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From: uunet!questrel!chris (Chris Cole)
Subject: rec.puzzles FAQ, part 13 of 15
Message-ID: <puzzles-faq-13_717034101@questrel.com>
Followup-To: rec.puzzles
Summary: This posting contains a list of
     Frequently Asked Questions (and their answers).
     It should be read by anyone who wishes to
     post to the rec.puzzles newsgroup.
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Date: Mon, 21 Sep 1992 00:09:46 GMT
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Archive-name: puzzles-faq/part13
Last-modified: 1992/09/20
Version: 3

==> logic/situation.puzzles.s <==
                  Answers to Jed's List of Situation Puzzles

   This is the list of answers to the puzzles in my situation puzzles
list.  See that list for more details.  This document also contains
variant setups and answers for some of the puzzles.

Section 1: "Realistic" situation puzzles.

1.1.  A bunch of people are on an ocean voyage in a yacht.  One afternoon,
they all decide to go swimming, so they put on swimsuits and dive off the
side into the water.  Unfortunately, they forget to set up a ladder on the
side of the boat, so there's no way for them to climb back in, and they
drown.
1.1a.  Variant answer: The same situation, except that they set out a
ladder which is just barely long enough.  When they all dive into the
water, the boat, without their weight, rises in the water until the ladder
is just barely out of reach.  (also from Steve Jacquot)

1.2.  The room is the ballroom of an ocean liner which sank some time ago.
The man ran out of air while diving in the wreck.
1.2a.  Variant which puts this in section 2: same statement, ending with
"a large window through which rays are coming."  Answer: the rays are
manta rays (this version tends to make people assume vampires are
involved, unless they notice the awkwardness of the phrase involving
rays).

1.3.  The husband killed himself a while ago; it's his ashes in an urn on
the mantelpiece that the wife looks at.  It's debatable whether this
belongs in section 2 for double meanings.

1.4.  A poor peasant from somewhere in Europe wants desperately to get to
the U.S.  Not having money for airfare, he stows away in the landing gear
compartment of a jet.  He dies of hypothermia in mid-flight, and falls out
when the landing gear compartment opens as the plane makes its final
approach.
1.4a.  Variant: A man is lying drowned in a dead forest.  Answer: He's
scuba diving when a firefighting plane lands nearby and fills its tanks
with water, sucking him in with the water.  He runs out of air while the
plane is in flight; the plane then dumps its load of water, with him in
it, onto a burning forest.  (from Jim Moskowitz)

1.5.  The man is a midget.  He can't reach the upper elevator buttons, but
he can ask people to push them for him.  He can also push them with his
umbrella.  I've usually heard this stated with more details: "Every
morning he wakes up, gets dressed, eats, goes to the elevator..."  Ron
Carter suggests a nice red herring: the man lives on the 13th floor of the
building.

1.6.  The sisters are Siamese twins.
1.6a.  Variant: A man and his brother are in a bar drinking.  They begin to
argue (as always) and the brother won't get out of the man's face, shouting
and cursing.  The man, finally fed up, pulls out a pistol and blows his
brother's brains out.  He sits down to die.  Answer: They are Siamese twins.
In the original story, the argument started when one complained about the
other's bad hygiene and bad breath.  The shooter bled to death (from his
brother's wounds) by the time the police arrived.  (from Randy Whitaker,
based on a 1987 _Weekly World News_ story)

1.7.  The man has hiccups; the bartender scares them away by pulling a
gun.

1.8.  The man used to be blind; he's now returning from an eye operation
which restored his sight.  He's spent all his money on the operation, so
when the train (which has no internal lighting) goes through a tunnel he
at first thinks he's gone blind again and almost decides to kill himself.
Fortunately, the light of the cigarettes people are smoking convinces him
that he can still see.
1.8a.  Variant: A man dies on a train he does not ordinarily catch.
Answer: The man (a successful artist) has had an accident in which he
injured his eyes.  His head is bandaged and he has been warned not to
remove the bandages under any circumstances lest the condition be
irreversibly aggravated.  He catches the train home from the hospital and
cannot resist peeking.  Seeing nothing at all (the same train-in-tunnel
situation as above obtains, but without the glowing cigarettes this time),
he assumes he is blinded and kills himself in grief.  I like this version
a lot, except that it makes much less sense that he'd be traveling alone.
(from Bernd Wechner)

1.9.  The man was in a ship that was wrecked on a desert island.  When
there was no food left, another passenger brought what he said was abalone
but was really part of the man's wife (who had died in the wreck).  The
man suspects something fishy, so when they finally return to civilization,
he orders abalone, realizes that what he ate before was his wife, and
kills himself.
1.9a.  Variant: same problem statement but with albatross instead of
abalone.  Answer: In this version, the man was in a lifeboat, with his
wife, who died.  He hallucinated an albatross landing in the boat which he
caught and killed and ate; he thought that his wife had been washed
overboard.  When he actually eats albatross, he discovers that he had
actually eaten his wife.
1.9b.  Variant answer to 1.9a, with a slightly different problem
statement: the man already knew that he had been eating human flesh.  He
asks the waiter in the restaurant what kind of soup is available, and the
waiter responds, "Albatross soup."  Thinking that "albatross soup" means
"human soup," and sickened by the thought of such a society (place in a
foreign country if necessary), he kills himself.  (from Mike Neergaard)

1.10.  He stood on a block of ice to hang himself.  The fact that there's
no furniture in the room can be added to the statement, but if it's
mentioned in conjunction with the puddle of water the answer tends to be
guessed more easily.

1.11.  He stabbed himself with an icicle.

1.12.  He jumped out of an airplane, but his parachute failed to open.
Minor variant wording (from Joe Kincaid): he's on a mountain trail instead
of in a desert.  Minor variant wording (from Mike Reymond): he's got a
ring in his hand (it came off of the ripcord).
1.12a.  Silly variant: same problem statement, with the addition that one
of the man's shoelaces is untied.  Answer: He pulled his shoelace instead
of the ripcord.
1.12b.  Variant answer: The man was let loose in the desert with a pack
full of poisoned food.  He  knows it's poisoned, and doesn't eat it -- he
dies of hunger.  (from Mike Neergaard)

1.13.  He was with several others in a hot air balloon crossing the
desert.  The balloon was punctured and they began to lose altitude.  They
tossed all their non-essentials overboard, then their clothing and food,
but were still going to crash in the middle of the desert.  Finally, they
drew matches to see who would jump over the side and save the others; this
man lost.  Minor variant wording: add that the man is nude.

