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Post Archives from the ‘Puzzles’ Category

Factorial plus one

Posted on: January 20th, 2013 by
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Let n be a positive integer.  Prove that n! + 1 is composite for infinitely many values of n. via Nick's Mathematical Puzzles: 111 to 120.

Aces in Bridge Hands

Posted on: January 18th, 2013 by
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Someone deals you a bridge hand 13 cards from a regular deck of 52 cards. You look at the hand and notice you have an Ace and say “I have an Ace”. What is the probability that you have another Ace?The cards are collected and different hand is dealt. This time you look at your hand and state “I have the Ace of Spades” which Continue reading the story "Aces in Bridge Hands"

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

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

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.

Fermat Theorem Puzzle

Posted on: January 12th, 2013 by

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"

Is Axiom of choice correct?

Posted on: January 11th, 2013 by

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

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.

Five Points on a Sphere

Posted on: January 8th, 2013 by
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Given any five points on a sphere, show that there exists a closed hemisphere containing four of them.

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