Hilbert’s New Hotel
An infinite number of people (numbered $1$, $2$, $3$, etc.) are lined up to get a room at Hilbert’s newest infinite hotel. The hotel contains an infinite...
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John Rhodes
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August 15, 2023
Cyclic Numbers
A cyclic number with $n$ digits has a very interesting property: When it is multiplied by $1, 2, 3, 4, \dots, n$, all the products have exactly...
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John Rhodes
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August 15, 2023
Prime Generating Integers
Consider the divisors of $30$: $1,2,3,5,6,10,15,30$. It can be seen that for every divisor $d$ of $30$, $d + 30 / d$ is prime. Find the sum...
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John Rhodes
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August 15, 2023
Largest Roots of Cubic Polynomials
Let $a_n$ be the largest real root of a polynomial $g(x) = x^3 – 2^n \cdot x^2 + n$. For example, $a_2 = 3.86619826\cdots$ Find the last...
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John Rhodes
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August 15, 2023
Maximal Coprime Subset
Define $\operatorname{Co}(n)$ to be the maximal possible sum of a set of mutually co-prime elements from $\{1,2,\dots,n\}$. For example $\operatorname{Co}(10)$ is $30$ and hits that maximum on...
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John Rhodes
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August 15, 2023
Distances in a Bee’s Honeycomb
Consider a honey bee’s honeycomb where each cell is a perfect regular hexagon with side length $1$. One particular cell is occupied by the queen bee. For...
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John Rhodes
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August 15, 2023
Risky Moon
A moon could be described by the sphere $C(r)$ with centre $(0,0,0)$ and radius $r$. There are stations on the moon at the points on the surface...
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John Rhodes
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August 15, 2023
Blood Tests
Each one of the $25$ sheep in a flock must be tested for a rare virus, known to affect $2\%$ of the sheep population. An accurate and...