1.14.  The radio program is one of the call-up-somebody-and-ask-them-a-
question contest shows; the announcer gives the phone number of the man's
bedroom phone as the number he's calling, and a male voice answers.  It's
been suggested that such shows don't usually give the phone number being
called; so instead the wife's name could be given as who's being called,
and there could be appropriate background sounds when the other man
answers the phone.

1.15.  He worked as a DJ at a radio station.  He decided to kill his wife,
and so he put on a long record and quickly drove home and killed her,
figuring he had a perfect alibi: he'd been at work.  On the way back he
turns on his show, only to discover that the record is skipping.
1.15a.  Variant: The music stops and the man dies.  Answer: The same,
except it's a tape breaking instead of a record skipping.  (from Michael
Killianey)  (See also #1.16, #1.19e, and #1.34a.)

1.16.  The woman is a tightrope walker in a circus.  Her act consists of
walking the rope blindfolded, accompanied by music, without a net.  The
musician (organist, or calliopist, or pianist, or whatever) is supposed to
stop playing when she reaches the end of the rope, telling her that it's
safe to step off onto the platform.  For unknown reasons (but with
murderous intent), he stops the music early, and she steps off the rope to
her death.
1.16a.  Variant answer: The woman is a character in an opera, who "dies"
at the end of her song.
1.16b.  Variant answer: The "woman" is the dancing figure atop a music
box, who "dies" when the box runs down.  (Both of the above variants would
probably require placing this puzzle in section 2 of the list.)
1.16c.  Variant: Charlie died when the music stopped.  Answer: Charlie was
an insect sitting on a chair; the music playing was for the game Musical
Chairs.  (from Bob Philhower)
(See also #1.15a, #1.19e, and #1.34a.)

1.17.  The man is a blind midget, the shortest one in the circus.  Another
midget, jealous because he's not as short, has been sawing small pieces
off of the first one's cane every night, so that every day he thinks he's
taller.  Since his only income is from being a circus midget, he decides
to kill himself when he gets too tall.
1.17a.  Slightly variant answer: Instead of sawing pieces off of the
midget's cane, someone has sawed the legs off of his bed.  He wakes up,
stands up, and thinks he's grown during the night.
1.17b.  Variant: A pile of sawdust, no net, a man dies.  Answer: A midget
is jealous of the clown who walks on stilts.  He saws partway through the
stilts; the clown walks along and falls and dies when they break.  (from
Peter R. Olpe)
1.17c.  Rough sketch of variant: There were a mirror and a bottle on the
table, and sawdust on the floor.  He came in and dropped dead.  Answer: He
was a midget, but he wasn't aware of it, because the table used to be too
high for him to see his reflection in the mirror, until someone shortened
its legs.  He was horrified by the discovery, and the shock killed him.
(vaguely remembered by Ivan A Derzhanski, who adds that this would be best
used as raw material for some elaboration.  I agree; it's pretty
implausible as is)

1.18.  The man is a lion-tamer, posing for a photo with his lions.  The
lions react badly to the flash of the camera, and the man can't see
properly, so he gets mauled.
1.18a.  Variant: He couldn't find a chair, so he died.  Answer: He was a
lion-tamer.  This one is kind of silly, but I like it, and it sounds
possible to me (though I'm told a whip is more important than a chair to a
lion-tamer).  (from "Reaper Man," with Karl Heuer wording)

1.19.  A blind man enjoys walking near a cliff, and uses the sound of a
buoy to gauge his distance from the edge.  One day the buoy's anchor rope
breaks, allowing the buoy to drift away from the shore, and the man walks
over the edge of the cliff.
1.19a.  Variant: A bell rings.  A man dies.  A bell rings.  Answer: A
blind swimmer sets an alarm clock to tell him when and what direction to
go to shore.  The first bell is a buoy, which he mistakenly swims to,
getting tired and drowning.  Then the alarm clock goes off.  In other
variations, the first bell is a ship's bell, and/or the second bell is a
hand-bell rung by a friend on shore at a pre-arranged time.
1.19b.  Variant answer to 1.19a: The man falls off a belltower, pulling
the bell-cord (perhaps he was climbing a steeple while hanging onto the
rope), and dies.  The second bell is one rung at his funeral.  Could also
be a variant on 1.19 (as suggested by Mike Neergaard): the bell-cord
breaks when he falls (and there's no second bell involved).
1.19c.  Variant answer to 1.19a: The man is a boxer.  The first bell
signals the start of a round; the second is either the end of the round or
a funeral bell after he dies during the match.  Could also be a variant on
1.19 (as suggested by Mike Neergaard): a boxing match in which the top
rope breaks, tumbling a boxer to the floor (and he dies of a concussion).
1.19d.  Variant: The wind stopped blowing and the man died.  Answer: The
sole survivor of a shipwreck reached a desert isle.  Unfortunately, he was
blind.  Luckily, there was a freshwater spring on the island, and he
rigged the ship's bell (which had drifted to the island also) at the
spring's location.  The bell rang in the wind, directing him to water.
When he was becalmed for a week, he could not find water again, and so he
died of thirst.  (from Peter R. Olpe)
1.19e.  Variant: The music stopped and the man died.  Answer: Same as
1.19a, but the blind swimmer kept a portable transistor radio on the beach
instead of a bell.  When the batteries gave out, he got lost and drowned.
(from Joe Kincaid)  (See also #1.15a, #1.16, and #1.34a.)

1.20.  The woman is the assistant to a (circus or sideshow) knife thrower.
The new shoes have higher heels than she normally wears, so that the
thrower misjudges his aim and one of his knives kills her during the show.

