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MMMMKnights&knavesMMM 
 You have twelve coins,
one of which is known is known to be counterfeit and therefore
heavier or lighter than all the others. You also have a balancepan
scale. In three weighings determine which coin is counterfeit and
whether it is heavier or lighter.
 You have ten big vats of
10gram coins. Except that one of the vats contains counterfeit
coins, which weigh only 9 grams. You have an accurate (numerical)
scale. In one wieghing, determine which vat contains the
counterfeit coins. What is the smallest number of coins this
requires?
 Same as above, except
that any number of vats (including none) may contain counterfeit
9gram coins. In one weighing, find the counterfeit vats, using as
few coins as possible.

Coin weighing
Light bulbs
Knights & knaves
Miscellaneous
River crossing
Census takers


 In the lobby of a hotel
there are three toggle switches. One is attached to a light bulb on
the third floor; the other two are attached to nothing at all. You
may work the three switches as often as you please, but you may
visit the third floor only once. How can you tell which switch is
attached to the light bulb?
 You have a tower with
1000 wires running from the top to the bottom. You also have a
battery and a lightbulb. The wires are coaxial, i.e., a battery  wires  lightbulb
connection will light the bulb. (You can tie wires together to form
one long wire.) Figure out which wires at the top of the tower
correspond to which wires at the bottom, in as few traversals of
the tower as possible (i.e., as little stair climbing as
possible).
 Same as above, except
that now you have wires that are not coaxial, but just wires. For
the light bulb to go on, you need two connections: battery  wires  bulb  wires 
battery. What is the new answer for least number of tower
traversals?

Coin weighing
Light bulbs
Knights & knaves
Miscellaneous
River crossing
Census takers


 There are three gods:
True, False, and Random. They answer accordingly. but not in
English! They say "da" and "ya" instead of "yes" and "no", and you
dont know which is which. With three questions determine who is
who.
 There are three gods:
And, Or, and Xor. They all answer the truth, but of all the
questions have been asked thus far they apply their namesake
operation. Identify using three questions.
 Same names as above, all
answer the truth. But you have to pick your questions beforehand,
and they answer the operation applied to all questions.
Identify in three.
 Same as above, except
that they apply their respective operations to all of the questions
not asked to them.
 There are three gods:
Past, Present, and Future. They'll answer the truth, but: Present
answers the question you are asking; Past answers the last question
you asked; Future answers the next question you will ask. If you
ask the first question to Past or the third question to Future,
they give a random answer. Because of possible time conflicts, you
must determine your questions ahead of time, rather than based on
previous answers. You are, however, allowed to choose who you ask
your three questions to dynamically. No time related questions are
allowed (i.e., "if the answer to my q2 was no, then... otherwise
..."). They answer with das and yas like before. With three
questions determine who is who.

Coin weighing
Light bulbs
Knights & knaves
Miscellaneous
River crossing
Census takers


 Suppose you start at the
Earth's equator and travel continuously northwest until you reach
the North Pole. What does your path look like? How long is your
path?
 A man leaves his camp
and walks one mile south. He then spots a bear 1 mile due east of
him, and shoots the bear with his rifle. He walks the 1 mile eat to
the bear, then carries the bear 1 mile north, arriving at his camp.
What color is the bear?
 Though the previous
problem has a unique correct answer, there is not a unique possible
location for the camp. What is the locus of all points on a
spherical earth satisfying this geometry  i.e., walking 1 mile
south, one mile east, then one mile north brings you to your
starting place?
 Consider the (perfectly
spherical) globe. You want a plane to circumnavigate it (around a
great circle). You have one airplane base and as many airplanes as
it takes, but each plane can only hold enough fuel to get it
halfway around the world. Airplanes can refuel instantaneously in
midair. How can you get one plane around the world (without any of
the planes crashing  they all must have enough fuel to get back to
the base), using as few airplanes as possible?
 You have an angry
couple: wife in a boat in the middle of a circular lake, and
husband on the shore. The man can run on land four times as fast as
the woman. Can the woman escape the lake (she can run faster than
the man on land) or does she get caught by the man?
 Replace "four" by other
values in the previous problem. Who wins? What is the transition
between wife winning and husband winning?
 Two cars start 50 miles
apart, heading towards each other. One travels at 30 mph, the other
at 20 mph. At the same time that the cars start driving, a bug that
can fly at 60 mph begins flying back and forth between the two
cars. How far has the bug flown by the time the two cars pass each
other?
 You have four ants at
the four corners of a square, each facing the one in front of him
in a cyclic order. They simultaneously begin walking towards the
and in front of them (i.e., A walks towards B, B towards C, C
towards D, and D towards A), always heading straight at the next
ant. How far have they travelled by the time they meet at the
center of the square?
 A deer and a lion are
trapped in a circular cage. They both walk at the same speed. Can
the lion catch the deer?
 A dog is chasing a cat.
They both run at the same speed. The cat initially heads in the
direction perpendicular to the dog, and always walks in that
direction; the dog always walks straight towards the cat. What
happens?
 Although Mt. Everest, at
29,028 feet above sea level, is the highest mountain in the world,
it is not the farthest from the center of the earth. The earth's
bulge at the equator pushes Chimborazo in Ecuador, at 20,561 feet
above sea level, farther. (How much?) Now suppose you run a water
pipe from Everest to Chimborazo. Which way would the water
flow?
 You have a cake that you
want to split fairly between n people, in the sense that each of
them thinks they have at least 1/n of the cake. (They may judge the
sizes of portions of the cake differently.) For two people, this
can be accomplished by having one person cut the cake exactly in
half according to his point of view, and then the second person
picks the bigger piece (according to his point of view). Then each
thinks they have at least half of the cake.
1. How do you divide a cake between three people?
2. How do you divide a cake between n
people?
 Subdivide a square into
8 acute triangles.
 Subdivide an obtuse
triangle into (finitely many) acute triangles.
 You have an Lshaped
piece of paper (i.e., six sides, all at right angles). Using a
straightedge and compass, draw a line cutting its area in
half.
 Repeat, using only a
straightedge.
 Repeat, using only a
straightedge and ensuring that you divide the paper into only two
(connected) pieces.
 You have an infinite
checkerboard, with a piece on every black square below the xaxis.
Pieces jump diagonally as in checkers, and you remove the piece
being jumped over. How high above the xaxis can you get a
piece?
 Repeat, with a piece on
every square below the xaxis and pieces jumping horizontally or
vertically.
 You have an integer
written out in decimal. When you chop off hte last digit and put it
on the front, this doubles the integer. What is the smallest
possible alue for the integer?
 Same as above, except
that it halves the integer.

