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LOGICAL PROBLEMS Edited
hen did he know?
PROBLEM # 1:
SOLUTION:
In a certain mythical community, politicians never tell the truth and non-politicians always tell the
It is very difficult to answer to this !uestion. ut rather impossi"le.
truth. A stranger meets three natives and ask the first of them,” Are you a politician? “he first native
/any pro"lems are not accurately descri"ed, and their miss description may prove so miss leading
answers the !uestion .he second native then reports that the first native denied "eing a politician
that no solution can part or parts of the description of the pro"lem need to "e re&ected or replaced.
.he third native says that the first native is a politician.
ut we cannot do this "y over selves.
#ow many of these three natives are politicians?
0rom the premises given in the pro"lem, we can prove that he wore red "ut we can also prove that he wore white hat.
SOLUTION: If the first native is a politician then he lies and denies "eing a politician, "ut if he tells the truth and denies "eing a politician, in either case, then, the first native denies "eing a politician.
(ormal 1ision %ne eye man lind man 2 4 2hite
2 2 2
$ince the second native reports that the first native denies "eing a politician, he tells the truth, and is
As at time, he could only wear one hat, either read or white, so it is impossi"le to construct a
therefore a non-politician.
conclusion. It indicates that pro"lem is incomplete.
R 2 2 3 4 3ed
2 R 2
3 R 2
2 2 R
he third native asserts that the first native is a politician, if the first native is a politician, then the third native speaks the truth and is therefore a non-politician, "ut if the third native lies he is,
$ome pro"lems in the real world more over even when they are descri"ed accurately may "e
therefore, a politician. #ence only one of the first and third native is a politician, and since the second
incomplete in that some thing not originally availa"le may "e essential for the solution. he solution
is a non-politician, there is only one politician among the three natives.
may depend on some additional scientific discovery or some previously. 5nimagined invention or e!uipment, or the search of some as yet une)plored territory. ut in the
PROBLEM # 2: %f three prisoners in a certain &ail, one had normal vision, the second had only one eye and the third was totally "lind .the &ailor told the prisoners that from three white hats and two red hats, he would select three and put then on the prisoners' heads. (one could see what color hat he wore. he &ailor offered freedom to the prisoner with normal vision if he could tell what color hat he wore. o prevent a weakly guess the &ailor threatened e)ecution e)ecution for any incorrect answer .the first prisoner could not tell what hat he wore. (e)t the &ailor made the second offer to the one eyed prisoner. prisoner. he second prisoner could not tell what he wore either. he &ailor did not "other, "ut he agreed to e)tend the same terms to that prisoner, "ut when he made the re!uest, the "lind prisoner said* “I do not need to have my sight+ from my friends with eyes have said, I clearly see my hat is
statement of the pro"lem, all the information that is sufficient for the solution must "e given+ other wise we feel that the mystery mystery writer or the pro"lem em maker has "een unfair to us.
PROBLEM # 3: %n a certain train, the crew consists of the "rakeman, the fireman and the engineer. heir names listed alpha"etically are 6ones, 3o"inson, and $mith. %n the train are also three passengers with corresponding corresponding names+ /r. 6ones, /r. 3o"inson and /r. $mith. he following facts are known. a7 /r. 3o"inson lives in 8etroit. "7 he "rakeman lives halfway "etween 8etroit and 9hicago. c7 /r. 6ones earns e)actly : ;<<<< a year.
d7 $mith once "eat the fireman at "illiards.
f7
/r. Gelly is the manager's neigh"or.
e7 he "rakeman's ne)t-door neigh"or, one of the three passengers mentioned, earns e)actly three
2ho holds each position?
times as much as the "rakeman.
SOLUTION:
f7 he passenger living in 9hicago has the same name as the "rakeman.
/anager
Assistant
$tenographer
9ashier
eller
9lerk
/r. lack
>
; Y
@
B
J
SOLUTION:
/r. 2hite
0irst of all we make a chart from the a"ove given information that is+
/rs. 9offee
C
D
E
>< Y
>>
>;
>@ Y
>
>B
>J
>C
>D
/iss Am"rose
>E
;<
;>
;;
;@
; Y
2hat is the engineer's name?
(ame 6ones 3o"inson $mith F 4 %G
/anager
rakeman > Y C
0ireman ; B Y D 4 9ancel
=ngineer @ E Y
/r. Gelly
;B
;J
;C Y
;D
;E
@<
/iss =arn show
@>
@;
@@
@
@B Y
@J
4 9ancel
F 4 %G
0rom Hd7, smith is not fireman as, once he "eat the fireman at "illiards. $o cancel D.
0rom Ha7, /rs. 9offee, /iss.=arnshow and /iss Am"rose cannot "e assistant /anager, so cancel
0rom He7, /r. 6ones is not his neigh"or "ecause he does not earn e)actly three times as much as
>,;< and @; and also /iss Am"rose and /iss =arnshow cannot "e /anager as they are unmarried
rakeman. ecause he earns :;<,<<< a year which is not divisi"le "y @.
so cancel >E and @>.
0rom Hf7, the name of rakeman is 6ones "ecause /r. 6ones is not reak man's neigh"or
0rom H"7, as cashier is male so cancel >J, ;;, and @, as they are females. Also cancel ;> and @@
And rakeman lives in half way "etween 8etroit and 9hicago. 2hile /r. 3o"inson lives in 8etroit. $o
"ecause /iss Am"rose and /iss =arn show are not married.
