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...
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John Rhodes
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August 16, 2023
Expressing an Integer as the Sum of Triangular Numbers
Gauss famously proved that every positive integer can be expressed as the sum of three triangular numbers (including $0$ as the lowest triangular number). In fact most...
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John Rhodes
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August 16, 2023
Planetary Gears
A circle $C$ of circumference $c$ centimetres has a smaller circle $S$ of circumference $s$ centimetres lying off-centre within it. Four other distinct circles, which we call...
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John Rhodes
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August 16, 2023
Square Subsets
For a set of positive integers $\{a, a+1, a+2, \dots , b\}$, let $C(a,b)$ be the number of non-empty subsets in which the product of all elements...