1.21.  Several men were shipwrecked together.  They agreed to survive by
eating each other a piece at a time.  Each of them in turn gave up an arm,
but before they got to the last man, they were rescued.  They all demanded
that the last man live up to his end of the deal.  Instead, he killed a
bum and sent the bum's arm to the others in a box to "prove" that he had
fulfilled the bargain.  Later, one of them sees him on the subway, holding
onto an overhead ring with the arm he supposedly cut off; the other
realizes that the last man cheated, and kills him.
1.21a.  Variant wording: A man sends a package to someone in Europe and
gets a note back saying "Thank you.  I received it."  Answer: This is just
a simpler version; the shipwreck situation is the same, and the man
actually did send his own arm.
1.21b.  Variant wording: Two men throw a box off of a cliff.  Answer:
Exactly the same situation as in 1.21a (one slight variation has a hand in
the box instead of a whole arm), with the two men being two of the fellow
passengers who had already lost their arms.
1.21c.  Variant wording: A man in a Sherlock Holmes-style cape walks
into a room, places a box on the table and leaves.  Answer: In this one
he's wearing the cape either to disguise the fact that he hasn't really
cut off his arm/hand as required, or else simply in order to hide his
now-missing limb.  (from Joe Kincaid)

1.22.  Both women are white; the one whose house this takes place in is
single.  A black friend of the other woman, the one who goes into the
bathroom, was recently killed, reportedly by the KKK.  The woman who goes
into the bathroom discovers a bloodstained KKK robe in the other's laundry
hamper, picks up a nail file from the medicine cabinet (or some other
impromptu weapon), and goes out and kills the other.
1.22a.  Variant: A man goes to hang his coat and realises he will die that
day.  Answer: The man (who is black) has car trouble and is in need of a
telephone.  He asks at the nearest house and on being invited in goes to
hang his coat, whereupon he notices the white robes of the Ku Klux Klan in
the closet.  (from Bernd Wechner)

1.23.  He is in a hotel, and is unable to sleep because the man in the
adjacent room is snoring.  He calls the room next door (from his own
room number he can easily figure out his neighbor's, and from the room
number, the telephone number).  The snorer wakes up, answers the phone.
The first man hangs up without saying anything and goes to sleep before
the snorer gets back to sleep and starts snoring again.
1.23a.  Slightly variant answer: It's a next-door neighbor in an apartment
building who's snoring, rather than in a hotel.  The caller thus knows his
neighbor and the phone number.

1.24.  It's the man's fiftieth birthday, and in celebration of this he
plans to kill his wife, then take the money he's embezzled and move on to
a new life in another state.  His wife takes him out to dinner; afterward,
on their front step, he kills her.  He opens the door, dragging her body
in with him, and all the lights suddenly turn on and a group of his
friends shout "Surprise!"  He kills himself.  (Note that the whole first
part, including the motive, isn't really necessary; it was just part of
the original story.)

1.25.  Abel is a prince of the island nation that he landed on.  A cruel
and warlike prince, he waged many land and naval battles along with his
father the king.  In one naval encounter, their ship sank, the king died,
and the prince swam to a deserted island where he spent several months
building a raft or small boat.  In the meantime, a regent was appointed to
the island nation, and he brought peace and prosperity.  When Prince Abel
returned to his kingdom, Cain (a native fisherman) realized that the peace
of the land would only be maintained if Abel did not reascend to his
throne, and killed the prince (with a piece of driftwood or some other
impromptu weapon).

1.26.  The drinks contain poisoned ice cubes; one man drinks slowly,
giving them time to melt, while the other drinks quickly and thus doesn't
get much of the poison.  The fact that they drink at different speeds
could be added to the statement, possibly along with red herrings such as
saying that one of the men is big and burly and the other short and thin.

1.27.  Joe is a kid who goes trick-or-treating for Halloween.

1.28.  He's a smuggler.  On the first cruise, someone brings the
contraband to his cabin, and he hides it in an air conditioning duct.
Returning to the U.S., he leaves without the contraband, and so passes
through customs with no trouble.  On the second trip, he has the same
cabin on the same ship.  Because it doesn't stop anywhere, he doesn't have
to go through customs when he returns, so he gets the contraband off
safely.

1.29.  Hans and Fritz do everything right up until they're filling out a
personal-information form and have to write down their birthdays.  Fritz'
birthday is, say, July 7, so he writes down 7/7/15.  Hans, however, was
born on, say, June 20, so he writes down 20/6/18 instead of what an
American would write, 6/20/18.  Note that this is only a problem because
they *claim* to be returning Americans; as has been pointed out to me,
there are lots of other nations which use the same date ordering.

1.30.  Another WWII story.  Greg is a German spy.  His "friend" Tim is
suspicious, so he plays a word-association game with him.  When Tim says
"The land of the free," Greg responds with "The home of the brave."  Then
Tim says "The terror of flight," and Greg says "The gloom of the grave."
Any U.S. citizen knows the first verse of the national anthem, but only a
spy would have memorized the third verse.  (Why Tim knew the third verse
is left as an exercise to the reader.)

1.31.  The dead man was the driver in a hit-and-run acccident which
paralyzed its victim.  The victim did manage to get the license plate
number of the car; now in a wheelchair, he eventually tracked down the
driver and shot and killed him.

1.32.  His home is a houseboat and he has run out of water while on an
extended cruise.
1.32a.  Variant wording: A man dies of thirst in his own home.  This
version goes more quickly because it gives more information; but it may be
less likely to annoy people who think the original statement is too vague.

1.33.  I'm told this is a true story.  Windows in Paris at that time were
apparently imperfectly flat; they could act as lenses.  One particularly
hot day, the sun shining in through such a window caused a woman's
lingerie (which she was wearing at the time, awaiting her husband's
return) to catch fire, and eventually the entire house caught and burned.

1.34.  He's leaving a hospital after visiting his wife, who's on heavy
life-support.  When the power goes out, he knows she can't live without
the life-support systems (he assumes that if the emergency backup
generator were working, the elevator wouldn't lose power; this aspect
isn't entirely satisfactory, so in a variant, the scene is at home rather
than in a hospital).
1.34a.  Variant: The music stops and a woman dies.  Answer: The woman is
confined in an iron lung, and the music is playing on her radio or stereo.
The power goes out.  (from Randy Whitaker)  (See also #1.15a, #1.16, and
#1.19e.)

1.35.  A large man comes home to the penthouse apartment he shares with
his beautiful young wife, taking the elevator up from the ground floor.
He sees signs of lovemaking in the bedroom, and assumes that his wife is
having an affair; her beau has presumably escaped down the stairs.  The
husband looks out the French windows and sees a good-looking man just
leaving the main entrance of the building.  The husband pushes the
refrigerator out through the window onto the young man below.  The husband
dies of a heart attack from overexertion; the young man below dies from
having a refrigerator fall on him; and the wife's boyfriend, who was
hiding inside the refrigerator, also dies from the fall.