/r.6ones is only passenger who lives in 9hicago. 9ancel and C and ; @.
0rom Hc7 cancel >, @ and "ecause they all are married.
(ow from the chart it is clear that $mith is engineer. 9ancel J while 3o"inson is the fireman.
0rom Hd7 cancel E and C "ecause at t he age of ;<, /r. 2hite cannot have son in law and grandson. 0rom He7 cancel ;@. $he is not teller.
PROBLEM # 4:
0rom Hf7 /r. Gelly is not manager so cancel ;B.
he employee of a small loan company is /r. lack, /r. 2hite, /rs. 9offee, /iss Am"rose, /r. Gelly
(ow from the chart we can easily say that /iss Am"rose is clerk so cancel J,>;,>D,@< and @J.
and /iss. =arn $haw. he positions they occupy are manager, assistant manager, cashier,
0rom this it concludes that /iss =arnshaw is teller then cancels B, >>, >C and ;E./r lack is
stenographer, teller and clerk. hough not necessary in that order.
assistant manager cancel D and ;J./r 2hite is cashier so cancel ;D. /r. Gelly is stenographer then cancels >B and /rs. 9offee is a manager of a small loan company.
he following facts are known* a7
he assistant manager is the manager's grandson.
PROBLEM# 5:
"7
he cashier is the stenographer's son in law.
enno orelli, Kenial host at #amtramcks most e)clusive nightclu", was shot and killed "y a
c7
/r. lack is a "achelor.
racketeer gang "ecause he fell "ehind in his protection payments. After considera"le effort on the
d7
/r. 2hit e is twenty years old.
part of the police, five suspects were "rought "efore the district attorney, who asked them what they
e7
/iss Am"rose is the teller's stepsister.
had to say for themselves. =ach of them made three statements, two true and one false. heir
c-
2hite is not a resident of rowns1il le, nor is (ash, who is not a "ar"er or a heating
statements were*
contractor. 2ith only the information given, determine the name of the city in which (ash
LEFTY: I did not kill orelli. I never owned a revolver in my life. $pike did it.
resides?
RED: I did not kill orelli. I never owned a revolver in my life. he others are all passing "uck. .
SOLUTION:
DOPEY: I am innocent .I never saw utch "efore. $pike is guilty.
0irst of all cancel >, C, >@, >E and ;B "ecause no man lives in the city having named similar to his
SPIKE: I am innocent. utch is the guilty one. Lefty did not tell the truth. 2hen he said I did it.
name.
BUTC: I did not kill orelli. 3ed is guilty one. 8oppy and I are old pals. 2ho had done it?
SOLUTION: $ince Lefty said that $pike did it, $pike first and third statements an e!uivalent in remaining, and they're fore either "oth true or "oth false. $ince one statement is false, they are "oth true.
2hite rown Meter #arper (ash 4 9ancel
2hite Mlains > J Y >> >J ;>
rown1ille ; C >; Y >C ;;
Meterurg @ D >@ >D ;@ Y F 4 %G
#arper0erry Y E > >E ;
(ash1ille B >< >B ;< Y ;B
0rom Ha7, cancel D and E 0rom H"7, #arper lives in (ash1ille, so cancel >J,>C,>D and B,><,>B.
8opey's third statement is therefore false and so her first two are true. herefore utch third
0romHc7, cancel ; and ;;.
statement is false and so his first two statements are true, of which the second reveals that 3ed i.e.
0rom the chart, it is clear that rown lives in the 2hiteMlains so+ cancel >> and ;>, and Meter lives in
the guilty man.
rown1ille. 0rom Hc7, (ash is not "ar"er and "ar"er does not live in peter "urg. $o white lives in #arper 0erry
PROBLEM # 6:
and (ash is the resident of Meter"urg.
0ive men who are "uddies in the last war are having reunion. hey are 2hite, rown, Meters, #arpen and (ash, who "y occupation are painter, writer, and "ar"er, neurologist and heating contractor. y
PROBLEM # !:
coincidence, they live in the cities of 2hite Mlains, rownville, Meterurg, #arper's0erry and
8anial Gilrain was killed on a lonely road, two miles from Manriac, /ichigan, at @*<
C
(ash1ille, "ut no man live in the city having a name similar to his, no does the name of his
of last uear. %tto, 9urrey, $lim, /ickey and Gid were arrested a week later in 8etroit and !uestioned.
occupation have the same initial as his name or the name of the city which he lives. he following
=ach of the five made four statements, three of which were true and one of which was false. %ne of
facts are known*
these prisoners killed Gilraine. heir statements were
a-
"-
he "ar"er doesn't live in the Meterurg's and rown's is neither a heating constructor nor a painter nor does he live in Meterurg or #arper's 0erry.
OTTO: I was in 9hicago when Gilrain was murdered. I never killed any one. he kid is the guilty one
/r. #arper lives in (ash1ille and is neither "ar"er nor writer
/ickey and I am a pal.
CURLY:I did not kill Gilraine. I never owned a revolver in my life. he kid knows me. I was in 8etriot
MICKEY:
the night of /arch >C.
>7
I did not kill G ilraine. H7
SLIM: 9urly kited when he said he never owned a revolver. he murer was committed in $t, Matrick
;7
he kid has never "een in Montiac. H7
Ns 8ay. %tto was in 9hicago at this time. %ne of us is guilty.
