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Four Colour Theorem

A Presentation By :-

Priya SharmaIX ‘B’

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PUZZLE-

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Hello! I am Arpita.I am in a bigtrouble. Could youhelp me?

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Ar!ita "ants to #reate a house $ut the !ro$lem isthat she is tol% to !aint ea#h room in a manner

su#h that no & a%'a#ent room ha(e same #olour)

*emem$er:- She has limite% #olours o+ !aint, souse the minimum no) o+ #olour)

 TE .USE

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Let’s try "ith #olour

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.$ser(ation

• /e #an see that it is not !ossi$le to%o so in #olour) Usin0 #olur ma%ee(ery room in the same #olour "hi#h

%oes not satis+y the #on%ition 0i(en)

• 1o" "e "ill ha(e to try "ith &#olours)

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Let’s try "ith & #olours

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o$ser(ation

• /e #an see that e(en $y usin0 &#olours rooms li2e hall- #orri%or,$e%room- #orri%or, (eran%ah- hall,

et#) are o+ same #olour "hi#h %oesnot satis+y our #on%ition)

• 1o" "e "ill ha(e to try "ith 3

#olours)

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Let’s try "ith 3 #olours

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o$ser(ation

• /e #an see that e(en a+ter usin0 3#olours, rooms li2e (eran%ah-#orri%or an% (eran%ah- "ashroom

are o+ same #olour "hi#h %oes notsatis+y our #on%ition)

• 1o" "e ha(e to a%% another #olour

to see i+ it "or2s)

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Let’s try "ith 4 #olours

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o$ser(ation

• /e #an see that usin0 4 #olours,there are no rooms le+t "ith same#olour an% ar!ita #an no" use only 4

#olours to !aint the house)

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Oh! Thank you so much. You helped me a lot! No I

can paint the house

according to the onerschoice and can also use

less colours.

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PUZZLE- &

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O"! #T$%&NT#' YO$( TO%AY# H.).I# A# *O++O)#,,.

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h) ") 0i(en $y tea#her

•  The +ollo"in0 %ra"in0s must $e#oloure% su#h that no & a%'a#entareas must ha(e same #olour)

Fi0ure not to s#ale

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Amit’s metho%

• Amit thin2s that 3 #olours areenou0h to sol(e the 5uestion) e "asa$le to %o the 6rst !art $ut "as stu#2

in the se#on%)

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 Tea#her’s metho%

• It is not ne#essary that e(ery 60ure"ill re5uire only 3 #olours) The 4#olour theorem states that, 0i(en any

se!aration o+ a !lane into #onti0uousre0ions, !ro%u#in0 a 60ure #alle%a ma!, no more than +our #olours are

re5uire% to #olour the re0ions o+ thema! so that no t"o a%'a#ent re0ionsha(e the same #olour)

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PUZZLE- 3

Oh no! I ha-e to go or the orld

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Oh no! I ha-e to go or the orldtour ne/t morning but Im

conused. I ha-e 0 eeks to -isite-ery country and I dont ant

to -isit any 1 ad2acent countries

in the same eek. )ill youplease help me out?

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/orl% ma!

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Let’s use the +our #olour theorem to

sol(e this

First "ee2

Se#on% "ee2

 Thir% "ee2

Fourth "ee2

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 8ou are so intelli0ent) 8ou %is#o(ere% a ne"

theorem)

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is it really true

• Is this theorem really %is#o(ere% $yus, or is alrea%y %is#o(ere%9

• I+ alrea%y %is#o(ere%, "ho %is#o(ere%it9 o" %i% he !ro(e% it9

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story o+ +our #olour theorem

•  The theorem "as 6rst %is#o(ere% $y Fran#isuthrie, "hen he "as tryin0 to #olour thema! o+ #ounties o+ En0lan%) e noti#e% that

only +our %i;erent #olours "ere nee%e%) Atthe time, uthrie

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story o+ +our #olour theorem

• .ne alle0e% !roo+ "as 0i(en $y Al+re% em!e inD, "hi#h "as "i%ely a##laime%G3H another"as 0i(en $y Peter uthrie Tait in ) It "asnot until that em!e

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story o+ +our #olour theorem

•  Tait, in , sho"e% that the +our#olor theorem is e5ui(alent to thestatement that a #ertain ty!e o+

0ra!h ?#alle% a snar2 in mo%ernterminolo0y@ must $e non-!lanar)GH

• In 43, u0o a%"i0er +ormulate%

the a%"i0er #on'e#ture ?a%"i0er43@, a +ar-rea#hin0 0eneraliNationo+ the +our-#olor !ro$lem that still

remains unsol(e%)

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story o+ +our #olour theorem

• Sin#e the !ro(in0 o+ the theorem, eO#iental0orithms ha(e $een +oun% +or 4-#olorin0ma!s re5uirin0 only .?n&@ time, "here n is thenum$er o+ (erti#es) In , 1eil*o$ertson,=aniel P) San%ers, Paul Seymour,an% *o$in Thomas #reate% a 5ua%rati#-time al0orithm, im!ro(in0 on a 5uarti#-timeal0orithm $ase% on A!!el an% a2en’s !roo+

?Thomas *o$ertson et al) @) This ne"!roo+ is similar to A!!el an% a2en

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./ IS TE P*..F I1TE*ESTI1

•  This is a theorem "hose !roo+ 2e!t#han0in0 an% e(ery !roo+ ha% some#ontro(ersy or o$'e#tion)

•

A theorem "hi#h starte% +rom the th #entury an% still mathemati#ians are"or2in0 to im!ro(e the !roo+)

• Proo+s #han0e% o(er time to time $ut ea#h

!roo+ is #orre#t ins!ite o+ o$'e#tions arise%)•  There is no one !erson "ho %is#o(ere% it)

>athemati#ians 0a(e !roo+ o(er !ast &

#enturies)

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PUZZLE- 4

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PuNNle $ase% on +our #olour theorem

• /hat i+ mars is %i(i%e% into areassu#h that ea#h area $elon0s to a#ountry an% they too are #oloure% in

the #olour o+ their #ountry an% $y thesame rule9 o" many #olours arere5uire% no"9

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mars

=i(i%e into !arts ea#h $elon0in0 to a #ountry an%#olour them to the #olour o+ the #ountry they $elon0 to)

*emem$er the rule also) o" many #olours are re5uire%no")

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&-% 60ure o+ mars %i(i%e% a##or%in0

to rule)

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.$ser(ation)

• Sin#e the 60ure is a 0ra!h, the +our#olour theorem "ill a!!ly here) Usin0atlas ea#h #ountry has $een 0i(en an

area +rom the 60ure seen earlier) Stillonly +our #olours "ere re5uire%) This!ro(es that "hate(er the no) o+

!artitions or area $e, +our #olourtheorem "ill a!!ly +or e(ery 0ra!hirres!e#ti(e o+ siNe)

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Thank Yo