhodes
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August 15, 2023
Kakuro
The above is an example of a cryptic kakuro (also known as cross sums, or even sums cross) puzzle, with its final solution on the right. (The...
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John Rhodes
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August 15, 2023
Consecutive Die Throws
Let $n$ be a positive integer. A 6-sided die is thrown $n$ times. Let $c$ be the number of pairs of consecutive throws that give the same...
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John Rhodes
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August 15, 2023
Sequence of Points on a Hyperbola
Let $H$ be the hyperbola defined by the equation $12x^2 + 7xy – 12y^2 = 625$. Next, define $X$ as the point $(7, 1)$. It can be...
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John Rhodes
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August 15, 2023
Prime Factors of $n^{15}+1$
Numbers of the form $n^{15}+1$ are composite for every integer $n \gt 1$. For positive integers $n$ and $m$ let $s(n,m)$ be defined as the sum of...
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John Rhodes
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August 15, 2023
$2 \times 2$ Positive Integer Matrix
A positive integer matrix is a matrix whose elements are all positive integers. Some positive integer matrices can be expressed as a square of a positive integer...