@7
I never saw %tto "efore. H07
MICKEY: I did not kill Garlian. he kid saw %tto "efore. 9urly was in 8etroit with me in the night of
7
9urly was in 8etriot wit h me on the night of /arch >C. H7
/arch >C. TE KID: I did not kill Gilrain. I have never "een in Montiac. I never saw 9urly "efore. %tto erred
TE KID:
when he said I am guilty.
>7
I did not kill kalrine. H7
2ho did it?
;7
I have never "een in Montiac H7
@7
I never saw curly "efore. H07
7
%tto erred when he said I am guilty. HF7
SOLUTION: 0irst of all I give num"ers to the different statements made "y different people. OTTO:
he kid's first and fourth statement is e!uivalent in meanings and therefore either "oth true of "oth
>7
I was in 9hicago when Gialraine was murdered. H7
false. $ince only one statement is false, so they are "oth true.
;7
I never killed anyone. H 7
%tto's third statement is therefore, false, and so his first, second and fourth statements are true.
@7
he kid is guilty one. H07
herefore, /ickey's third statement is false and so his first, second and fourth statements are true.
7
/ickey and I are pals. H7
As /ickey's second statement is true, therefore kid's second statement is also true. he kid's third
CURLY:
statement is false as his other three statements are true.
>7
I did not kill Gilraine. H07
As kids third statement is f alse therefore, 9urly's third statement is true.
;7
I never owned a revolver. H7
9urly's fourth statement is true "ecause /ickey's fourth statement is true.
@7
he kid knows me. H7
$lims third statement is true "ecause %tto first statement is true.
7
I was in the 8etroit that night. H7
As /arch >C is the $t. Matrick's 8ay so slims second statement is true. $lim's fourth statement is also true "ecause all others e)cept 9urly and slims are confirmed to "e innocent. As slims second third
SLIM:
and fourth statements are true so his first statement is false.
>7
9urly lie d when he said he never owned a revolver. H07
herefore 9urly's second statement is true. And his first statement is false .so he is guilty and he
;7
he murder was committed on $t.Matrick's day. H7
killed the Gilraine.
@7
%tto was in 9hicago that night.H7
7
%ne of us is guilty. H7 PROBLEM #": A woman recently hosted a politician meeting to which she invited five guests. he name of the si) people who sat own at the circular ta"le were A"rams, an&o, 9lives, 8umont, =kwll and fish. %ne
was of them, was deaf and one was very talkative, one was terri"ly fat, one simply hated 8umont, one had a vitamin deficiency and one was the hostess.
CART B$:
a7
he person who hated 8umont sat directly opposite an&o.
8eaf
"7
he deaf one sat opposite 9live, who sat "etween the one who had a vitamin deficiency and who hated 8umont.
c7
he fat one sat opposite A"ram, ne)t to the deaf and to left of the one who hated 8umont.
d7
he person who had vitamin deficiency sat "etween 9live and the who sat opposite the
Fish 8umont
alkative
person who hated 8umont. e7
0ish, who was a good friend of every one sat ne)t to the fat person and opposite the
Fat
hostess.
Banjo
Identify each of these people, matching names and description?
SOLUTION:
4 9ancel
F 4 %G
8= A0
ALG II1=
0A
%(= 2#%
%(=
#A1I(K
#A=8
1IA/I(
#%$=$$
Hat Dumont
Vitamin Defi
=kwall
A"rams
Clive
A"rams
>
;
@
85/%(
8=0I9I=(9F B F
J
an&o
C
D
F
E
><
>>
>;
9live
>@
>
>B
>J
>C
>D F
8umont
>E
;<
;> F
;;
;@
;
=kwall
;B
;J
;C
;D F
;E
@<
0rom Ha7, ango does not hat 8umont. As man who hated 8umont is sat opposite to ango so,
0ish
@> F
@;
@@
@
@B
@J
cancel ><.
#ostess
0rom H"7, it is indicated that 9live is not deaf, does not have vitamin deficiency and does not hat 8umont, so cancel >@,>J and >C. 0rom Hc7, it shows that A"rams is not fat, deaf and does not hat 8umont. $o cancel >, @
And .
not the correct way. ecause when he asks to add :; to :;C rather than adding it. y doing so you'll
0rom chart , unto now, it is proved that A"ram has vitamin deficiency. $o cancel ;,J,>>,>C,;@,;E
get the actual amount received "y the clerk or paid "y them to the clerk.
and @B.
An other way to e)plain the twist made "y the writer or pro"lem maker in the !uestion is that he
$ince ango is not fat so, cancel E.
su"tracts :@ from :@< and gets :;C, and then he adds :; to get the original result of :@<. It is the
0rom He7, it is clear that 0ish cannot hat any one so, cancel @.
rule of mathematics that whatever you are addingOsu"tracting, you have to su"tractOadd the same
As he is not fat nor he is hostess therefore cancel @@ and @J.
figure or num"er to neutraliPe the effect of t hat figure. $o, how can you get the same result "y adding
(ow from chart , it is proved that 0ish is deaf and 9live is hostess. $o cancel @;, C,>E and ;B H0or
one thing and su"tracting another?
0ish7 and >, >B ----, >;,;,@< and @J H0or 9live7.
Another method to solve the !uestion is that the clerk receives :;B for the room Hreturns :B7. he
9ancel ;; "ecause 8umont cannot hat himself. $o it is concluded that =kwall hats 8umont. $o
decision of dollars can "e made like this+
cancel ;J, ;C. ango is talkative and 8umont is fat man.
