Андрей Михайлович Райгородский - Задачи о раскрасках

Jlenurn lllKOJia «COBpeMeHHaJI MaTeMaTMKa»
)(y6Ha, MIOJib

2019

A. M. Pa:Hropo.n;cKM:H

3a,n:aq11 o pacKpacKax

MocKBa
MQHMO
2020

113,ZJ;aTeJihCTBO

519.1
EEK 22.15
P18

Y,LJ;K

1. Bee,n;eHHe
B

3Toii KHHre MhI paccKruKeM o HecKOJihKHX KJiaccHqecKHx rrpo6JieMax

COBpeMeHHOH KOM6HHaTOpHKH H TeopHH rpaoB, CB.H3aHHhlX c IIOH.HTHeM

PaiiropO,ll;CKHH A. M.

P18

3a,n;aqH o pacKpaCKax. -M.: MUHMO,

2020. -32

c.

pacKpaCKH. HaBepH.HKa MHOrHe qf1TaTeJIH HeO,ZJ;HOKpaTHO CTaJIKHBaJIHCh c
3THM rroH.HTHeM. MHorHe CJihIIIIaJIH, HarrpHMep, o 3HaMeHHTOH rHrroTe3e

ISBN 978-5-4439-1472-5

qeThipex KpacoK: Ha mo6oii «,n;ocTaToqHo xopoIIIeH» KapTe MHpa CTpaHhI

B

MOrYT 6bITh TaK IIOKpaIIIeHhI, qTo6bI coce,n;HHe CTpaHhl HMeJIH pa3Hhie :QBe-

KHHre paccKa3bIBaeTC.H o HeCKOJibKHX KJiaccWieCKHX npo6JieMax co-

BpeMeHHoi1 KOM6HHaTop11KH 11 TeOplili rpaoB, CBH3aHHblX c IlOHHTlieM
pacKpacKH. 0Ha OCHOBaHa Ha Kypce, KOTOpbIH aBTOp npO'lliTaJI B ,z:i;y6He Ha
JieTHeH IIIKOJie «COBpeMeHHaH MaTeMaTHKa» B HJOJie

2019 ro,ll;a.

~ cTapIIIeKJiaccHHKOB H CTY,ll;eHTOB MJia,ll;IllliX KypCOB.

Ta, a Bcero :QBeToB 6hmo He 6oJihIIIe qeThipex.

A em;e MHOrHe 3Ha:EOT 3a,n;aqy

0 xpoMaTHqecKOM qflCJie IIJIOCKOCTH: HaHTH HaHMeHhIIIee qf1CJIO :QBeTOB, B
KOTOpbie MO)KHO TaK IIOKpaCHTh BCIO IIJIOCKOCTh, qTo6bI KruK,Zl;hie ,ZJ;Be ToqKH,

EEK 22.15

paCCTO.HHHe Me:>K,ZJ;y KOTOphIMH paBHO e,n;HHH:Qe, 6bmH IIOKpaIIIeHhl B pa3Hhle :QBeTa (CM.

B

[1], [2]).

3TOH KHHre MhI paccKruKeM He 06 3THX 3a,ll;aqax. Ho Te 3aJJ;aqH, o KO-

TOphIX Mhl rroroBOpHM, ceiiqac HrpaIOT KOJIOCCaJihttyIO pOJih B ,ZJ;HCKpeTHOH
MaTeMaTHKe, HaXO,ll;.HTC.H B caMOM ee :QeHTpe -

Ha CThIKe H,ZJ;eH H MeTO,ZJ;OB.

,lJ;JI.H

,n;ocTIDKeHHR MaKCHMaJihHO-

B

KHHre Macca KpacHBhIX pacc)')l(,lJ;eHHH.

ro «KaTapcHCa» IIOJie3HO BJia,ll;eTh OCHOBaMH KOM6HHaTOpHKH (CM., HarrpHMep,

[3]-[7]),

[8], [9]), TeopHH rrpe,n;eJIOB H aCHMII[10]), TeopHH Bepo.HTHOCTeH (CM. [SJ,
[12]). 0,n;HaKo He CTOHT rryraThC.H TaKOro

TeOpHH rpaoB (CM.

TOTHK B ,Zl;HCKpeTHOM aHaJIH3e (CM.

[11])

H JIHHeHHOH aJire6pbI (CM.

