What are some "aha!"-type puzzles, puzzles whose simple solutions are not obvious?

Below is a list of "aha!"-type puzzles.  These puzzles have simple
solutions that are not obvious.  The best "aha!"-type puzzles are
unsolvable except by the simple solution.  Some can be solved by brute
force; these are marked with an asterisk ('*') in the list below.

[ Editor: the list gives answers to only three of the puzzles; they
are indented under the questions. ]

Invariants

*One cup contains tea, another coffee.  A teaspoon of tea is moved to
the coffee cup and thoroughly mixed.  Then a teaspoon of the mixture is
moved to the tea cup.  Which cup now contains more of its original
contents?

*A chessboard has opposite corners removed.  Can the remaining squares
be covered by 31 dominoes, each of which covers two adjacent squares?

Logic

There is a list of n sentences.  The ith sentence says "Exactly i of
these sentences are false."  How many true sentences are on this list?

Green, Brown and Blue are wearing ties.  Green says to the person
wearing the blue tie, "None of us is wearing a tie of the color of our
name."  Who is wearing what color?

A merchant charges $80 for a radio (which is double the cost).  A
customer buys a radio, presenting a $100 bill.  The merchant cannot
break the bill, so he takes it next door and gets 5 $20 bills from the
bank.  The merchant gives him a radio and $20 in change.  The $100 bill
turns out to be a counterfeit.  The merchant has to make this up to the
bank.  How much is the merchant out?

If half of 5 were 3, what would a third of 10 be?

Geometry - Symmetry

An island is two kilometers offshore.  An athlete is three kilometers
inland and three kilometers from the point on the shore nearest to the
island.  If the athlete can run at 10 kph and swim at 2 kph, what is
the shortest time she can take to get to the island?

*Two face diagonals on a cube meet at a corner.  What is the angle
between them?

A farmer wishes to fence in the maximum possible area with 100 meters
of fencing.  There is a straight cliff that can be used as one side of
the fence.  What is the largest area that can be fenced in?

What is the largest ellipse that can be inscribed in a right triangle
whose legs are a and b?

        Project ellipse to circle; right triangle to equilateral
        triangle.  Area of ellipse = a * b * pi / (6 * sqrt(3)).

What is the volume of the intersection of two right circular cylinders
of equal radius whose axes intersect at right angles?

        16/3 r^3.

A billiard ball is hit with no "English," hits all four bumpers, and
returns to its original position.  How far did it travel?

        Twice the diagonal of the table.

Probability

*A deck of cards is thoroughly shuffled.  What are the odds that all
but one card has returned to its original position?

An urn contains two balls.  One is black; the other is black or white
with even odds.  You pick a ball from the urn.  It is black.  You
replace the ball.  You pick another ball from the urn.  What are the
odds that it is black?

Operations Research

A couple needs to mow the lawn, vacuum the carpet and feed the baby.
Each job takes one person 1/2 hour.  What is the shortest time in which
all three jobs can be accomplished?

*Two surgeons must each operate on two patients, and there is a
shortage of surgical gloves.  What is the least number of gloves they
can use?

Reasoning Backward

If a plane starts flying southwest at the North Pole, where does it
meet the Equator?

Two joggers and a dog start together, but since one jogger jogs at
5 kph and the other jogs at 6 kph, they soon separate.  The dog runs
back and forth between the two joggers at 10 kph.  After 1/2 hour,
where is the dog?

Geometry - Dissection

Divide a square into five congruent pieces.

What is the shortest line that divides a rhombus into two congruent
pieces?

What is the length of the curve of minimum length that bisects the
area of an equilateral triangle with edge 1 cm?

Pigeonhole Principle

*A drawer contains ten black socks and sixteen blue socks.  How many
socks have to be removed from the drawer to ensure that at least one
pair of the same color has been removed?

*If people have up to 100,000 hairs on their head, how many people, at
a minimum, in a city of 200,000 people have the same number of hairs as
someone else in the village?

*97 teams are playing in an elimination tournament.  How many games
must be played to determine an overall winner?

Physics

Each edge of a cube is a 1-ohm resistor.  What is the resistance
between two diagonally opposite vertices of the cube?

*A fisherman's hat blows off while he is rowing upstream.  He does not
notice for ten minutes.  He then misses his hat and begins rowing
downstream.  If he rows at 6 kph is still water, and the river is
flowing at 3 kph, how long will it take him to recover his hat?

*Three edges of a square sheet of metal are kept at 0 degrees, while
the fourth is kept at 100 degrees.  Assuming no radiative heat loss,
what is the temperature of the middle of the sheet?

Weighing Problems

*You have twelve coins, one of which is counterfeit and lighter or
heavier than the others, and a beam balance.  What is the least number
of weighings in which the counterfeit coin can be found?

*You have ten bottles containing a hundred pills.  All the pills in one
bottle weigh 1 gram less than the pills in the other bottles.  What is
the least number of weighings on a scale that are required to identify
the odd bottle?

*How many weighings with a beam balance are required to rank five
different weights from lightest to heaviest?

Meta Puzzle - Geometry

*Two people walk to the top of a cylindrical lighthouse, one via a
straight ladder up the middle and the other via a helical ladder up the
inside wall.  If helical ladder winds around the tower once and they
both arrive at the same time, what is the ratio of their speeds?

*A cylindrical hole is drilled through a sphere.  If the hole is six
centimeters long, what is the volume of the remaining material?

Meta Puzzle - Logic

*The moderator takes a set of 8 stamps, 4 red and 4 green, known to the
logicians, and loosely affixes two to the forehead of each logician so
that each logician can see all the other stamps except those 2 in the
moderator's pocket and the two on her own head.  He asks them in turn
if they know the colors of their own stamps:
  A: "No"
  B: "No"
  C: "No"
  A: "No"
  B: "Yes"
What are the colors of B's stamps?

[ Editor: the list is unsigned in the collected copy.  Several of these
puzzles have entries of their own in the archive, e.g.
logic/smullyan/stamps, geometry/hole.in.sphere and logic/weighing. ]