As the clerk receives :;B keep :J on one side and divide the rest :; "y three. It means that each person paid :D to the clerk. (ow add :@ Hhat were returned to the three people "y the "ell man7 to
PROBLEM # %:
:;. he result is :;C. Add :; Hof charity7 we get :;E. (ow add that :C which was kept on one side
hree people went into a hotel and rented a room for : @<, each paying :>< as his share. Later, the
and the result comes :@<. It means that :C was with the clerk.
clerk discovered that the price of the room was only :;B. #e handed the "ellman five :> "ills and asked him to return them to the three people. he "ellman not knowing how to divide :B among three people, instead gave each person :>and the rest to charity. hree people originally paid a total of :;C for the room .Add to that the :; that the "ellman gave away and you have a total e)penditure of :;E instead of :@<. 2hat happened to the other dolor?
SOLUTION: he three people paid :@< to the clerk "ut the original e)penditure for rented room received "y the clerk was :;B. #e handed the "ellman five :> "ills and asked him to return it to the three people. he "ellman not knowing how to divide :B among three people, instead gave each person :> and rest to charity.
he three people paid :>< each, "ut each then received :> "ack, so the total e)penditure for the room that they actually paid was : ;C including : ;that was given to charity "y the "ellman. his is the point where the writer or pro"lem maker twists the !uestion "y asking to add to the : ;C, that :; that the "ell man give away, and you'll then have a total e)penditure of :;E instead of :@<. ut it is
PROBLEM#1&: A &eweler has ten diamonds, nine of them e)actly the same weight, the tenth slightly different. hey are all mi)ed together and his pro"lem is to select the one that is different and to tell whether it is lighter or heavier than the others. #ow can he do t his "y making only three uses of his "alance?
SOLUTION: (um"er the diamonds > to ><, and proceed as follows+ 0irst weight >,;,@,,B,J,C,D
A$
I' t(e) *+,+-.e:
he odd diamond must "e E or ><. 2eight E against C. If E go up, it is the odd diamond and light. If E go down, it is the odd diamond and heavy, if they "alance, the odd diamond must "e ><. hen weight >< against >+ if >< goes up, it is light, if goes down, it is heavy.
B$
I' d/ -/t *+,+-.e:
$uppose that >,;,@, go down, while B,J,C,D go up. hen weight >,;,D,E against @,,C,><. If >,;,D,E
In a certain "ank there are eleven distinct positions namely, in decreasing order Hrank7, Mresident,
go down the odd diamond must "e either > or ; and heavy or C and light. hen weight C against ;+ if
0irst 1ice -Mresident, $econd 1ice Mresident, third 1ice Mresident, 9ashier, eller, Assistant eller,
either goes down, it is the odd diamond and heavy. If they "alance, the odd diamond is C and light. If,
ook Geeper, 0irst $tenographer, $econd $tenographer and 6anitor. hese eleven positions are
on the second weighing @,, C,>< go down, the odd diamond must "e either @ or , and heavy or D
occupied "y the following, listed alpha"etically* /r. 8ale, /r. =vans, /rs. rown, /r. Krant, /iss.
and light. 2eight @ against + if either goes down, it is the odd diamond and heavy+ if they "alance,
#ill, /r. 6ones, /rs. Gane and /r. Long. 9oncerning them, only the following facts are known+
the odd diamond is D and light.
A.
he hird 1ice Mresident is the pampered grand son of the Mresident and is disliked "y "oth /rs. rown and Assistant eller.
PROBLEM#11: Imagine a room with four walls+ a nail placed in the center of each wall, as well as in the ceiling and floor, si) nails in all. he nails are connected to each nail "y a separate string. hese nails strings o"viously make many triangles, since any three nails may "e considered the ape)es of a triangle. 9an the colors of the strings "e distri"uted so that no one triangle has all three sides Hstrings7 of the same color? If so, how? And if not, why not?
SOLUTION: It is not possi"le to distri"ute the strings so that no one triangle has all three sides Hstrings7 of the same color.
9onsider any one nail, say the one on a wall we call A. 0rom it stretch five strings and of these five at least three must "e of the same color, since only two colors H"lue and red7 are availa"le. $uppose that three of the strings from the nail in the wall A are red, that they go to the other three walls , 9 and 8. (ow consider the triangle formed "y the nails on these three other walls , 9 and 8. hey must not all "e of the same color, so they cannot all "e "lue. $o at least one of the strings connecting , 9 and 8 is red, it must complete a triangle of three red strings. H$uppose the string connecting and 8 is red one. hen there will "e a triangle of three red strings connecting A, and 8, etc7. (o matter which nail we "egin with, there is no way to avoid at least one triangle all of whose sides are strings of the same color.
PROBLEM #12:
.
he Assistant eller and the second $tenographer shared e!ually in their father's estate.
9.
he second 1ice Mresident and the Assistant eller wear the same style of hat.
8.
/r. Krant told /iss. #ill to send him a stenographer at once.
=.
he Mresident's nearest neigh"ors are /rs. Gane, /r.Krant and /r. Long.
0.
he first 1ice Mresident and the cashier live at the e)clusive achelor's clu".
K.
he 6anitor, a miser, has occupied the same room since "oyhood.
#.
/r. Adams and the $econd $tenographer are leaders in the social life of the younger unmarried set.
I.
he $econd 1ice Mresident and the ookkeeper were once engaged to "e married to each other.
6.
he fashion a"le eller is the son-in-law of the 0irst $tenographer.
G.