1.36.  Let's say "she" is named Suzy, and "they" are named Harry and Jane.
Harry is an elderly archaeologist who has found a very old skeleton, which
he's dubbed "Jane" (a la "Lucy").  Suzy is a buyer for a museum; she's
supposed to make some sort of purchase from Harry, so she invites him to
have a business dinner with her (at a restaurant).  When she calls to
invite him, he keeps talking about "Jane," so Suzy assumes that Jane is
his wife and says to bring her along.  Harry, offended, calls Suzy's boss
and complains; since Suzy should've known who Jane was, she gets fired.

1.37.  The man is delivering a pardon, and the flicker of the lights
indicates that the person to be pardoned has just been electrocuted.

1.38.  The murderer sets the car on a slope above the hot dog stand where
the victim works.  He then wedges an ice block in the car to keep the
brake pedal down, and puts the car in neutral, after which he flies to
another city to avoid suspicion.  It's a warm day; when the ice melts, the
car rolls down the hill and strikes the hot dog man at his roadside stand,
killing him.

1.39.  There's a car wash on that corner.  On rainy days, the rain reduces
traction.  On sunny days, water from the car wash has the same effect.  If
rain is threatening, though, the car wash gets little business and thus
doesn't make the road wet, so I can take the corner faster.

1.40.  The object she throws is a boomerang.  It flies out, loops around,
and comes back and hits her in the head, killing her.  Boomerangs do not
often return so close to the point from which they were thrown, but I
believe it's possible for this to happen.
1.40a.  Silly variant answer: She's in a submarine or spacecraft and
throws a heavy object at the window, which breaks.

1.41.  He is a passenger in an airplane and sees the bird get sucked into
an engine at 20,000 feet.

1.42.  They're the remains of a melted snowman.

1.43.  One of the brothers (A) confesses to the murder.  At his trial, his
brother (B) is called as the only defense witness; B immediately
confesses, in graphic detail, to having committed the crime.  The defense
lawyer refuses to have the trial stopped, and A is acquitted under the
"reasonable doubt" clause.  Immediately afterward, B goes on trial for the
murder; A is called as the only defense witness and HE confesses.  B is
declared innocent; and though everyone knows that ONE of them did it, how
can they tell who?  Further, neither can be convicted of perjury until
it's decided which of them did it...  I don't know if that would actually
work under our legal system, but someone else who heard the story said
that his father was on the jury for a VERY similar case in New York some
years ago.  Mark Brader points out that the brothers might be convicted of
conspiracy to commit perjury or to obstruct justice, or something of that
kind.

1.44.  He is a mail courier who delivers packages to the different foreign
embassies in the United States.  The land of an embassy belongs to the
country of the embassy, not to the United States.

1.45.  A man was shot during a robbery in his store one night.  He
staggered into the back room, where the telephone was, and called home,
dialing by feel since he hadn't turned on the light.  Once the call went
through he gasped, "I'm at the store.  I've been shot.  Help!" or words to
that effect.  He set the phone down to await help, but none came; he'd
treated the telephone pushbuttons like cash register numbers, when the
arrangements of the numbers are upside down reflections of each other.
The stranger he'd dialed had no way to know where "the store" was.

1.46.  The dead man was playing Santa Claus, for whatever reason; he
slipped while coming down the chimney and broke his neck.
1.46a.  Variant answer: The dead man WAS Santa Claus.  This moves the
puzzle to section 2.

1.47.  The man was struck by an object thrown from the roof of the Empire
State Building.  Originally I had the object being a penny, but several
people suggested that a penny probably wouldn't be enough to penetrate
someone's skull.  Something aerodynamic and heavier, like a dart, was
suggested, but I don't know how much mass would be required.
1.47a.  Variant: A man is found dead outside a large marble building with
three holes in him.  Answer: The man was a paleontologist working with the
Archaeological Research Institute.  He was reviving a triceratops frozen
in the ice age when it came to life and killed him.  This couldn't
possibly happen because triceratops didn't exist during the ice age.
(from Peter R. Olpe)

1.48.  The man died from eating a poisoned popsicle.

1.49.  The man was a sword swallower in a carnival side-show.  While he
was practicing, someone tickled his throat with the feather, causing him
to gag.

1.50.  A mosquito bit me, and I swatted it when it later landed on my
ceiling (so the blood is my own as well as the mosquito's).

1.51.  The man is a lighthouse keeper.  He turns off the light in the
lighthouse and during the night a ship crashes on the rocks.  Seeing this
the next morning, the man realizes what he's done and commits suicide.
1.51a.  Variant, similar to #1.15: The light goes out and a man dies.
Answer: The lighthouse keeper uses his job as an alibi while he's
elsewhere committing a crime, but the light goes out and a ship crashes,
thereby disproving the alibi.  The lighthouse keeper kills himself when he
realizes his alibi is no good. (From Eric Wang)
1.51b.  Variant answer to 1.51a: Someone else's alibi is disproven.  (A
man commits a heinous crime, claiming as his alibi that he was onboard a
certain ship.  When he learns that it was wrecked without reaching port
safely, he realizes that his alibi is disproven and commits suicide to
avoid being sent to prison.)  (From Eric Wang)

1.52.  They were skydiving.  He broke his arm as he jumped from the plane
by hitting it on the plane door; he couldn't reach his ripcord with his
other arm.  She pulled the ripcord for him.
1.52a.  Sketch of variant answer: The ring was attached to the pin of a
grenade that he was holding.  Develop a situation from there.

1.53.  The man is a travel agent.  He had sold someone two tickets for an
ocean voyage, one round-trip and one one-way.  The last name of the man
who bought the tickets is the same as the last name of the woman who
"fell" overboard and drowned on the same voyage, which is the subject of
the article he's reading.

1.54.  The man is a beekeeper, and the bees attack en masse because they
don't recognize his fragrance.  Randy adds that this is based on something
that actually happened to his grandfather, a beekeeper who was severely
attacked by his bees when he used a new aftershave for the first time in 10
or 20 years.

1.55.  He is a guard / attendant in a leper colony.  The letter (to him)
tells him that he has contracted the disease.  The key is the cigarette
burning down between his fingers -- leprosy is fairly unique in killing off
sensory nerves without destroying motor ability.  (Randy was told this by
Gary Haas and Chris Englehard)

1.56.  The man was a famous artist.  A woman who collected autographs saw
him dining; after he left the restaurant, she purchased the check that he
used to pay for the meal from the restaurant manager.  The check was
therefore never cashed, so the artist never paid for the meal.

