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"Hat" Riddles - Next 10 of 3408.

Riddle: You have 52 playing cards, 26 red, and 26 black. You draw cards one by one. A red card pays you a dollar. A black one fines you a dollar. You can stop any time you want. Cards are not returned to the deck after being drawn. What is the optimal stopping rule in terms of maximizing your expected payoff? Also, what is the expected payoff following this optimal rule?
Answer: The solution to this problem is, in my opinion the most difficult to understand of all the puzzles. Indeed I was unable to solve it and didn't receive a complete solution until two years after originally posting it. The final solution, in the form of the spreadsheet was sent to me by Han Zheng. For this reason I have left on the page the thoughts i had before I had the final solution as they represent an easier to understand and more simplistic approach. Also the reasoning may help you arrive at the final solution by yourself or help you understand it. I would recommend reading that answer before you dive into the full answer. But an important thing to note are that as the player we can't lose this game as we can gamble till all the cards are drawn and our net position is zero. From our earlier analysis it is clear we need a dynamic quit rule. A singal value is not sufficent. We must, at each stage consider what cards are remaining, and therefor the probability of a positive or negative outcome from drawing again. For the explanation i will ask you first to consider a deck containing only 6 cards, 3 +ve & 3 -ve (note i'm no longer calling the cards black and red, it confuses me.)
Riddle: What did the koala say to the bear on the phone?
Answer: I’ll koala you later.
Riddle: I'm done to boats, to cargo, to loads; When indoors I'm, in a way, a narrow road. What am I?
Answer: Haul / Hall
Riddle: I have one eye but cannot see. No teeth, and yet I bite. My voice can whisper softly or can keep you up all night. P.S. This riddle is meant to be heard, not read. As such, please read it aloud and figure it out from there, to grasp the answer. What am I?
Answer: "Wind" - Reasoning: "Eye" sounds like "i", and the fact that the answer "cannot see" hints that this is the case. Wind, especially high wind, can feel biting, and in literature, the wind is often said to bring a "biting cold", or similar. The voice of the wind is depicted as being quiet or loud, depending on how strong the wind is. Loud winds are often associated with keeping people up during the night.
Riddle: I get fiercely hot. I vary in sizes. Without me, the moon is all we'll see. It's impossible to walk into a room without at least one of me in them. What am I?
Answer: A lightbulb.
Riddle: What do you get when you dot the I and cross the T?
Answer: “You get it.”
Riddle: I have no eyes, no legs or ears, and I help move the earth. What am I?
Answer: A worm.
Riddle: What lives undisturbed, dies when disturbed and spies when you're disturbed?
Answer: A mouse.
Riddle: What is that which, supposing it's greatest breadth to be four inches, length nine inches, and depth three inches and contains a solid foot?
Answer: A shoe.
Riddle: A boy is five feet tall and records his height by hammering a nail in a tree and carving his hame. Five years later he returns to the same tree and finds that it has grown on average 16 inches per year. How high now, is the nail from the ground?
Answer: It is still five feet tall because only the top of the tree grows. The trunk only gets bigger by circumference.