/r. 6ones regularly gives /r. =vans his discarded clothing to wear., without the elderly ookkeeper knowing a"out the gift.
$how how to match correctly the eleven against the eleven positions occupied?
0rom Ha7 it is clear that females cannot "e the third 1ice Mresident. $o cancel >B, @C, D>, ><@, BE and also /rs. rown cannot "e then Mresident and Assistant eller so, cancel >;, >D. As /iss. 8ale and /iss. #ill are unmarried so, they cannot "e Mresident, cancel @ and CD. 0rom Hd7, it is clear that /iss. #ill and /r. Krant cannot "e stenographer. $o cancel CB, CJ and DJ, DC.0rom He7, /rs. Gane, /r. Krant and /r. Long cannot "e president. $o cancel JC, ><< and >>>. 0rom Hf7, it means that 0irst 1ice Mresident and 9ashier are men so, cancel >@, @B, BC,CE, ><> and >J, @D, J< D;, ><, as they all are females. 0rom Hg7, 6anitor is a man so+ cancel all females for 6anitor ;;, , JJ, and DD, >><. 0rom Hh7, it is clear that /r. Adams and second stenographer are unmarried "ecause they are the leaders of the unmarried set and leader of such group can not "e married so, cancel >< for /r. Adams and ;>, JB, >, BD, ><; and >E, J@, >C, @E, and J>, D@, >. 0rom Hk7, /r. 6ones and /r. =vens cannot "e ookkeeper so, cancel B;, EJ.
It is know, up to, confirmed that /rs. rown is 0irst $tenographer so, cancel >>E, >, B@, J, EC. After canceling a"ove, we get that /rs. Gane is Assistant eller so+ cancel C, ;E, <, B>, J;, C@, D, EB and >>C. After this we confirmed /rs. 0ord as Mresident. hey're fore cancel ;@, B and DE. (ow from the chart, we can see that two options are left for /iss. #ill either ; nd 1ice Mresident or ookkeeper and as in HI7 we are told that once ; nd 1ice Mresident and ookkeeper were engaged to "e married to each other, it means that one of them is male and other is female. 2hile in Hc7 we are told that Assistant eller and ; nd 1ice Mresident wear the same style of hat and as it is confirmed that 4 9ancel
F 4 %G
Assistant eller is female H/rs. Gane7 so, ; nd 1ice Mresident must "e female "ecause no male can wear the same type of hat as female wears. It means that /iss. #ill is then ; nd 1ice Mresident and not the ookkeeper. $o cancel @, ;B, @J, C, JE, E>, >>@ and also DB.
As ;nd 1ice Mresident is female so, ookkeeper is then male, cancel >.
Is the third native a politician?
nd
0rom the chart it is clear that /iss. 8ale is ; $tenographer as all other "o)es are crossed. hen cancel @;, B, ED, >;<.
SOLUTION: (o the third is not a politician. If the third native was a politician, it means he is telling a lie "y saying
In Hd7, /r. Krant orders /iss. #ill to send him stenographer. As /iss. #ill is ; nd 1ice Mresident so,
that “that's not true either”. /eans the first t wo natives are telling truth.
/rs. Krant must "e supervisor .$o he is 0irst 1ice Mresident. 9ancel ;, ;, J, E<, >>; C<, C>, C;,
ut the first two native cannot tell the truth at the same time "ecause one is saying* “ 2e are all
C, CC.
politicians” and the second one is saying that* “6ust two of us are politician”.
In Hk7, we are told that /r. 6ones regularly gives /r. =vans his discarded clothing to wear. It means
$o the third native is not telling lie, he is telling the truth and non-politicians always tell the truth.
that /r. 6ones is a f ashiona"le person who regularly used to discard his clothing in order to remain in
hird native is non-politician.
fashion. $o he is a fashiona"le eller "y H&7. $o cancel E;, E@, EE ;D, B<, >>J. 0rom Hk7 it is clear that /r. =vans is a miser as he uses discarded clothes of /r. 6ones and in Hg7 we
HMro"lemQ>@7 Micture a checker"oard Hor chess"oard having eight rows and eight columnsR7
are told that 6anitor is a miser so, /r. =vans is 6anitor. $o cancel D, E, >>, @@, >;>. If so, how? And if not, why not? In He7, it is told that /r. Long and /r. Krant are Mresident's nearest neigh"ors and in Hf7 it is said that 0irst 1ice Mresident and the 9ashier live at the achelor's clu" and it is confirm that 0irst 1ice
SOLUTION:
Mresident is /r. Krant and as /r. Long lives near to /r. Krant so, he must "e cashier as cashier lives
It is not possi"le to leave the upper left and the lower eight corners empty. %ne dominoes is left such
at the =)clusive achelor's 9lu", near to /r. Krant. $o cancel >>,>>D and B, ;C.
that it could not "e settled his is "ecause the empty corners are diagonal to each other.
In Hk7, we are told that ookkeeper is elder than /r. 6ones, who is married and in Hh7 we are told that
If we rearrange "y changing there places i.e. moving the whole "oard to one side and then up or
/r. Adams is the leader in the social life of younger unmarried set. $o when we compare these two
down. hey "oth comes diagonal to each other and the remaining could not "e covered with
statements Hk h7 we come to the conclusion that /r. Adam is younger and ookkeeper is elder, so
dominoes filling two.
he cannot "e the ookkeeper. hus he is the third 1ice Mresident , cancel ;J D. #ence /r. 9amp is
%ne domino comes diagonal+ if we are allowed to place one domino diagonally then it is possi"le.
then the ookkeeper.