1.57.  The movie is at a drive-in theatre.


Section 2: Double meanings, fictional settings, and miscellaneous others.

2.1.  The man is a heroin addict, and has contracted AIDS by using an
infected needle.  In despair, he shoots himself up with an overdose,
thereby committing suicide.

2.2.  The man walks into a casino and goes to the craps table.  He bets
all the money he owns, and shoots craps.  Since he is now broke, he
becomes despondent and commits suicide.

2.3.  Kids getting their pictures taken with Santa.  I see #2.1, #2.2, and
#2.3 as different enough from each other to merit separate numbers,
although they all rely on the same basic gimmick of alternate meanings of
the word "shoot."

2.4.  It's the cabin of an airplane that crashed there because of the
snowstorm.
2.4a.  Variant wording: A cabin, on the side of a mountain, locked from
the inside, is opened, and 30 people are found dead inside.  They had
plenty of food and water.  (from Ron Carter)

2.5.  He's a priest; he is marrying them to other people, not to himself.

2.6.  The "island" is a traffic island.

2.7.  A baseball game is going on.  The base-runner sees the catcher
waiting at home plate with the ball, and so decides to stay at third base
to avoid being tagged out.
2.7a.  Variant: Two men are in a field.  One is wearing a mask.  The other
man is running towards him to avoid him.  Answer: the same, but the
catcher isn't right at home plate; the runner is trying to get home before
the catcher can.  (from Hal Lowery, by way of Chris Riley)  This phrasing
would allow the puzzle to migrate to section 1, but I don't like it as
much.

2.8.  The man is an astronaut out on a space walk.

2.9.  The man was an amateur mechanic, the book is a Volkswagen service
manual, the beetle is a car, and the pile of bricks is what the car fell
off of.

2.10.  The Eagle landed in the Sea of Tranquility and will likely remain
there for the foreseeable future.

2.11.  It's a wolf pack; they've killed and eaten (most of) the man.

2.12.  The dead man is Superman; the rock is Green Kryptonite.  Invent a
reasonable scenario from there.

2.13.  This is a post-holocaust scenario of some kind; for whatever
reason, the man believes himself to be the last human on earth.  He
doesn't want to live by himself, so he jumps, just before the telephone
rings...  (of course, it could be a computer calling, but he has no way of
knowing).

2.14.  The one who looks around sees his own reflection in the window
(it's dark outside), but not his companion's.  Thus, he realizes the other
is a vampire, and that he's going to be killed by him.

2.15.  The "bicycles" are Bicycle playing cards; the man was cheating at
cards, and when the extra card was found, he was killed by the other
players.
2.15a.  Variant: There are 53 bees instead of 53 bicycles.  Answer: The
same (Bee is another brand of playing cards).
2.15b.  Variant: There are 51 instead of 53.  Answer: Someone saw the guy
conceal a card, and proved the deck was defective by turning it up and
pointing out the missing ace.  Or, the game was bridge, and the others
noticed the cheating when the deal didn't come out even.  The man had
palmed an ace during the shuffle and meant to put it in his own hand
during the deal, but muffed it.  (both answers from Mark Brader)

2.16.  A chess game; knight takes pawn.
2.16a.  Variant: It's the year 860 A.D., at Camelot.  Two priests are
sitting in the castle's chapel.  The queen attacks the king.  The two
priests rise, shake hands, and leave the room.  Answer: The two priests
are playing chess; one of them just mated by moving his queen.  (from
Ellen M. Sentovich)
2.16b.  Variant: A black leader dies in Africa.  Answer: The black leader
is a chess king, and the game was played in Africa.  (from Erick
Brethenoux)

2.17.  It's a model train set.
2.17a.  Variant: The Orient Express is derailed and a kitten plays nearby.
Answer: The Orient Express is a model train which has been left running
unattended.  The kitten has playfully derailed it.  (from Bernd Wechner)

2.18.  It's a game of Monopoly.

2.19.  The sun is shining; there's no rain.

2.20.  It's daytime; the sun is out.

2.21.  Alice is a goldfish; Ted is a cat.
2.21a.  A very common variant uses the names Romeo and Juliet instead, to
further mislead audiences.  For example: Romeo is looking down on Juliet's
dead body, which is on the floor surrounded by water and broken glass.
(from Adam Carlson)
2.21b.  Minor variant: Tom and Jean lay dead in a puddle of water with
broken pieces of glass and a baseball nearby.  Answer: Tom and Jean are both
fish; it was a baseball, rather than a cat, that broke their tank.  (from
Mike Reymond)

2.22.  Friday is a horse.
2.22a.  Variant with the same basic gimmick: A woman comes home, sees
Spaghetti on the wall and kills her husband.  Answer: Spaghetti was the
name of her pet dog.  Her husband had it stuffed and mounted after it made
a mess on his rug.  (Simon Travaglia original)

2.23.  Bruce is a horse.

2.24.  Should be done orally; the envelope is an evelope of dye, and she's
dying some cloth, but it sounds like "opens an envelope and dies" if said
out loud.

2.25.  The native chief asked him, "What is the third baseman's name in
the Abbot and Costello routine 'Who's on First'?"  The man, who had no
idea, said "I don't know," the correct answer.  However, he was a major
smartass, so if he had known the answer he would have pointed out that
What was the SECOND baseman's name.  The chief, being quite humorless,
would have executed him on the spot.  This is fairly silly, but I like it
too much to remove it from the list.

2.26.  The men have gone spelunking and have taken an Igloo cooler with
them so they can have a picnic down in the caves.  They cleverly used dry
ice to keep their beer cold, not realizing that as the dry ice sublimed
(went from solid state to vapor state) it would push the lighter oxygen
out of the cave and they would suffocate.

==> logic/smullyan/black.hat.p <==
Three logicians, A, B, and C, are wearing hats, which they know are either
black or white but not all white. A can see the hats of B and C; B can see
the hats of A and C; C is blind.  Each is asked in turn if they know the color
of their own hat.  The answers are:
	A: "No."
	B: "No."
	C: "Yes."
What color is C's hat and how does she know?

