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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 #384: Coin Flipping April 6, 2012

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Albert has $10$ different coins, weighted to have a $\frac{1}{3}$, $\frac{1}{5}$, $\frac{1}{7}$, $…$ $\frac{1}{21}$ chance of landing heads, respectively. If he flips all of them, what is the probability that he obtains an odd number of heads?

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 #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.

Problem of the Day #373: Palindromey Strings March 26, 2012

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A palindromey string is a string of lowercase letters that is either a palindrome or the concatenation of two palindromey strings. Find the number of palindromey strings of length $12$.

Problem of the Day #370: Rows of Counters March 23, 2012

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Alex is playing a game in which he manipulates a row of black and white counters. To change the row, he is allowed to remove any black counter and replace it with either a white counter and a black counter (in that order), or two black counters. He is not allowed to perform any other manipulations.

If he starts with a single black counter, how many distinct rows of $10$ counters can he arrange?

Problem of the Day #368: Paths through a Grid, Part II March 21, 2012

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Albert starts at the top left corner of a $16$ by $16$ grid and is allowed to move up, down, or right for each step. How many ways are there for him to reach the bottom right corner, given that no cell is traversed more than twice?

Problem of the Day #367: Non-Repetitive Paths through a Grid March 20, 2012

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Albert starts at the top left corner of a $16$ by $16$ grid and is allowed to move either down one cell or right one cell for each step. He is not allowed to move in the same direction more than $3$ times in a row. How many ways are there for Albert to reach the bottom right corner?