%therwise it is not possi"le.
PROBLEM# 13: In the same mythical community descri"ed in =)ercise >, a stranger meets
three other people
H 4 =mpty place7 •
Hnatives7 and asks them, “#ow many of you are politicians?” he first native replies, “2e are all politicians.” he second native says,”(o, 6ust two of us are politicians.” he third native then says, “ hat is not true either.”
8raw ta"le •
• •
Fou are presented with a set of twelve metal "alls, apparently identical in every
respect siPe, color
and so on. Infact eleven of them are identical, "ut one of them is odd it differ from all the rest in weight only it is either heavier or lighter than all the others. Fou are given a "alance scale, on which the "alls can "e weighted against one another. If the same num"er of "alls are put on each side of the "alance and the odd "all is on one side, that side will go down if the odd "all is •
heavier or up if the odd "all is lighter the two sides will "e "alanced if the odd "all is not among them. hose weighted and the same num"ers of "alls are placed on each side. Fou are allowed three weighing only any removal or additions of a "all constitute a separate weighing.
PROBLEM#14: In the same mythical community descri"ed in e)ercise >, a strangerRRRR Is the third native a politician?
Four challenge is this, device a set of three weighing that will ena"le you to identify the odd "all wherever it may lie in a random mi)ing of the twelve "alls and that will ena"le you to determine whether the odd is heavier or lighter than the rest. 0irst la"el the "all i.e. >, ;, @, , B, J, C, D, E and a, ", c, d.
SOLUTION: (o, the third is not a politician. If the third native was a politician, it means he is t elling a lie "y saying that” it isn't true either” means that the first two natives are telling truth "ut the first two natives cannot tell the truth at the same time "ecause one is saying “we are all politicians” and the second one is saying that,” &ust two of us are politicians”. $o the third native is not telling a lie, he is telling the truth. And the non-politician always tells the truth. hird native is a non-politician.
SOLUTION: 0irst weight >, ;, @, against B, J, C, D. If "alanced then it means that odd "all is among a, ", c, d .now weight a " against c, @ or c in com"ination with any num"er among > to D . If they are "alanced then the odd "all is d. now add d against any "all among > to D. if d goes up it means it is lighter or if it goes down it is heavier. If a, "-side is heavier then it means that either a or " is odd and heavier or c is odd and light. (ow weight a against ". the one which is heavier is odd. If "oth a and " are "alanced then c is odd and light. If a, " side goes up it means that either a or " is odd or c is odd and heavy so weight a against ". the one which goes up is odd and if "oth are "alanced then c is odd and heavy. (ow suppose that weighing >, ;, @, against B, J, C, D Hthe first weight7 is not "alanced it means that odd "all is among them.
PROBLEM#15:
$uppose >, ;, @, pans go down while B, J, C, D goes up it means that the odd "all is among > to D and a, ", c, d "alls are right. hen weight >, ;, D, a against @, , C, " if >, ;, D,a goes down the odd
CALLENGING 0UESTION:
"all must "e either > or ; and heavy or C and light. hen weight C and ;. If either goes down it is the odd "all and heavy if they "alance the odd "all is C and light if in ;
nd
weighing @, , C, " go down the
odd "all must "e @ or and heavy or D and light then weight @ against if either goes down it is the odd "all and heavy, if they are "alanced the odd "all is D and light.
P/*,e N/ 1* %n a certain train, t he crew consists of three men, the "rakeman, the fireman, and the engineer. heir names listed alpha"etically are 6ones, 3o"inson and $mith. %n the train are also three passengers with corresponding names, /r. 6ones, /r. 3o"inson and /r. $mith. he following facts are known. a7 /r. 3o"inson lives in 8etroit. "7 he "rakeman lives halfway "etween 8etroit and 9hicago. c7 /r. 6ones earns e)actly :;<,<<< a year. d7 $mith once "eat the fireman at "illiards. e7 he "rakeman's ne)t-door neigh"or, one of the three passengers mentioned, earns e)actly three times as much as the "rakeman. f7 he passenger living in 9hicago has the same name as the "rakeman. (+t +6 t(e e-7i-ee86 -+e9 M+)+ T/;iee86 A-6e y elimination process, we can deduct that* • •
• •
$mith is not the fireman /r. 3o"inson lives in 8etroit so 3o"inson cannot "e the "rakeman since the man who has the same name lives in 9hicago. /r. 6ones earns :;<,<<< that is three times as much as his neigh"or, the "rakeman. $ince 3o"inson is not the "rakeman and $mith is not the fireman "ut 6ones cannot "e the neigh"or of /r. 6ones if he were the "rakeman, $mith must "e the "rakeman and /r. $mith must live in 9hicago.
