Wandering Robots
It was quite an ordinary day when a mysterious alien vessel appeared as if from nowhere. After waiting several hours and receiving no response it is decided...
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John Rhodes
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August 15, 2023
Verifying Primes
Let $q$ be a prime and $A \ge B >0$ be two integers with the following properties: $A$ and $B$ have no prime factor in common, that...
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John Rhodes
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August 15, 2023
Unfair Race
$n$ runners in very different training states want to compete in a race. Each one of them is given a different starting number $k$ $(1\leq k \leq...
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John Rhodes
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August 15, 2023
Idempotent Matrices
A matrix $M$ is called idempotent if $M^2 = M$. Let $M$ be a three by three matrix : $M=\begin{pmatrix} a & b & c\\ d &...
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John Rhodes
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August 15, 2023
Super Pandigital Numbers
A positive number is pandigital in base $b$ if it contains all digits from $0$ to $b – 1$ at least once when written in base $b$....
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John Rhodes
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August 15, 2023
Snowflakes
A snowflake of order $n$ is formed by overlaying an equilateral triangle (rotated by $180$ degrees) onto each equilateral triangle of the same size in a snowflake...
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John Rhodes
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August 15, 2023
Prime Mountain Range
A mountain range consists of a line of mountains with slopes of exactly $45^\circ$, and heights governed by the prime numbers, $p_n$. The up-slope of the $k$th...
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John Rhodes
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August 15, 2023
Reciprocal Games II
Tom has built a random generator that is connected to a row of $n$ light bulbs. Whenever the random generator is activated each of the $n$ lights...