H306HJIH.H rrpe,n;MeTOB. Ha caMOM ,n;eJie OT KOM6HHaTOpHKH tty)l(Hhl TOJihKO
qflCJia coqeTaHHH (6HHOMHaJihHhie K03H:QHeHThI)' OT TeOpHH rpaoB TOJihKO 6a30BOe orrpe,n;eJieHHe rpaa, OT TeopHH Bepo.HTHOCTeH -

TOJihKO

ee KOM6HHaTopHhie acrreKThI (Bh16op cnyqaliHoro o6oeKTa H3 KOHeqHoro
MHO)KeCTBa, 6pocaHHe MOHeThl, MaTeMaTHqecKoe O)KH,ll;aHHe rrpocTeHIIIeH

~aHHOH BeJIHqflHhI)' OT aCHMIITOTHK -

rrpocTo IIOHHMaHHe, HarrpHMep,

Toro,qTo

n2 + n,....., n 2

rrpH

n--+ oo

B TOM CMhICJie, qTo rrpe,n;eJI OTHOIIIeHH.H BeJIHqflH n

(tty,

2

+ n H n2 paBeH e,n;HHHIJ;e

MO)KeT, Wb IIOCJIO)KHee, HO Mhl Bee aKKypaTHO H IIOCJie,n;oBaTeJihHO

HaIIOMHHM)' OT JIHHeHHOH aJire6pb1 -

TOJihKO rrpe,n;cTaBJieHHe 0 TOM, qTo

6brnaeT rrpocTpaHCTBO ]Rn' a B HeM BeKTOphI, MaTpH:QhI, CKaJI.HpHbie rrpoH3Be,n;eHH.H...

B

o6m;eM, H3JIO)KeHHe 6yJJ;eT ,ZJ;OCTYIIHO a6coJIIOTHO JI1060MY

3aKOHqHBIIIeMY BTOpOH Kypc H oqeHh MHOrHM CTapIIIeKJiaCCHHKaM, o6yqaIOIIJ;HMC.H B

Kp~ax,

MaTeMaTHqecKHX KJiaccax H BCeB03MO)KHhIX JieTHHX H

He TOJihKO JieTHHX OJIHMIIHa,ZJ;HhIX H He TOJihKO OJIHMIIHa,ll;HhIX IIIKOJiax.

©
©

A. M., 2020.
2020.

Paiiropo,n;CKHH
MUHMO,

KHHra OCHOBaHa Ha KYPCe, KOTOphIH aBTOp rrpoqHTaJI B ,LJ;y6He Ha JieTHeH IIIKOJie «COBpeMeHHa.H MaTeMaTHKa» B HIOJie

3

2019

ro,n;a, a TaK)Ke Ha

MHOroqHCJieHHhIX ero )Ke JieKI.J;H51X Ha BCeB03MO)l(HhlX c6opax (B «CHpHy-

Tenepb H3yqMM Bepo.HTHOCTb Toro, qTo XOrnR 6bL

oaH.a

ce», «KOMaH,ll;e», «KOMnhIOTeppHH» H ,n;p.), Ha pa3JIHqHhIX IIIKOJiax (KHpoBCKruI JIMill, KoM6HHaTOpHKa H aJiropHTMhI H ,n;p .) H yttHBepcHTeTax no Bceii

TI.HTepKa I.J;eJIHKOM
15

CH,D;MT B o,n;HoM Ka6HHeTe . Pa3YMeeTc.H, 3TO Bepo.HTHOCTb co6bITHR

LJ A i .

i =l

PoccHH.

30 qe5 .JiyqlIIHX qlfc15 TIRTepoK. Ko-

HOe co6bITMe, KOTOpoe COCTOHT B TOM, qTo HH o,n;Ha TI.HTepKa u;eJIMKOM He

HeqHo, 3TH TI.HTepKH JiyqIIIHX MOrYT KaK yro,n;Ho nepeceKaTbC.H: 3apaHee Mhl

CH,D;HT B O,ll;HOM H3 Ka6HHeTOB. Ho Be,ZJ;b 3TO pOBHO TO, qTo Hy)l(HO! «HM

,l(JUI 3aTpaBKH paccMoTpHM cJie,n;YJOrn;YJO 3a,n;aqy, B KJiacce yqaTC.H
JIOBeK.

yfa

JIOBHKOB,

HHX OT6HpaIOTC.H

5

JiyqlIIHX KOM6HHaToprn;HKOB,

5 JiyqIIIHX Bepo.HTHOCTHHKOB

H TaK ,n;aJiee. Bcero

3HaqHT, c

nOJlO:HCUmellbH.OU

BepO.HTHOCTblO HMeeT MeCTO rrpOTHBOnOJIO)l(-

BCe-

o,n;Ha TIRTepKa I.J;eJIHKOM He CH,D;HT»- 3TO )Ke TO )Ke caMoe, qTo «B Ka.)1(-

IIIKOJibHHKOB no ,ll;BYM Ka6HHeTaM,

,ll;OM Ka6HHeTe eCTb XOTR 6bI O,ll;HH rrpe,n;c'i'aBHTeJib Ka)l(,ll;OH H3 TI.HTepOK».

