Scatterstone Nim
Alice and Bob are playing a modified game of Nim called Scatterstone Nim, with Alice going first, alternating turns with Bob. The game begins with an arbitrary...
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John Rhodes
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August 16, 2023
Open Chess Positions
A position in chess is an (orientated) arrangement of chess pieces placed on a chessboard of given size. In the following, we consider all positions in which...
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John Rhodes
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August 16, 2023
Counting Products
Consider the set $S$ of all possible products of $n$ positive integers not exceeding $m$, that is $S=\{ x_1x_2\cdots x_n \mid 1 \le x_1, x_2, \dots, x_n...
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John Rhodes
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August 16, 2023
Counting Binary Matrices
A binary matrix is a matrix consisting entirely of $0$s and $1$s. Consider the following transformations that can be performed on a binary matrix: Swap any two...
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John Rhodes
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August 16, 2023
Gcd Sum
$G(N)=\sum_{j=1}^N\sum_{i=1}^j \gcd(i,j)$. You are given: $G(10)=122$. Find $G(10^{11})$. Give your answer modulo $998244353$. The computation of G(10^11) would be extremely cumbersome to compute directly, it would involve...
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John Rhodes
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August 16, 2023
Two Heads Are Better Than One
An unbiased coin is tossed repeatedly until two consecutive heads are obtained. Suppose these occur on the $(M-1)$th and $M$th toss. Let $P(n)$ be the probability that...
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John Rhodes
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August 16, 2023
Lambda Count
The lambda-calculus is a universal model of computation at the core of functional programming languages. It is based on lambda-terms, a minimal programming language featuring only function...
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John Rhodes
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August 16, 2023
Riffle Shuffles
A riffle shuffle is executed as follows: a deck of cards is split into two equal halves, with the top half taken in the left hand and...