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?