Squarefree Factors
Consider the number $54$. $54$ can be factored in $7$ distinct ways into one or more factors larger than $1$: $54$, $2 \times 27$, $3 \times 18$,...
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John Rhodes
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August 15, 2023
Subsequence of Thue-Morse Sequence
The Thue-Morse sequence $\{T_n\}$ is a binary sequence satisfying: $T_0 = 0$ $T_{2n} = T_n$ $T_{2n + 1} = 1 – T_n$ The first several terms of...
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John Rhodes
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August 15, 2023
Scary Sphere
Given two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in three dimensional space, the Manhattan distance between those points is defined as$|x_1 – x_2| + |y_1...
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John Rhodes
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August 15, 2023
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...