qTo6hI B Ka)l(,ZJ;OM Ka6HHeTe 6bIJI XOTH 6hI O,ll;HH rrpe,n;cTaBHTeJih Ka)l(,ZJ;OH H3

11 3TO BbITIOJIHeHo c noJIO)l(HTeJibHOH Bepo.HTHOCTbIO. ECJIM BepoTHOCTb

TI.HTepoK? HanpHMep, eCJIH BCe TI.HTepKH COBna,n;aIOT, TO, oqeBH,D;HO, pac-

paeea.n;KH e H)7)KHbIM CBOHeTBOM nOJIO)l(HTeJibHa, TO TaKruI pacea,n;Ka TOqHo

He 3HaeM, KTO OKa)l(eTC.H CHJibHee B KaKOM H3 HarrpaBJieHHH. Bonpoc:

zaa JlU

MO)l(HO TaK pacca,n;HTh HaIIIHX

30

ca,n;Ka B03MO)l(Ha. Ho Be,ZJ;b eCTh orpoMHOe KOJIHqecTBO ,n;pyrHX BapHaHTOB!

ecTb. KaK ee MCKaTb -

Mo)l(eT, eCJIH TI.HTepKH pacnpe,n;eJIHTC.H 6oJiee XHTpO, TO OKa)l(eTCH, qTo, KaK

Bcer,n;a.

HH paCCa)l(HBaH

30

IIIKOJihHHKOB no ,ll;BYM Ka6HHeTaM, ofornaTeJihHO HaH-

,n;eTC.H Ka6HHeT, B KOTOpOM o,n;Ha H3 TI.HTepoK HaXO,ll;HTCH u;eJIHKOM?
0TBeT Ha BOnpoc BCe-TaKH nOJIO)l(HTeJibHhIH: ,n;a, TaKruI pacca,n;Ka

[13] , [14]).

0TMeTHM, qTo B HaIIIeM

peweH.UU

qTo Beero IIIKOJibHMKOB HMeHHO

BCezaa

B03MO)l(Ha. PeIIIeHHe oqeHh rrpocToe H KpacHBOe ! OHO OCHOBaHO Ha Bepo.HTHOCTHOM MeTO,n;e B KOM6HHaTOpHKe (cp.

,n;pyro:H Bonpoc. Ho 3a,n;aqa peIIIeHa: paeca,n;Ka ecTb

B npHHI.J;Hne 3,ZJ;eCb

30.

HMr,n;e He HenOJib30BaJIC.H TOT cpaKT,

COOTBeTeTBeHHO, B03HHKaeT Bonpoc:

a MO)l(HO JIM, no-rrpe)l(HeMY npeHe6perru1 HCXO,ll;HbIM KOJIHqecTBOM IIIKOJibHHKOB, BMeCTO

15

B3.HTb 60Jib1IIee qHCJIO H noJiyqHTb TOT )Ke pe3yJibTaT c

B03MO)l(HOeTbIO pacca,n;KH? ,l(o IIIeCTHa.n;u;aTH ,n;oTRHYTb JierKo. B eaMOM ,n;e-

Ai

ern;e MO)l(HO 6hIJIO 6hI o6oHTHCh 6e3 CChlJIOK Ha TeopHIO BepO.HTHOCTeH, HO

Jie, Bee eo6bITHR

6y,n;eT Jiyqllle, eCJIH Mhl cpa3y BOCnOJih3yeMC.H HMeHHO Bepo.HTHOCTHOH Tep-

peeeqeHMH Haxo,n;HTe.H ,n;ypau;KruI pacca,n;Ka, rrpH KOTopo:H BCe IIIKOJibHMKH

MHHOJIOrHeii.

CH,ZJ;.HT B nepBOM Ka6MHeTe. Ho TOr,n;a BepO.HTHOCTb o6'he,ll;HHeHH.H

11TaK, HaM ,n;aHhI

c.nJllaUHYJO

15

TIRTepoK. 06o3HaqHM HX

Mv ... , M 15 • PaccMoTpHM

pacca.n;Ky IIIKOJibHHKOB. llTo 3TO 3HaqHT?

Boo6~e

roBopJI, qTO

yro,n;Ho, Be,n;b CJiyqaiiHOCTh MO)l(HO onpe,n;eJUITh no-pa3HOMY· Ho MhI 6y,n;eM nOHHMaTh CJiyqaiiHOCTh MaKCHMaJihHO npOCTO: Ka)l(,ZJ;bIH IIIKOJibHHK
u

6

OTnpaBJUleTC.H B rrepBbIH Ka HHeT c --">