Bozo Sort
Bozo sort, not to be confused with the slightly less efficient bogo sort, consists out of checking if the input sequence is sorted and if not swapping...
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John Rhodes
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August 15, 2023
Stone Game III
Two players, Anton and Bernhard, are playing the following game. There is one pile of $n$ stones. The first player may remove any positive number of stones,...
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John Rhodes
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August 15, 2023
A Huge Binomial Coefficient
The binomial coefficient $\displaystyle{\binom{10^{18}}{10^9}}$ is a number with more than $9$ billion ($9\times 10^9$) digits. Let $M(n,k,m)$ denote the binomial coefficient $\displaystyle{\binom{n}{k}}$ modulo $m$. Calculate $\displaystyle{\sum M(10^{18},10^9,p\cdot...
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John Rhodes
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August 15, 2023
Comfortable Distance
There are $N$ seats in a row. $N$ people come after each other to fill the seats according to the following rules: If there is any seat...
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John Rhodes
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August 15, 2023
Bézier Curves
A cubic Bézier curve is defined by four points: $P_0, P_1, P_2,$ and $P_3$. The curve is constructed as follows: On the segments $P_0 P_1$, $P_1 P_2$,...
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John Rhodes
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August 15, 2023
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...