M+)+ T/;iee86 A-6e /e-: >7 /s. Am"rose is not the teller+ ;7 /rs. 9offee is the manager since she is called /rs therefore can "e the only one to have grand-son+ @7 /iss arnshaw is the teller+ and /s. Am"rose is the clerk. Me-: he cashier is a man+ the assistant manager is a man thus >7 /r. Gelly is the stenographer+ ;7 /r. 2hite is the cashier since he is the son-in-law of the stenographer+ /r. lack is the manager's grand-son. P/*,e N/ 3* enno orelli, genial host at #amtrameck's most e)clusive nightclu" was shot and killed "y a racketeer gang "ecause he fell "ehind in his protection payments. After considera"le effort on the part of the police, five men were "rought "efore the 8istrict Attorney, who asked them what they had to say for themselves. =ach of the men made three statements, two true and one false. heir statements were* L=0F* “I did not kill orelli. I never owned a revolver. $pike did it.” 3=8* “I did not kill orelli. I never owned a revolver. he other guys are all passing the "uck.” 8%M=F* “I am innocent. I never saw utch "efore. $pike is guilty.” $MIG=* “I am innocent. utch is the guilty man. Lefty lied when he said I did it.” 59#* “I did not kill orelli. 3ed is the guilty man. 8opey and I are old pals.” 2hodunnit? M+)+ T/;iee>6 A-6e TRUE •
ANSER: M
•
•
P/*,e N/ 2: he mem"ers of a small loan company are /r. lack, /r. 2hite, /rs. 9offee, /iss Am"rose, /r. Gelly and /iss =arnshaw. he positions they occupy are manager, assistant manager, cashier, stenographer, teller, and clerk, though not necessarily in that order. he assistant manager is the manager's grandson, the casher is the stenographer's son-in-law+ /r. lack is a "achelor. /r. 2hite is ;; years old+ /iss Am"rose is the teller's step-sister+ and /r. Gelly is the manager's neigh"or. (/ (/,d6 e+.( =/6iti/-9 0irst, let's eliminate the males* >7 /r. lack is a "achelor so he can't "e the manager since the manager is a grand-father+ ;7 /r. 2hite is only ;; years old so he can't "e the manager+ and @7 /r. Gelly is the manager's neigh"or+ thus no man holds the manager's position.
•
L=0F* I did not kill orelli. I never owned a revolver. 3=8* he other guys are all passing the "uck $MIG=* I am innocent. Lefty lied when he said I did it. 59#* I did not kill orelli. 3ed it the guilty man.
FALSE • •
• • •
$pike did it. I did not kill orelli. I never owned a revolver. utch is the guilty man 8opey and I are old pals $pike is guilty
8%M=F* I am innocent. I never saw utch "efore S/,ti/-: $ince utch says the truth* “I did not kill orelli” and lies when he says “8opey and I are old pals” is false, so the other statement “3ed is the guilty man,” must "e true. •
RED i6 t(e 7i,t) /-e
P/*,e N/ 4: /rs. Adams, /rs. aker, /rs. 9att, /rs. 8odge, /rs. =nnis and that dowdy /rs. 0isk all went shopping one morning at the =mporium. =ach woman went direly to the floor carrying the article which she wanted to "uy, and each woman "ought only one article. hey "ought a "ook, a dress, a hand"ag, a necktie, a hat, and a lamp.
P/*,e N/ ?: 8aniel Gilraine was killed on a lonely road, two miles f rom Montiac, at @*@< A./., /arch >C, >EB;. %tto, 9urly, $lim, /ickey, and the Gid were arrested a week later in 8etroit and !uestioned. =ach of the five made four statements, three of which were true and one of which was false. %ne of the these men killed Gilraine. 2hodunnit? heir statements were*
All the women e)cept /rs. Adams entered the elevator on the main floor. wo men also entered the elevator. wo women, /rs. 9att and the one who "ought the necktie, got off at the second floor, met her hus"and returning the necktie which one of the other women had given him. If "ooks are sold on the main floor, and /rs. =nnis was the si)th person to get out of the elevator, what did each of these women "uy?
%% said* “I was in 9hicago when Gilraine was murdered. I never killed anyone. he Gid is the guilty man. /ickey and I are pals.” 953LF said* “I did not kill Gilraine. I never owned a revolver in my life. he Gid knows me. I was in 8etroit the night of /arch >Cth.” $LI/ said* “9urly lied when he said he never owned a revolver. he murder was committed on $t. Matrick's day. %tto was in 9hicago at this time. %ne of us is guilty.” /I9G=F said* “I did not kill Gilraine. he Gid has never "een in Montiac. I never saw %tto "efore. 9urly was in 8etroit with me on the night of /arch >Cth. #= GI8 said* “I did not kill Gilraine. I have never "een in Montiac. I never saw 9urly "efore. %tto lied when he said I am guilty.”
M+)+ T/;iee86 A-6e • • • • • •
/rs. =nnith "ought the lamp* Bth floor /rs. Adams "ought the "ook* /ain floor /rs. 9att "ought the hand"ag* ;nd floor /r. 0isk "ought the hat* Jth floor /rs. 8odge "ought the necktie* ;nd floor /rs. aker "ought the dress* @rd floor
M+)+ T/;iee86 A-6e •
•
P/*,e N/ 5: 0ive men who were "uddies in the late war are having a reunion. hey are 2hite, rown, Meters, #arper, and (ash, who "y occupation are printer, writer, "ar"er, neurologist, and heating-contractor. y coincidence, they live in the cities of 2hite Mlains, rownsville, Meters"urg, #arper's 0erry, and (ashville, no man lives in the city having a name similar to his, nor does the name of his occupation have the same initial as his name or the name of the city in which he lives. he "ar"er doesn't live in Meters"urg, and rown is neither heating-contractor nor printer-nor does he live in Meters"urg or #arper's 0erry. /r. #arper lives in (ashville and is neither "ar"er nor writer. 2hite is not a resident of rownsville, nor is (ash, who is not a "ar"er, nor a heating-contractor. If you have only the information given a"ove, can you determine the name of the city in which (ash resides? M+)+ T/;iee86 A-6e • • • • •
rown lives in 2hite Mlains. 2hite lives in #arper's 0erry and is "ar"er. (ash lives in Meters"urg. #arper lives in (ashville. Meter lives in rownsville.
