Suppose we have a clock whose hour and minute hands are identical. How times times per day will we find it impossible to tell the time, provided we always know whether it’s a.m. or p.m.?
A boat floats in a swimming pool. In the boat is a block of ice. If the block is dumped into the water and melts, does the water level rise or fall?
When eight certain cards are removed from a standard poker deck, it becomes impossible to deal a straight flush. What are the cards? (Assume the deck contains no jokers.)
On a 12 × 12 grid, some squares are infected and some are healthy. On each turn, a healthy square becomes infected if it has two or more infected orthogonal neighbors. (In the example above, the black squares are infected, the white squares are healthy, and the gray squares will be infected next turn.) What’s the smallest number of initially infected squares that can spread an infection over the whole board?
A more challenging version of the Counterfeit Coin puzzle from 2011:
You have 12 coins, one of which has been replaced with a counterfeit. The false coin differs in weight from the true ones, but you don’t know whether it’s heavier or lighter. How can find it using three weighings in a pan balance?
This, now, is straightforward and business-like: A. applied to B. for a loan of $100. B. replied, ‘My dear A., nothing would please me more than to oblige you, and I’ll do it. I haven’t $100 by me, but make a note and I’ll indorse it, and you can get the money from the bank.’ A. proceeded to write the note. ‘Stay,’ said B., ‘make it $200. I want $100 myself.’ A. did so, B. indorsed the paper, the bank discounted it, and the money was divided. When the note became due, B. was in California, and A. had to meet the payment. What he is unable to cipher out is whether he borrowed $100 of B., or B. borrowed $100 of him.
— Henry C. Percy, Our Cashier’s Scrap-Book, 1879
In his 1936 collection Brush Up Your Wits, British puzzle maven Hubert Phillips relates that his brother-in-law felt himself cursed with an unintelligent maid. “I have just overheard her taking a ‘phone call,” he told Phillips, “and this is what I heard:
“‘Is Mr. Smith at home?’
“‘I will ask him, sir. What name shall I give him?’
“‘What’s that, sir?’
“‘Would you mind spelling it?’
“‘Q for quagga, U for umbrella, O for omnibus, I for idiot –‘
“‘I for what, sir?’
“‘I for idiot, T for telephone. Q, U, O, I, T, Quoit.’
“‘Thank you, sir.'”
Why did he accuse her of unintelligence?
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.)
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?
By Peder Andreas Larsen. White to mate in two moves.
(I’ve added a guide to chess notation.)
A puzzle by Sam Loyd. The red strips are twice as long as the yellow strips. The eight can be assembled to form two squares of different sizes. How can they be rearranged (in the plane) to form three squares of equal size?
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.
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.
A circular table stands in a corner, touching both walls. A certain point on the table’s edge is 9 inches from one wall and 8 inches from the other. What’s the diameter of the table?