A king has 100 identical servants, each with a different rank between 1 and 100. At the end of each day, each servant comes into the king’s quarters, one-by-one, in a random order, and announces his rank to let the king know that he is done working for the day. For example, servant 14 comes in and says, “Servant 14, reporting in.”

One day, the king’s aide comes in and tells the king that one of the servants is missing, though he isn’t sure which one.

Before the other servants begin reporting in for the night, the king asks for a piece of paper to write on to help him figure out which servant is missing. Unfortunately, all that’s available is a very small piece that can only hold one number at a time. The king is free to erase what he writes and write something new as many times as he likes, but he can only have one number written down at a time.

The king’s memory is bad and he won’t be able to remember all the exact numbers as the servants report in, so he must use the paper to help him.

How can he use the paper such that once the final servant has reported in, he’ll know exactly which servant is missing?

A man has 5 pieces of chain that must be joined into a long chain. He can open ring 3 (first operation), link it to ring 4 (second operation), then unfasten ring 6 and link it to ring 7, and so on.

He could complete his task in 8 operations, but he wants to do it in 6. How does he do it?

Here is a light switch. Note the order of the positions. If the light is now at medium and it is switched 3922 times what will be the position of the switch?

Three Gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter.

Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one God. The Gods understand English, but will answer all questions in their own language, in which the words for yes and no are “da” and “ja”, in some order. You do not know which word means which.

1. It could be that some God gets asked more than one question (and hence that some God is not asked any question at all).
2. What the second question is, and to which God it is put, may depend on the answer to the first question. (And of course similarly for the third question.)
3. Whether Random speaks truly or not should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely.
4. Random will answer “da” or “ja” when asked any yes-no question.

What would your three questions be?

“Who shot her?” cried Rogers as he rushed into the hospital three minutes after his ex-wife died from a bullet through her head.

“Just a minute, Mr. Rogers,” said Professor Stiggins. “We’ll have to ask you a few questions-routine, you know. Although divorced for the past six months, you have been living in the same house with your ex-wife, have you not?”

“That’s right,” replied Rogers.

“Well, yesterday, when I told her I was going on a business trip, she threatened to commit suicide. In fact, I grabbed a bottle of iodine from her as she was about to drink it. When I left last evening at seven, however, telling her I was spending the night with friends in Sewickley, she made no objection. Returning to town this afternoon,” continued Rogers, “I called my home and the maid answered.”

“Just what did she say?” inquired Stiggins.

“‘Oh, Mr. Rogers, they took poor mistress to St. Ann’s Hospital abbout half an hour ago. Please hurry to her.’ “She was crying, so I couldn’t get anything else out of her; then I hurried here. Where is she?”

“The nurse will direct you,” said Stiggins with a nod.

“A queer case, this, Professor,” said Inspector Kelley. “These moderns are a little too much for me, I’m afraid. A man and woman living together after being divorced six months!”

“A queer case indeed, Inspector,” mused the professor, “and you’d better detain Mr. Rogers. If he didn’t shoot her himself, I’m confident he knows who did.”

Why did the professor advise the Inspector to detain Rogers?