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Monthly Archives for March, 2013

Are you good at arithmetic?

Posted on: March 21st, 2013 by
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What are the first 10 decimal digits in the decimal representation of the fractional part of (1+\sqrt{2})^{2013}? Note: The fractional part of a non-negative real number is the part of the number that appears after the decimal point.

Setting an exam paper

Posted on: March 20th, 2013 by
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There's a class of 60 students taking a class. The professor wants to give a lot of flexibility to the students in choosing their final exam paper. Each student must list 6 topics of his/her choice. Further, he must declare one of 2 different days on which he wants to take the finals. The professor has exactly 1 question per topic. The professor is free Continue reading the story "Setting an exam paper"

A trigonometric identity

Posted on: March 19th, 2013 by
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Prove that

\prod_{k=1}^{\infty}\cos \left(\frac{\pi x}{2^k} \right) = \frac{\sin \pi x}{\pi x}

via ST Aditya Hint: ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Use Nyquist sampling theorem for integer samples.

Winning at Wimbledon

Posted on: March 18th, 2013 by

As a result of temporary magical powers, you have made it to the Wimbledon finals and are playing Roger Federer for all the marbles. However, your powers cannot last the whole match. What score do you want it to be when they disappear, to maximize your chances of hanging on for a win?

Boxes in boxes

Posted on: March 17th, 2013 by
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At many train stations, post offices and currier services around the world, the cost of sending a rectangular box is determined by the sum of its dimensions; that is, length plus width plus height. Prove that you can’t “cheat” by packing a box into a cheaper box.

Zebra crossing river

Posted on: March 16th, 2013 by
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A zebra was going to the river. On the way the zebra encountered five giraffes. Each giraffe had five monkeys on their neck. How many animals were going to the river? via Brain Teasers. Answer: ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Just one - the zebra. The other 30 animals (5 giraffes and 25 monkeys) were going the other way.

Weighing coins

Posted on: March 15th, 2013 by
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There are nine coins, one real and eight fake. Four of the fake coins weigh the same and are lighter than the real coin. The other four fake coins weigh the same and are heavier than the real coin. Find the real coin in seven weighings on the balance scale (In fact, it is possible to do so in only 6 weighings). - via Tanya's Continue reading the story "Weighing coins"

Sampling uniformly at random from a stream

Posted on: March 14th, 2013 by
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Suppose x_1,x_2,\dots x_t, \dots is a stream of objects. The goal is to produce a stream y_1,y_2,\dots y_t\dots such that at each time t\geq 1, y_t is uniformly distributed among x_1,x_2\dots x_t. However you are allowed to use only O(1) memory. How can this be achieved? Hint: You don't have to make y's independent of each other. -- Via prof. Ashish Goel's class on randomized algorithms.

Chocolate blocks

Posted on: March 13th, 2013 by

You are given an m×n block of chocolate which you wish to break into mn unit squares. At each step, you may pick up one piece of chocolate and break it into two pieces along a straight line. What is the minimum number of steps required? via Australian Mathematical Society

Graphing equations

Posted on: March 12th, 2013 by
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This series of puzzles attempt to develop intuition regarding graphing equations. First question: Plot the curve y={1\over x^{10}+1}. Guess it's shape intuitively. Hint: 0.5^{10}\approx 0.001. Second question: Plot the 3-d surface z={1\over x^{10}+y^{10}+1}. Hint: Use the first question. Third question: Plot the 3-d surface z={1\over x^{10}+\left(y-\sin(2\pi x)\right)^{10}+1} via prof. Ernst Boyd, an amazing lecturer and a juggler.

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