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Frequently Asked Questions (FAQS);faqs.475



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.