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Problem of the Day #408: Rolling Dice April 30, 2012

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Find the expected number of rolls of a fair six-sided die before the sequence of rolls contains $1, 2, 3, 4, 5, 6$ (in that order) as a subsequence.

Problem of the Day #406: True or False April 28, 2012

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Albert is taking a test where each of the $1000$ questions has one of two answers: true or false. Having not prepared for the test, he does not know the answer to a single problem and resorts to intelligent guessing. He knows that the number of questions on the test with answer true is a power of two, with each power of two having an equal probability and each possible distribution being equally likely for a certain true count. Assuming Albert guesses optimally, how many questions is he expected to answer correctly?

Problem of the Day #394: Permutations with Shared Elements April 16, 2012

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Find the probability that two random permutations of the integers from $1$ to $N$ share exactly one element.

Problem of the Day #390: Pawn-Only Chess Game April 12, 2012

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Suppose you are playing a game of chess where only pawn moves are allowed. Find the number of possible positions after 3 moves from each player.

Problem of the Day #383: Infinite Square Roots April 5, 2012

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Find $\sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \cdots}}}}$ to three decimal places of precision.

Problem of the Day #382: Four Prime Factors April 4, 2012

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Find the sum of the prime factors of $1,015,074,782$, given that there are exactly four of them and one of them is $499$.

Problem of the Day #380: Filling a Grid, Part II April 2, 2012

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Given a grid with $R$ rows and $C$ columns, what is the maximum number of cells that can be colored such that no colored cell is adjacent to more than $X$ other colored cells? Solve the problem for $X = 0, 1, 2, 3$ and express your answer in terms of $R$ and $C$. Two cells are adjacent if they share an edge.

Problem of the Day #379: Easy Problem April 1, 2012

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Dear Loyal Viewers,

It’s been more than a year since we started “Math Problem of the Day” and it has been an amazing experience for all of us. However, we regret to inform you that we will be stopping our daily “problem of the day” in search of other pursuits.

For our last problem of the day, please convert $137,665,927,309,005,970,969,842,064$ to base $36$.

Problem of the Day #378: Filling a Grid, Part I March 31, 2012

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Given a grid with $R$ rows and $C$ columns, what is the maximum number of cells that can be colored such that no colored cell is adjacent to more than two other colored cells? Express your answer in terms of $R$ and $C$. Two cells are adjacent if they share an edge.

Problem of the Day #374: Increasing Sequence Numbers March 27, 2012

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Find the number of integers satisfying the property that the concatenation of all the odd-indexed digits (the leftmost digit is index 1), starting from the leftmost digit, followed by all the even-indexed digits, starting from the rightmost even-indexed digit, forms a strictly increasing sequence.