There are 1 million closed school lockers in a row, labeled 1 through 1,000,000. You first go through and flip every locker open. Then you go through and flip every other locker (locker 2, 4, 6, etc…). When you’re done, all the even-numbered lockers are closed. You then go through and flip every third locker (3, 6, 9, etc…). “Flipping” mean you open it if it’s closed, and close it if it’s open. For example, as you go through this time, you close locker 3 (because it was still open after the previous run through), but you open locker 6, since you had closed it in the previous run through. Then you go through and flip every fourth locker (4, 8, 12, etc…), then every fifth locker (5, 10, 15, etc…), then every sixth locker (6, 12, 18, etc…) and so on. At the end, you’re going through and flipping every 999,998th locker (which is just locker 999,998), then every 999,999th locker (which is just locker 999,999), and finally, every 1,000,000th locker (which is just locker 1,000,000). At the end of this, is locker 1,000,000 open or closed?

Five potential donors are sitting on a bench from left to right outside the surgery room and are waiting for the doctor. Their ages are 6, 10, 31, 47 and 61. Their heights are 41, 49, 61, 66 and 75 inches. Their weights are 41, 76, 97, 126 and 166 pounds. Determine the position, blood group, age, height and weight of each of them with the help of the following ten hints. The person on the far right is 37 years older than Jimmy, and is 61 inches tall. Jimmy weighs 56 pounds more than his height. Alexandrio weighs 76 pounds and is 75 inches tall. Justin is type AB and weighs 56 pounds less than Jimmy. The person in the center is 10 years old, is blood type AO and weighs 97 pounds. Andrew, who is the first, is 66 inches tall, and weighs 100 pounds more than his height. The person who is blood type O, is 25 years older than the person to the left of them. Kian is 61 years old. The person who is blood type A, is 55 years younger than Kian and is not next to the person who is type AO. The person who is next to the 10 year old but not next to the person who is 66 inches tall, is blood type B, and weighs 126 pounds.

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. It could be that some God gets asked more than one question (and hence that some God is not asked any question at all). 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.) 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. Random will answer “da” or “ja” when asked any yes-no question. What would your three questions be?

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?

One day, a father went to his three sons and told them that he would die soon and he needed to decide which one of them to give his property to. He decided to give them all a test. He said, “Go to the market my sons, and purchase something that is large enough to fill my bedroom, but small enough to fit in your pocket. From this I will decide which of you is the wisest and worthy enough to inherit my land.” So they all went to the market and bought something that they thought would fill the room, yet was still small enough that they could fit into their pockets. Each son came back with a different item. The father told his sons to come into his bedroom one at a time and try to fill up his bedroom with whatever they had purchased. The first son came in and put some pieces of cloth that he had bought and laid them end to end across the room, but it barely covered any of the floor. Then the second son came in and laid some hay, that he had purchased, on the floor but there was only enough to cover half of the floor. The third son came in and showed his father what he had purchased and how it could fill the entire room yet still fit into his pocket.The father replied, “You are truly the wisest of all and you shall receive my property.” What was it that the son had showed to his father?

Four prisoners named P1, P2, P3 and P4 are arrested for a crime, but the jail is full and the jailer has nowhere to put them. He eventually comes up with the solution of giving them a puzzle and if they answer correctly they can go free but if they fail they are to be executed. The jailer makes prisoners P1, P2 and P3 stand in a single file. Prisoner P4 is put behind a screen. The arrangement looks like this: P1 P2 P3 || P4 The ‘||’ is the screen. The jailer tells them that there are two black hats and two white hats; that each prisoner is wearing one of the hats; and that each of the prisoners is only able to see the hats in front of them but not on themselves or behind. Prisoner P1 can see P2 and P3. Prisoner P2 can see P3 only. The fourth man, P4, behind the screen can’t see or be seen by any other prisoner. No communication between the prisoners is allowed. If any prisoner can figure out and tell the jailer the color of the hat he has on his head all four prisoners go free. If any prisoner gives an incorrect answer, all four prisoners are executed. How the prisoners can escape, regardless of how the jailer distributes the hats? You can assume that the prisoners can all hear each other if one of them tries to answer the question. Also, every prisoner thinks logically and knows that the other prisoners think logically as well.

Once upon a time there was a kingdom. A king and a clown lived in this kingdom. Unfortunately they hated each other so they agreed that they will poison each other one day. There are only twelve vials of poison in whole kingdom and they are locked in one chamber in the castle. The poisons have numbers from 1 to 12. The higher the number the stronger the poison. The effect on the human body is simple – you drink the poison, you die. Each stronger poison neutralizes all weaker poisons which means that poison 12 neutralizes all poisons, poison 11 neutralizes all poisons except poison 12, etc. If you drink poison 11 followed by poison 12 nothing happens. If you drink poison 12 and then poison 11 you die. The king enters the chamber first and takes all the even numbered poisons (2, 4, 6, 8, 10, 12). The clown then enters and takes the odd numbered poisons. They meet in the throne hall where each fills one cup and hands it over to the other who immediately drinks it. Now each fills the cup once again, now for himself, and drinks it (hoping to save his own life). Both the king and the clown primarily want to survive but want to poison the other. There is one dose of each poison – it’s not possible to divide it. The poisons are fluids without color or smell and they have the same consistency as water. The clown survived and the king died from the poison. What did the clown do?

Pirate Pete had been captured by a Spanish general and sentenced to death by his 50-man firing squad. Pete cringed, as he knew their reputation for being the worst firing squad in the Spanish military. They were such bad shots that they would often all miss their targets and simply maim their victims, leaving them to bleed to death, as the general’s tradition was to only allow one shot per man to save on ammunition. The thought of a slow painful death made Pete beg for mercy. “Very well, I have some compassion. You may choose where the men stand when they shoot you and I will add 50 extra men to the squad to ensure someone will at least hit you. Perhaps if they stand closer they will kill you quicker, if you’re lucky,” snickered the general. “Oh, and just so you don’t get any funny ideas, they can’t stand more than 20 ft away, they must be facing you, and you must remain tied to the post in the middle of the yard. And to show I’m not totally heartless, if you aren’t dead by sundown I’ll release you so you can die peacefully outside the compound. I must go now but will return tomorrow and see to it that you are buried in a nice spot, though with 100 men, I doubt there will be much left of you to bury.” After giving his instructions the general left. Upon his return the next day, he found that Pete had been set free alive and well. “How could this be?” demanded the general. “It was where Pete had us stand,” explained the captain of the squad. Where did Pete tell them to stand?

“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. “Had any trouble recently?” “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?