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Graduate
By
Shiva Kintali
on Aug. 29, 2012 | Updated Jan. 16, 2017
Random Intervals
There are \(n\) points on a line. These points are paired up at random to form \(n/2\) intervals. Prove that the probability that among these intervals there is one which intersects all the others i…
Mathematics
Puzzles
Probability
Puzzles
counting
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High School
By
Shiva Kintali
on May 19, 2013 | Updated Jan. 16, 2017
Basics of counting
Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of \(S\) are there that satisfy \(A \subseteq B\) ?
Mathematics
Combinatorics
basics
counting
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Graduate
By
Shiva Kintali
on Aug. 8, 2012 | Updated Jan. 16, 2017
Gallai Identities
Consider the following parameters of an undirected graph \(G\) on \(n\) vertices. \(\nu(G)\) is the size of a maximum matching of \(G\). \(\tau(G)\) is the size of a minimum vertex cover of \(G\). …
Mathematics
Graph Theory
edge cover
independent set
matching
vertex cover
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High School
By
Shiva Kintali
on Aug. 1, 2012 | Updated Jan. 16, 2017
Two Secret Integers
I am thinking of two integers, each of them is more than 1 and their sum is less than 100. I tell my friend A the sum of these two numbers, and another friend B, product of these two numbers. Then the…
Puzzles
Puzzles
math puzzle
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High School
By
TrueShelf Inc.
on Sept. 27, 2013 | Updated Jan. 16, 2017
Color of the last ball
In a large urn there are \(w\) white balls and \(b\) black balls. Beside the urn is a big pile (infinite number) of black balls. Now, we do the following. First, two balls are drawn at random from the…
Puzzles
Puzzles
invariant
0
Undergraduate
By
Vibhav Pant
on March 4, 2014 | Updated Jan. 16, 2017
Prove $\sum_{z=0}^{\infty}(\zeta(z)-1)=1$
Riemann's zeta function \(\zeta(z)\) is defined as \(\zeta(z)=1+\frac{1}{2^z}+\frac{1}{3^z}+\cdots=\sum_{k=0}^{\infty}\frac{1}{k^z}\) Prove that \(\sum_{z=0}^{\infty}(\zeta(z)-1)=1\)
Mathematics
Discrete Mathematics
sums
0
High School
By
Shiva Kintali
on June 10, 2012 | Updated Jan. 16, 2017
Find the faulty Ball
There are 12 balls. They all look alike but one of them is faulty; it weights differently. It is not known, if this ball is heavier or lighter than the other balls. How do you find the faulty ball by …
Puzzles
Puzzles
counting
interview question
0
Undergraduate
By
Shiva Kintali
on June 9, 2012 | Updated Jan. 16, 2017
Linear inequalities satisfied with equality
Let \(Ax \leq b\) be a system of linear inequalities. Describe a linear program to determine which inequalities among \(Ax \leq b\) are always satisfied with equality.
Optimization
Linear Programming
optimization
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Undergraduate
By
Shiva Kintali
on Nov. 6, 2012 | Updated Jan. 16, 2017
Guessing Right
Players 1 and 2 each choose a member of the set \({1,2,\dots,K}\). If the players choose the same number then player 2 pays \(\\)1$ to player 1; otherwise no payment is made. Each player maximizes his…
Mathematics
Game Theory
nash equilibrium
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Murphy's Law
Let \(A_1, A_2, \ldots ,A_n\) be independent events, and let \(T\) be the number of these events that occur. Show that the probability that none of the events occur is at most \(e^{-E[T]}\), where…
Mathematics
Probability
expectation
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