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Monthly Archives for August, 2013

Filling fruits

Posted on: August 31st, 2013 by

How many ways can you find to fill n fruits in a basket subject to the following constraints: 1. The number of apples must be even. 2. The number of bananas must be a multiple of 5. 3. The number of oranges is at most 4. 4. The number of pears is at most 1. via Math Exchange

Infinite prisoners and hats

Posted on: August 30th, 2013 by

Suppose there are infinite prisoners. Each prisoner wears a red or a blue hat. Every prisoner can see every other prisoner's hats. No prisoners know their hat color. Each prisoner can either attempt to guess his hat color or abstain from guessing. However, in order for the prisoners to succeed, an infinite number of prisoners should make a guess and all the prisoners who guess Continue reading the story "Infinite prisoners and hats"

Largest gcd distance

Posted on: August 29th, 2013 by

Consider a directed graph whose nodes are positive integers. There is an edge from a to b if a<b and a is relatively prime to b. Given two integers x,y with x<y, d(x,y) is the number of edges in the shortest path from x to y. What is the largest value of d(x,y)? - via a friend at Packard

Balls in a box

Posted on: August 28th, 2013 by
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There are n balls in a box. Each ball is of a different color. At each iteration, you draw 2 balls at random, and paint the second ball so that it's the same color as the first; then you replace the balls. You keep repeating this until all the balls in the box are of the same color. What is the expected number of iterations? - Continue reading the story "Balls in a box"

Minimum sufficient statistics

Posted on: August 27th, 2013 by
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Let X,Y be discrete finite-valued random variables with joint pmf p(x,y). Let T(x) be a function of X satisfying the Markov chain X-T(X)-Y that has the minimum cardinality of the range, |{T(x):x is in support of X}|. Let f(x) be any other function that satisfies the Markov Chain X-f(X)-Y. Show that T(x) is a function of f(x).

Boat ladder

Posted on: August 26th, 2013 by
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A ladder with 6 rungs is attached to a boat. The rungs are 1 foot apart. The bottom most rung is 1 foot above the surface of the water. During high tide, the water rises 12 inches. How many rungs are above the surface of the water during high tide? - via youtube

A card game with 9 cards

Posted on: August 25th, 2013 by

9 cards numbered 1,2,...9 are placed on a table. You and your friend take turns to pick a card. The first person to own three cards whose sum is 15 wins. If all the cards are exhausted and no one owns 3 cards summing to 15, the game is a draw. Is there a winning strategy for either player? - via Packard 2nd floor white board Continue reading the story "A card game with 9 cards"

A convex puzzle

Posted on: August 24th, 2013 by

Suppose n points are selected independently and uniformly at random on a line segment of length 1. These n points divide the line segment into n+1 segments. What is the probability that it is possible to rearrange these n+1 segments into a convex polygon? - via ST Aditya at Stanford

Fill in numbers

Posted on: August 23rd, 2013 by

Consider the image below. Fill in the circles with numbers from 3 to 9, each number appearing exactly once so that each line sums to 14. puzzleimage Via Math puzzles

Projector illumination 3D

Posted on: August 22nd, 2013 by
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A projector in 3D space can illuminate an octant (it's field of view is just wide enough to illuminate exactly 1 octant if placed at the origin). Show that 8 projectors at 8 arbitrary points in space can be rotated so as to illuminate the entire 3D space. - via Math Puzzles

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