==> logic/smullyan/black.hat.s <==
A must see at least one black hat, or she would know that her hat is black
since they are not all white.  B also must see at least one black hat, and
further, that hat had to be on C, otherwise she would know that her
hat was black (since she knows A saw at least one black hat).  So C knows
that her hat is black, without even seeing the others' hats.

==> logic/smullyan/fork.three.men.p <==
Three men stand at a fork in the road.  One fork leads to Someplaceorother;
the other fork leads to Nowheresville.  One of these people always answers
the truth to any yes/no question which is asked of him.  The other always
lies when asked any yes/no question.  The third person randomly lies and
tells the truth.  Each man is known to the others, but not to you.
What is the least number of yes/no questions you can ask of these men and
pick the road to Someplaceorother?

==> logic/smullyan/fork.three.men.s <==
It is clear that you must ask at least two questions, since you might be
asking the first one of the randomizer and there is nothing you can tell
from his answers.

Start by asking A "Is B more likely to tell the truth than C?"

If he answers "yes", then:
   If A is truthteller, B is randomizer, C is liar.
   If A is liar, B is randomizer, C is truthteller.
   If A is randomizer, C is truthteller or liar.

If he answers "no", then:
   If A is truthteller, B is liar, C is randomizer.
   If A is liar, B is truthteller, C is randomizer.
   If A is randomizer, B is truthteller or liar.

In either case, we now know somebody (C or B, respectively) who is either
a truthteller or liar.  Now, use the technique for finding information from
a truthteller/liar, viz.:

You ask him the following question: "If I were to ask a person of the opposite
type to yourself if the left fork leads to Someplacerother, would he say yes?"

If the person asked is a truthteller, he will tell you what a liar would
say, which would be the wrong information.  If the person asked is a liar,
he will either tell you what a liar would say, or he will lie about what a
truthteller would say.  In either case, he will report the wrong information.
If the answer is yes, take the right fork, if no take the left fork.

==> logic/smullyan/fork.two.men.p <==
Two men stand at a fork in the road.  One fork leads to Someplaceorother; the
other fork leads to Nowheresville.  One of these people always answers the
truth to any yes/no question which is asked of him.  The other always lies
when asked any yes/no question.  By asking one yes/no question, can you
determine the road to Someplaceorother?

==> logic/smullyan/fork.two.men.s <==
The question to ask is: "Will the other person say the right fork leads to
Someplaceorother?"  If the person asked says yes, then take the left fork,
else take the right fork.  

If the person asked is the truthteller, then he correctly reports that the
liar will misinform you about the right fork.  If he is the liar, then he
lies about what the truthteller will say.  Either way, you should go the
opposite direction from the way that the person asked says the other person
will answer.

The fact that there are two is a red herring - you only need one of 
either type.  You ask him the following question: "If I were to ask a
person of the opposite type to yourself if the left fork leads to
Someplacerother, would he say yes?"

If the person asked is a truthteller, he will tell you what a liar would
say, which would be the wrong information.  If the person asked is a liar,
he will either tell you what a liar would say, or he will lie about what a
truthteller would say.  In either case, he will report the wrong information.
If the answer is yes, take the right fork, if no take the left fork.

This solution also removes the problem that the men may not know the
other's identity.

It is possible, of course, that the liars are malicious, and they will tell
the truth if they figure out that you are trying to trick them.


==> logic/smullyan/integers.p <==
Two logicians place cards on their foreheads so that what is written on the
card is visible only to the other logician.  Consecutive positive integers
have been written on the cards.  The following conversation ensues:
    A: "I don't know my number."
    B: "I don't know my number."
    A: "I don't know my number."
    B: "I don't know my number."
    ... n statements of ignorance later ...
    A or B: "I know my number."
What is on the card and how does the logician know it?

==> logic/smullyan/integers.s <==
If A saw 1, she would know that she had 2, and would say so.  Therefore,
A did not see 1.  A says "I don't know my number."
If B saw 2, she would know that she had 3, since she knows that A did not see
1, so B did not see 1 or 2.  B says "I don't know my number."
If A saw 3, she would know that she had 4, since she knows that B did not
see 1 or 2, so A did not see 1, 2 or 3.  A says "I don't know my number."
If B saw 4, she would know that she had 5, since she knows that A did not
see 1, 2 or 3, so B did not see 1, 2, 3 or 4.  B says "I don't know my number."
... n statements of ignorance later ...
If X saw n, she would know that she had n + 1, since she knows that ~X did not
see 1 ... n - 1, so X did see n.  X says "I know my number."

And the number in n + 1.

==> logic/smullyan/liars.et.al.p <==
Of a group of n men, some always lie, some never lie, and the rest sometimes
lie.  They each know which is which.  You must determine the identity of each
man by asking the least number of yes-or-no questions.

==> logic/smullyan/liars.et.al.s <==
The real problem is to isolate the sometimes liars.

Consider the case of three men:
Ask man 1: "Does man 2 lie more than 3?"
If the answer is yes, then man 2 cannot be the sometimes liar.
Proof by analyzing the cases:
Case 1: Man 2 is not the sometimes liar.
Case 2: Man 2 is the sometimes liar, man 1 is the truth teller, and man 3 is
	the liar.  Then man 1 would not say that man 2 lies more than man 3.
Case 3: Man 2 is the sometimes liar, man 3 is the truth teller, and man 1 is
	the liar.  Then man 1 would not say that man 2 lies more than man 3.
QED.
Similarly, if the answer is no, then man 3 cannot be the sometimes liar.
Now ask the symmetric question of whichever man has been eliminated as the
sometimes liar.  The answer will now allow you to determine the identity
of the sometimes liar.  To determine the identity of the two remaining men, ask
some question like "Does 1=1?" which is always true.

This is not the only way to solve this problem.  You could have asked the
question which is always true (or false) second, which would now establish
the identity of either the liar or the truth teller.  Then ask the third
question of this man to find out which of the other two is the sometimes
liar.

This problem requires three questions, whether or not they are yes-or-no
questions.  In order to identify all three men, you must identify the
sometimes liar.  You cannot identify the sometimes liar in one question
since you may be asking it of the sometimes liar, and any answer from him
conveys no information at all.  Therefore at least two questions are
necessary to identify the sometimes liar.  Once the sometimes liar is
identified, you still need one more question at least to identify the
remaining men.  Therefore, three questions are required.

Suppose we have two truth-tellers, two liars, and two randomizers.
The answer is 8.  A proof follows.