•
Te: $LI/* he murder was committed on $t. Matrick's day. %tto was in 9hicago at this time. %ne of us is guilty. Te: /I9G=F* 9urly was with me in 8etroit on the night of /arch >Cth. F+,6e* 953LF lied when he said he said he never owned a revolver.
A-6e: C,) i6 7i,t) P/*,e N/ !: A woman recently gave a tea party to which she invited five guests. he names of the si) women who sat down at the circular ta"le were /rs. A"rams, /rs. an&o, /rs. 9live, /rs. 8umont, /rs. =kwall, and /rs. 0ish. %ne of them was deaf, one was very talkative, one was terri"ly fat, one simply hated /rs. 8umont, one had a vitamin deficiency, one was the hostess. he woman who hated /rs. 8umont sat directly opposite /rs. an&o. he deaf woman sat opposite /rs. 9live, who sat "etween the woman who had a vitamin deficiency and the woman who hated /s. 8umont. he fat woman sat opposite /rs. A"rams, ne)t to the deaf woman and to the left of the woman who hated /rs. 8umont. he woman who had a vitamin deficiency sat "etween /rs. 9live and the woman who sat opposite the woman who hated /rs. 8umont. /rs. 0ish, who was a good friend of everyone, sat ne)t to the fat woman and opposite the hostess. 9an you identify each of these lovely women? M+)+ T/;iee86 +-e:
M+)+ T/;iee>6 A-6e: >.
is either /rs. =kwall or /rs. 0ish. ut since /rs. 0ish is a good friend with every"ody, ) is /rs. =kwall. 2e have two women left* /rs. 8umont and /rs. 0ish. ut since /rs. 0ish sits ne)t to the fat woman, the woman must "e /rs. 8umont and the deaf woman /rs. 0ish.
P/*,e N/ ": 0ive men are in a poker game* rown, Merkins, urner, 6ones, and 3eilly. heir "rands of cigarettes are Luckies, 9amels, Gools, %ld Kolds, and 9hesterfields, "ut not necessarily in that order. At the "eginning of the game, the num"er of cigarettes possessed "y each of the players was ;<, >B, D, J and @, "ut not necessarily in that order.
rown smokes at least ; cigarettes since each one has at least one left. rown smokes %ld Kolds. ;. 3eil ly smokes 9amels * DO; 4 or B-> 4 @. urner smokes /enthol J-B 4 > . 2e have Merkins or 6ones who smoke either 9hesterfield or Luckies. ut Merkins does not smoke Luckies, he smokes 9hesterfield 4 Hy 7OB. B. 9hesterfield cigarettes 4 Hy 7OB. J. $o 6ones smokes Luckies which is Hy 7OB S at least ; cigarettes. If we follow the alge"ra formula and we su"stitute y "y ;< we will o"tain H;<7OB 4 >J. 5ndou"tedly y 4 >B in order to satisfy the statement that 6ones smoked at least ; cigarettes more than any"ody else. C. hus 9hesterfield smoked >BO; 4 C.B S ;.B 4 ><. D. And 6ones smoked at least >; cigarettes. E. Merkins smokes 9hesterfield. #e has >B at the "eginning, smokes half of the cigarettes C.B so he has C.B cigarettes plus ; T cigarettes 4 >< cigarettes. ><. 6ones smokes Luckies and he smoked at least >; cigarettes.
P/*,e N/ %* In a certain mythical community, politicians always lie, and non-politicians always tell the truth. A stranger meets three natives, and asks the first of them if he is a politician. he first native answers the !uestion. he second native then reports that the first native denied "eing a politician. hen the third native asserts that the first native is really a politician. #ow many of these three natives are politicians?
8uring the game, at a certain time when no one was smoking, the following conditions o"tained* • • •
•
•
• • •
Merkins asked for three cards. 3eilly had smoked half of his original supply or one less than urner smoked. he 9hesterfield man originally had as many more, plus half as many more, plus ; and half more cigarettes than he now has. he man who was drawing to an inside straight could taste only the menthol in his fifth cigarette, the last one he smoked. he man who smokes Luckies had smoked at least two more than anyone else, including Merkins rown drew as many aces as he originally had cigarettes. (o one had smoked all his cigarettes. he 9amel man asks 6ones to pass rown's matches.
#ow many cigarettes did each man have to "egin with and of what "rand?
P/*,e N/ 1&* %f three prisoners in a certain &ail, one had normal vision, the second had only one eye, and the third was totally "lind. All were of at least average intelligence. he &ailer told the prisoners that from three white hats and two red hats he would select three and put them on the prisonersU heads. =ach was prevented from seeing what color hat was placed on his own head. hey were "rought together, and the &ailer offered freedom to the prisoner with normal vision if he could tell what color hat was on his head. he prisoner confessed that he couldnUt tell. (e)t the &ailer offered freedom to the prisoner with only one eye if he could tell what color hat was on his head. he second prisoner confessed that he couldnUt tell. he &ailer did not "other making the offer to the "lind prisoner, "ut agreed to e)tend the same terms to him when he made the re!uest. he "lind prisoner then smiled "roadly and said* VI do not need to have my sight+ 0rom what my friends with eyes have said, I clearly see my hat is
