Brood War

A newlywed couple are planning their family. They’d like to have four children, a mix of girls and boys. Which is more likely: (1) two girls and two boys or (2) three children of one sex and one of the other? (Assume that each birth has an equal chance of being a boy or a girl.)

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Black and White

meredith chess problem

By William Meredith. White to mate in two moves.

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Flip Sum

A problem from the 1999 St. Petersburg City Mathematical Olympiad:

Fifty cards are arranged on a table so that only the uppermost side of each card is visible. Each card bears two numbers, one on each side. The numbers range from 1 to 100, and each number appears exactly once. Vasya must choose any number of cards and flip them over, and then add up the 50 numbers now on top. What’s the highest sum he can be sure to reach?

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All Relative

A problem from Dick Hess’ All-Star Mathlete Puzzles (2009):

A man points to a woman and says, “That woman’s mother-in-law and my mother-in-law are mother and daughter (in some order).” Name three ways in which the two can be related.

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Ballot Measure

A Russian problem from the 1999 Mathematical Olympiad:

In an election, each voter writes the names of n candidates on his ballot. Each ballot is then placed into one of n+1 boxes. After the election, it’s noted that each box contains at least one ballot, and that if one ballot is drawn from each box, these n+1 ballots will always have a name in common. Show that for at least one box, there’s a name that appears on all of its ballots.

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