For brevity, "T" means truth-teller, "L" liar, "R" randomizer, "P" predictable
(either T or L).  Define a _pattern_ to be one of the C(6,2)=15 permutations
of RRPPPP (each of which has C(4,2)=6 interpretations of the Ps as 2 Ts and 2
Ls).  For any question Q, let !Q denote the question "If I were to ask you Q,
would you answer Yes?".  Note that question !Q directed toward any P will
yield a truthful answer to question Q; in other words, a "Yes" answer to !Q
means that either Q is true or the respondent is an R, whereas "No" means that
either Q is false or the respondent is an R.

Ask #1, !"Are both Rs in the set {#2, #3, #4}?".  "No" implies that at most
one of {#2, #3, #4} is an R.  "Yes" implies that at most one of {#2, #5, #6}
is an R.  Without loss of generality, assume the former.

Ask #2, !"Is #3 an R?".  "No" implies that #3 is a P.  "Yes" implies that #4
is a P.

Having identified someone as a P, there are at most C(5,2)=10 possible
patterns, and hence at most 10*6=60 possible results.  We can determine which
one reflects reality with at most 6 more questions with a binary search.  (At
each step, bisect the set of possible answers, and ask the question !"Is the
correct pattern in the first subset?".)

Now, let's show that it can't be done in 7.

After asking your first two questions, renumber if necessary so that the first
question was directed to #1 and the second to #2.  (If you asked the same
person twice, you're even worse off than in the analysis below.)  You have no
way to rule out the possibility that both are Rs, so pattern RRPPPP yields 6
possibilities.  Of the four patterns RPRPPP RPPRPP RPPPRP RPPPPR, your first
question gave no information and the second had one bit; so at best you can
eliminate half of these 4*6 possibilities, leaving 12.  Similarly for the four
patterns PRRPPP PRPRPP PRPPRP PRPPPR there remain at least 12 possibilities.
Of the remaining 6 patterns PPRRPP PPRPRP PPRPPR PPPRRP PPPRPR PPPPRR, your
two bits of information can eliminate 3/4 of the 6*6, leaving 9.  Thus, after
two questions there are at least 6+12+12+9=39 arrangements that could have
given the answers you heard; your five remaining questions have only 32
possible replies, so you can't distinguish them.

==> logic/smullyan/painted.heads.p <==
While three logicians were sleeping under a tree, a malicious child painted
their heads red.  Upon waking, each logician spies the child's handiwork as
it applied to the heads of the other two.  Naturally they start laughing.
Suddenly one falls silent.  Why?

==> logic/smullyan/painted.heads.s <==
The one who fell silent, presumably the quickest of the three, reasoned
that his head must be painted also.  The argument goes as follows.
Let's call the quick one Q, and the other two D and S.  Let's assume
Q's head is untouched.  Then D is laughing because S's head is painted,
and vice versa.  But eventually, D and S will realize that their head
must be painted, because the other is laughing.  So they will quit
laughing as soon as they realize this.  So, Q waits what he thinks is
a reasonable amount of time for them to figure this out, and when they
don't stop laughing, his worst fears are confirmed.  He concludes that
his assumption is invalid and he must be crowned in crimson too.


==> logic/smullyan/priest.p <==
A priest takes confession of all the inhabitants in a small town.  He
discovers that in N married pairs in the town, one of the pair has
committed adultery.  Assume that the spouse of each adulterer does not
know about the infidelity of his or her spouse, but that, since it is
a small town, everyone knows about everyone else's infidelity.  In
other words, each spouse of an adulterer thinks there are N - 1
adulterers, but everyone else thinks there are N adulterers.  The
priest, who is an Old Testament type, decides that he should do
something about the situation.  He cannot break the confessional, but
being an amateur logician of sorts, he hits upon a plan to do God's
work.  He announces in Mass one Sunday that the spouse of each
adulterer has the moral obligation to punish his or her adulterous
spouse by publicly denouncing them in church, and that he will make
time during his next Sunday service for this, and continue to do so
until all adulterers have been denounced.  Is the priest correct?  Will
this result in every adulterer being denounced?

==> logic/smullyan/priest.s <==
Yes.  Let's start with the simple case that N = 1.  The offended spouse
reasons as follows: the priest knows there is at least one adulterer,
but I don't know who this person is, and I would if it were anyone
other than me, so it must be me.  What happens if N = 2?  On the first
Sunday, the two offended spouses each calmly wait for the other to get
up and condemn their spouses.  When the other doesn't stand, they
think:  They do not think that they are a victim.  But if they do not
think they are victims, then they must think there are no adulterers,
contrary to what the priest said.  But everyone knows the priest speaks
with the authority of God, so it is unthinkable that he is mistaken.
The only remaining possibility is that they think there WAS another
adulterer, and the only possibility is: MY SPOUSE!  So, they know that
they too must be victims.  So on the next Sunday, they will get up.
What if N = 3?  On the first Sunday, each victim waits for the other
two to get up.  When they do not, they assume that they did not get up
because they did not know about the other person (in other words, they
hypothesize that each of the two other victims thought there was only
one adulterer).  However, each victim reasons, the two will now realize
that they must be two victims, for the reasons given under the N = 2
case above.  So they will get up next Sunday.  This excuse lasts until
the next Sunday, when still no one gets up, and now each victim
realizes that either the priest was mistaken (unthinkable!) or there
are really three victims, and I am ONE!  So, on the third Sunday, all
three get up.  This reasoning can be repeated inductively to show that
no one will do anything (except use up N - 1 excuses as to why no one
got up) until the Nth Sunday, when all N victims will arise in unison.

By the way, the rest of the town, which thinks there are N adulterers,
is about to conclude that their perfectly innocent spouses have been
unfaithful too.  This includes the adulterous spouses, who are about to
conclude that the door swings both ways.  So the priest is playing a
dangerous game.  A movie plot in there somewhere?

==> logic/smullyan/stamps.p <==
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 her stamps, and what is the situation?

