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Monthly Archives for January, 2013



Primate Intelligence

Posted on: January 16th, 2013 by
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You walk into a room in which there are three primates: a chimpanzee, an orangutan, and a gorilla. The chimpanzee is holding a banana in each hand, the orangutan is holding a big stick, and the gorilla is holding nothing. Which primate in the room is the smartest? Hint: View the comments for hints. via Wu Riddles

Determinant of binary matrix

Posted on: January 15th, 2013 by
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An N by N matrix M has entries in {0,1} such that all the 1's in a row appear consecutively. Show that determinant of M is -1 or 0 or 1. via ST Aditya

De Bruin Card Trick

Posted on: January 15th, 2013 by
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A dealer(magician) starts with 16 cards on a deck arranged in some particular order. 4 people are seated in a table around the magician. The magician asks each person in turn to cut the deck at any particular point. Then he distributes the top 4 cards one at a time to each person in the table. Then he concentrates for a while. Since the mental Continue reading the story "De Bruin Card Trick"

Points on a circle

Posted on: January 14th, 2013 by
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what is the probability that n points distributed uniformly at random on a circle lie within the same semicircle ? via ST Aditya For hint: View comments. If you have interesting solutions, please post the solution as comments.

Benford's law

Posted on: January 14th, 2013 by
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Benford's law is an interesting anomaly regarding the distribution of the first digit in a collection of data. "Now, we have noted elsewhere that by definition the first digit of a decimal number cannot be a zero.  That leaves nine ciphers, though, and one might expect each of them to appear about one out of nine times in that first position.  For each, about 11% would make sense.  Continue reading the story "Benford's law"

Fermat Theorem Puzzle

Posted on: January 12th, 2013 by
2

A computer scientist claims that he proved somehow that the Fermat theorem is correct for the following 3 numbers:
  • x=2233445566,
  • y=7788990011,
  • z=9988776655
He announces these 3 numbers and calls for a press conference where he is going to present the value of N (to show that

x^N + y^N = z^N

and that the guy from Princeton was wrong). As the press conference starts, a 10-years old boy raises Continue reading the story "Fermat Theorem Puzzle"

Simple proofs of the Isoperimetric Inequality

Posted on: January 12th, 2013 by
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The isoperimetric inequality states that "Among all planar shapes with the same perimeter the circle has the largest area." Here is the simplest geometric proof that I know of: http://www.cut-the-knot.org/do_you_know/isoperimetric.shtml There are other proofs too: http://cornellmath.wordpress.com/2008/05/16/two-cute-proofs-of-the-isoperimetric-inequality/ and http://forumgeom.fau.edu/FG2002volume2/FG200215.pdf Acknowledgement: Thanks to the late prof. Thomas Cover for pointing out a simple proof. Let me know if any of these proofs  can be extended to dimension 3 or higher dimensions.

Is Axiom of choice correct?

Posted on: January 11th, 2013 by
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Puzzle #0: There are n+1 people in a line, and each has a 0/1 number on his hat. Each player can look to the numbers of the players in front of him. So, if x_i is the number of player i, then player i knows x_{i+1},\hdots,x_n. Now, from Continue reading the story "Is Axiom of choice correct?"

Beat your friend at a casino

Posted on: January 10th, 2013 by
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Suppose your friend invests $1 in a casino and earns $X, where X is a Random variable with mean 1, i.e. E[X]=1. Suppose you know the cdf F(x) of the random variable X. Find the cdf of a random variable Y with mean 1 (E[Y]=1) so as to maximize \Pr\{Y>=X\}. For hint: View comments on this article. Acknowledgement: Thanks to the late prof. Thomas Cover for sharing Continue reading the story "Beat your friend at a casino"

Yue's "swap" problem

Posted on: January 8th, 2013 by
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Given a regular 13-gon, a red-blue coloring of the vertices is said to be "symmetric" if there exists an axis of symmetry which passes through the center of the 13-gon. A "swap" is an operation in which we exchange the colors of any two vertices.  I claim that any red-blue coloring of the vertices is at most one swap away from being symmetric. Prove/disprove my claim.

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