Is it possible to prove statistically that there must be at least two people in China who have the same number of hairs on their heads? Try to stick to pure probability and not to assumptions such as, "There must be many bald people in China." Also, is it possible to prove statistically that there must be at least two people in China with the same arrangement of teeth (i.e., missing or existing in the same positions)? Again, try to stick to pure probability and not to assumptions such as, "There must be many old people with no teeth, or people with no missing teeth." (I got these two nice puzzles from my friend, SQL Server MVP Marcello Poletti.)
The answer to the first puzzle is yes. There are more than a billion people in China, and there are fewer than a billion hairs on a human head. Because there are fewer hairs on a human head than there are people in China, it's impossible that every person in China has a unique number of hairs. Therefore, there must be at least one number that occurs at least twice; in other words, there must be at least one set of at least two people in China with the same number of hairs on their heads.
The answer to the second puzzle is no. It can't be proven that there must be at least two people in China with the same arrangement of teeth. Humans have as many as 32 teeth. You can represent any teeth arrangement (missing/existing) with a 32-bit bitmap. The number of possible combinations is 232 - more than 4 billion. Because there are more possible combinations of teeth arrangements than the number of people in China, it's possible that all Chinese have unique teeth arrangements.
OCTOBER'S PUZZLE: 2 MATHEMATICIANS
Two mathematicians (let's call them M and N) - once good friends - meet after a long time to have a drink together. M asks, "Are you married? Any kids? Do you still live in that old apartment building?" N replies, "Yes, I'm married with three kids, and we live in a private house now." M asks, "How old are your kids?" N replies, "Let me answer with a riddle: The product of the ages of my kids is 36. Now, see that bus over there? The sum of my kids' ages is equal to that bus number." M thinks for a moment, then says, "I don't have sufficient information to solve the puzzle." N replies, "Oh, yes, you're right, I forgot to mention that one of my kids was born before we bought the house." Soon after N provides this last bit of information, M solves the puzzle and tells N the correct ages of the kids. Can you figure out the solution? Also, how would the solution change if N's additional piece of information was that one of his kids was born after he bought the house?