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

## Monthly Archives for August, 2013

## Infinite prisoners and hats

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"## Balls in a box

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?
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**Continue reading the story**"Balls in a box"## Minimum sufficient statistics

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

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?
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## A card game with 9 cards

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"## Fill in numbers

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.
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