==> logic/smullyan/stamps.s <==
B says: "Suppose I have red-red. A would have said on her 
second turn: 'I see that B has red-red. If I also have red-red, then all
four reds would be used, and C would have realized that she had green-green.
But C didn't, so I don't have red-red.  Suppose I have green-green. In that
case, C would have realized that if she had red-red, I would have seen
four reds and I would have answered that I had green-green on my first
turn.  On the other hand, if she also has green-green [we assume that
A can see C; this line is only for completeness], then B would have seen
four greens and she would have answered that she had two reds.  So C would
have realized that, if I have green-green and B has red-red, and if
neither of us answered on our first turn, then she must have green-red.
  "'But she didn't. So I can't have green-green either, and if I can't have 
green-green or red-red, then I must have green-red.'
  So B continues: "But she (A) didn't say that she had green-red, so
the supposition that I have red-red must be wrong.  And as my logic applies
to green-green as well, then I must have green-red."
  So B had green-red, and we don't know the distribution of the others
certainly.
  (Actually, it is possible to take the last step first, and deduce
that the person who answered YES must have a solution which would work
if the greens and reds were switched -- red-green.)

==> logic/timezone.p <==
Two people are talking long distance on the phone; one is in an East-
Coast state, the other is in a West-Coast state.  The first asks the other
"What time is it?", hears the answer, and says, "That's funny.  It's the
same time here!"

==> logic/timezone.s <==
One is in Eastern Oregon (in Mountain time), the other in
Western Florida (in Central time), and it's daylight-savings
changeover day at 1:30 AM.  

==> logic/unexpected.p <==
Swedish civil defense authorities announced that a civil defense drill would
be held one day the following week, but the actual day would be a surprise.
However, we can prove by induction that the drill cannot be held.  Clearly,
they cannot wait until Friday, since everyone will know it will be held that
day.  But if it cannot be held on Friday, then by induction it cannot be held
on Thursday, Wednesday, or indeed on any day.

What is wrong with this proof?

==> logic/unexpected.s <==
This problem has generated a vast literature (see below).  Several
solutions of the paradox have been proposed, but as with most paradoxes
there is no consensus on which solution is the "right" one.

The earliest writers (O'Connor, Cohen, Alexander) see the announcement as
simply a statement whose utterance refutes itself.  If I tell you that I
will have a surprise birthday party for you and then tell you all the
details, including the exact time and place, then I destroy the surprise,
refuting my statement that the birthday will be a surprise.

Soon, however, it was noticed that the drill could occur (say on Wednesday),
and still be a surprise.  Thus the announcement is vindicated instead of
being refuted.  So a puzzle remains.

One school of thought (Scriven, Shaw, Medlin, Fitch, Windt) interprets
the announcement that the drill is unexpected as saying that the date
of the drill cannot be deduced in advanced.  This begs the question,
deduced from which premises?  Examination of the inductive argument
shows that one of the premises used is the announcement itself, and in
particular the fact that the drill is unexpected.  Thus the word
"unexpected" is defined circularly.  Shaw and Medlin claim that this
circularity is illegitimate and is the source of the paradox.  Fitch
uses Godelian techniques to produce a fully rigorous self-referential
announcement, and shows that the resulting proposition is
self-contradictory.  However, none of these authors explain how it can
be that this illegitimate or self-contradictory announcement
nevertheless appears to be vindicated when the drill occurs.  In other
words, what they have shown is that under one interpretation of "surprise"
the announcement is faulty, but their interpretation does not capture the
intuition that the drill really is a surprise when it occurs and thus
they are open to the charge that they have not captured the essence of
the paradox.

Another school of thought (Quine, Kaplan and Montague, Binkley,
Harrison, Wright and Sudbury, McClelland, Chihara, Sorenson) interprets
"surprise" in terms of "knowing" instead of "deducing."  Quine claims
that the victims of the drill cannot assert that on the eve of the last
day they will "know" that the drill will occur on the next day.  This
blocks the inductive argument from the start, but Quine is not very
explicit in showing what exactly is wrong with our strong intuition
that everybody will "know" on the eve of the last day that the drill
will occur on the following day.  Later writers formalize the paradox
using modal logic (a logic that attempts to represent propositions
about knowing and believing) and suggest that various axioms about
knowing are at fault, e.g., the axiom that if one knows something, then
one knows that one knows it (the "KK axiom").  Sorenson, however,
formulates three ingenious variations of the paradox that are
independent of these doubtful axioms, and suggests instead that the
problem is that the announcement involves a "blindspot":  a statement
that is true but which cannot be known by certain individuals even if
they are presented with the statement.  This idea was foreshadowed by
O'Beirne and Binkley.  Unfortunately, a full discussion of how this
blocks the paradox is beyond the scope of this summary.

Finally, there are two other approaches that deserve mention.  Cargile
interprets the paradox as a game between ideally rational agents and finds
fault with the notion that ideally rational agents will arrive at the same
conclusion independently of the situation they find themselves in.  Olin
interprets the paradox as an issue about justified belief: on the eve of
the last day one cannot be justified in believing BOTH that the drill will
occur on the next day AND that the drill will be a surprise even if both
statements turn out to be true; hence the argument cannot proceed and the
drill can be a surprise even on the last day.

For those who wish to read some of the literature, good papers to start with
are Bennett-Cargile and both papers of Sorenson.  All of these provide
overviews of previous work and point out some errors, and so it's helpful to
read them before reading the original papers.  For further reading on the
"deducibility" side, Shaw, Medlin and Fitch are good representatives.  Other
papers that are definitely worth reading are Quine, Binkley, and Olin.

D. O'Connor, "Pragmatic Paradoxes," Mind 57:358-9, 1948.
L. Cohen, "Mr. O'Connor's 'Pragmatic Paradoxes,'" Mind 59:85-7, 1950.
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    -- tycchow@math.mit.edu.

==> logic/verger.p <==
A very bright and sunny Day
The Priest didst to the Verger say:
"Last Monday met I strangers three
None of which were known to Thee.
I ask'd Them of Their Age combin'd
which amounted twice to Thine!
A Riddle now will I give Thee:
Tell Me what Their Ages be!"

So the Verger ask'd the Priest:
"Give to Me a Clue at least!"
"Keep Thy Mind and Ears awake,
And see what Thou of this can make.
Their Ages multiplied make plenty,
Fifty and Ten Dozens Twenty."

The Verger had a sleepless Night
To try to get Their Ages right.
"I almost found the Answer right.
Please shed on it a little Light."
"A little Clue I give to Thee,
I'm older than all Strangers three."
After but a little While
The Verger answered with a Smile:
"Inside my Head has rung a Bell.
Now I know the answer well!"


Now, the question is:

How old is the PRIEST??
               ======