Coin weighing
Light bulbs
Knights & knaves
Miscellaneous
River crossing
Census takers


 You hve a cup of coffee
and a cup of milk. You take a teaspoon of milk and put it in the
coffee. You take a teaspoon of the milkcoffee mixture and put it
back in the milk. Now, is there more milk in the coffee or more
coffee in the milk?
 You have an
oldfashioned milk bottle (think: 2liter bottle with a flat
bottom), with a neck narrower than the body. It contains milk, with
a layer of cream on top, in the neck. Now you shake up the bottle
so the milk and cream are mixed together. Does the pressure on the
bottom of the bottle increase, decrease, or say the
same?
 What happens to a match
in zero gravity?
 You have a helium
balloon in a car. The car turns right. What happens to the
balloon?
 A farmer has a fox,
chicken, and sack of grain that he wants to take across a river. He
can only take one of the three with him in the rowboat, but if the
fox is left alone with the chicken it will eat it, and if the
chicken is left alone with the grain it will eat it. How does he
get across without losing any of his possessions?
 Four people want to
cross a rickety bridge at night, but they only have one flashlight.
Any time people cross the bridge, they must carry the flashlight
with them; but the bridge only holds two people. The people walk at
different rates: it takes them 1, 2, 5, and 10 minutes,
respectively, to cross the bridge; if two people cross the bridge
together they must travel at the slower of their rates. How can
they cross taking the least amount of time?
 Three innocent
bystanders and three cowardly axe murderers are trying to cross a
river, but they only have a boat that can hold two people. If at
any time the axe murderers outnumber the innocents (on some side of
the river), they will slaughter them. The axe murderers are,
however, friendly, so that they will follow the bystanders'
instructions (with regards to the boat). How can everyone get
across the river safely?

Coin weighing
Light bulbs
Knights & knaves
Miscellaneous
River crossing
Census takers


 A census taker
approaches a house and asks the occupants how many children they
have. They reply, "We have three children. The product of their
ages is 36 and the sum of their ages is my house number." The
census taker looks at their house number, comes back, and says, "I
still can't figure it out." They respond, "My oldest daughter as
red hair." The census taker says "Thank you" and walks away. How
old are the children?
 Replace 36 by 72 in the
above problem.
 C(ensus taker) and
(Mrs.)S, two old friends, see a lady with her two children. S knows
the lady and her children well.
C: How old is the lady?
S: The product of their ages is 2450 and the sum is
your age.
C: Oh, I cannot figure out.
S: Of course, you can't. I am older than the lady.
C: OK.
What the the ages of all five people involved?
 I pick two numbers
between 3 and 98, inclusive, and tell their sum to S(teve) and the
product to P(aula).
S: I know you don't know what the two numbers are.
P: Well, in that case, I know what the two numbers
are.
S: OK, now I know what the two numbers are.
What are the two numbers?

Coin weighing
Light bulbs
Knights & knaves
Miscellaneous
River crossing
Census takers

puzzles.html ~ Andrei
Gnepp ~
gnepp@fas.harvard.edu
$Date: 19981109 21:29:2105 $
