You’re taking care of a friend’s house while he’s on vacation. One hot day you pull the chain on a ceiling fan, and when it doesn’t respond you realize the house has temporarily lost power. Unfortunately, you have to leave now, and you’ll be away for several days.

You know that the fan was in the “off” position before you pulled the chain, and that pulling the chain successively will cycle it through its remaining settings (“off,” “high,” “medium,” etc.). You don’t know how many settings there are, but you’re sure there aren’t more than 4.

How can you ensure that the fan will be in the “off” position when power is restored? (Assume that you can’t simply cut the fan’s power.)

One hundred people stand in a line, all facing in the same direction. Each is wearing a red or a blue hat, assigned at random. Each person can see all the hats before him in the line, but not his own or those of the people behind him. Starting at the back of the line, each person in turn must guess the color of his own hat. Each person can hear all the prior guesses. If the group are allowed to discuss strategy beforehand, how many can be sure of guessing correctly?

By Louis Friedländer. White to mate in two moves.

Young telegraph operator Joseph Orton Kerbey was enlisted as a spy for the federal forces during the Civil War. In 1861, laid up in a sick bed in Richmond, he needed a way to communicate his latest discoveries to his friends in the north. The message would have to appear innocent and contain the key to its own decipherment. Here’s what he sent:

He directed it, not to his father’s name and address, but to a friend in the telegraph office at Annapolis. What was the hidden message?

Prove that, if we choose nine vertices of a regular icosagon, some three of these will form an isosceles triangle.

By Eugène Ferber. White to mate in two moves.

Speaking of Lewis Carroll — and further to Wednesday’s logic exercise — here’s the king of all Carroll’s logic problems. What’s the strongest conclusion that can be drawn from these premises?

1. When the day is fine, I tell Froggy “You’re quite the dandy, old chap!”
2. Whenever I let Froggy forget that 10 pounds he owes me, and he begins to strut about like a peacock, his mother declares “He shall not go out a-wooing!”
3. Now that Froggy’s hair is out of curl, he has put away his gorgeous waistcoat.
4. Whenever I go out on the roof to enjoy a quiet cigar, I’m sure to discover that my purse is empty.
5. When my tailor calls with his little bill, and I remind Froggy of that 10 pounds he owes me, he does not grin like a hyena.
6. When it is very hot, the thermometer is high.
7. When the day is fine, and I’m not in the humor for a cigar, and Froggy is grinning like a hyena, I never venture to hint that he’s quite the dandy.
8. When my tailor calls with his little bill and finds me with an empty pocket, I remind Froggy of that 10 pounds he owes me.
9. My railway shares are going up like anything!
10. When my purse is empty, and when, noticing that Froggy has got his gorgeous waistcoat on, I venture to remind him of that 10 pounds he owes me, things are apt to get rather warm.
11. Now that it looks like rain, and Froggy is grinning like a hyena, I can do without my cigar.
12. When the thermometer is high, you need not trouble yourself to take an umbrella.
13. When Froggy has his gorgeous waistcoat on, but is not strutting about like a peacock, I betake myself to a quiet cigar.
14. When I tell Froggy that he’s quite a dandy, he grins like a hyena.
15. When my purse is tolerably full, and Froggy’s hair is one mass of curls, and when he is not strutting about like a peacock, I go out on the roof.
16. When my railways shares are going up, and when it’s chilly and looks like rain, I have a quiet cigar.
17. When Froggy’s mother lets him go a-wooing, he seems nearly mad with joy, and puts on a waistcoat that is gorgeous beyond words.
18. When it is going to rain, and I am having a quiet cigar, and Froggy is not intending to go a-wooing, you had better take an umbrella.
19. When my railway shares are going up, and Froggy seems nearly mad with joy, that is the time my tailor always chooses for calling with his little bill.
20. When the day is cool and the thermometer low, and I say nothing to Froggy about his being quite the dandy, and there’s not the ghost of a grin on his face, I haven’t the heart for my cigar!

Unfortunately, Carroll died before he was able to publish the solution — but he warned that it contains “a beautiful ‘trap.'”

1. Find an expression for the number 1 that uses each of the digits 0-9 once.
2. Do the same for the number 100.
3. Write 31 using only the digit 3 five times.
4. Express 11 with three 2s.
5. Express 10 with two 2s.
6. Express 1 with three 8s.
7. Express 5 with two 2s.

A logic exercise by Lewis Carroll. What conclusion can be drawn from these premises?

1. Animals are always mortally offended if I fail to notice them.
2. The only animals that belong to me are in that field.
3. No animal can guess a conundrum unless it has been properly trained in a Board-School.
4. None of the animals in that field are badgers.
5. When an animal is mortally offended, it rushes about wildly and howls.
6. I never notice any animal unless it belongs to me.
7. No animal that has been properly trained in a Board-School ever rushes about wildly and howls.

How are the following numbers arranged?

0, 2, 3, 6, 7, 1, 9, 4, 5, 8