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1 Lecture 3 Warm up problem: The Couriers Problem (6.3 Ecco): There are five parts to a top secret code: A, B, C, D, and E. The code must be sent across enemy territory amongst 8 couriers safely to our agent. The enemy will attack 2 of the couriers receiving the parts of the code they contain. We can let them intercept any four parts of the code but not all 5 parts. By making 3 copies of the code how should the different parts be split amongst the 8 couriers to ensure a safe delivery. A C 1) 1\ p 3) D '+) 8 D S) B {; 6) 8 f" 7) <:. E ) c... The Pigeonhole Principle: f p pigeons enter h pigeonholes and if p is greater than nh for some integer n, then at least one pigeonhole contains more than n pigeons. ( Ex.. h 3 ' \ L cv ) P-- ;o p e Or'lS H () le-s 1[7 <> & C> "' 1)- tr ()» ) - j 5 p j G ('( n::: 3 ;;;.? p:::o ><f :::. l'"lh =/ One.. hoi has af leas+!f- p:yeons

2 a) Does a solution exist with 7 couriers? b) Prove this cannot be done with 6 couriers. c) Suppose we can split the code up into as many parts as we want; then what is the minimum number of couriers needed? a) Ves l5 i > p (\ r +-s A E: Cov r ier s- d 7 One.. C<Du t t'cr has a + lectsf- par -1- <; b) C) i) A B C 2) D A 3) D A f) D 8 r;;) [ B 6) E c 7) E c N OVJ D. E jvj us + ) p C.\ ; "ej VV ' f h,_, <... A:;11 :vt by P-HP. on e._ covf'ie..f' a+- lea$+- 3 rclr +-s ) bu-1- nol}/ f"le>"t b(!.., sep dra+ed -" l) A B (._ V O 3) D q.) 5). 6) See, (_Ce:> b- 3 (.solu+-loh) (Y' u.5+ h l\ ve. D) E c...clt/ E?..

3 Example 1: f you pick five different numbers from the integers 1 to 8, show two of them must add up to nine. l...e+.il C!l 4> n umber s Pi r s f-hccr. odd.f() 9 x ( [ ( 8 -> J 2 of<. 7] x,_ g 5 x.7 't 13 Xs or. b 1 c A...s l ( 4- One- hoiq... has f-\}/o d.'5, +,- n c f- f?vm be...r.s wh ich ajj Example 2: Prove that at a party, if certain guests shake hands with certain other guests, there will always be a pair of people who shook the same number of hands. (t is possible that someone didn't shake hands with anyone.) Le + f he. re be- h2: 2 yue-<;. 1-s. C A S l 5 o(y)e..- one.- shoo k no htr.njs ==> f"'lo o ne..- \,\ e. vet yone.s h4nj \ / -1-o q_ ) n - 'L h - i -- (. AS 2. () one:- s hoek /'7 e t' +h r o +he-rs r1 - i A+ 2 3 uesi-.s 5 J ol< e. -1- h e... 5ct(Vl #-

4 Example 3: Prove that in a collection of n + 1 distinct integers, there are distinct integers x and y such that x - y is a multiple of n. L e.+ X ) 2.. l.. ) X nt-f Mod ra.,... n-1 n '. one. vo.lv in n fwo :=) Xi - XJ ;:: (V t>j n ::=;> Xi - XJ = fyl h -f' E> r Example 4: f you pick five points on the surface of an orange, then there is a way to cut the orange so that four of the points will lie on the same hemisphere (suppose a point exactly on the cut belongs to both hemispheres). Or o.ny e. tor d.ny + \,VCD pdin+s; po tn-1-.s M 4 k -e. a. t$.) \::::;} 3 poin1-s tp fa, > G e.,, \..'-.._,, # - One... he(v) is ph ere. h cls +wo (Y7 ()('e... poln+-s four "'"'

5 Example 5: a) Given an 8 by 8 grid of squares, with two opposite comers missing, can you cover the grid with dominos? Why or why not? A domino takes up exactly two board squares. 32_ black Ssua.re.-S Qr"'l.J '" \}.) h ite...,..,- JoM:nos, _ j:-:1.1 h o le..s, _ f1il un-vse.j ----., l::j? 3i 1:#/ 3 (= l f.j::l f%1 # "'. Th No, 31 dominos 11 ee.j ej J c.. an c.o ver (VOSf' 3 Q ho ( t2-.5

6 b) Given a checker board (8 by 8) if any two squares of opposite colors are missing can the board be covered with dominos? y< 1""',; /'\ L 1\ "" K n> Ji /' ',, ",,, lih,, 1(\,, '. 'f' " > / 1'\ v _/... \._ / L-/,ll' 5 TEP +hto CJ_9 h +he...- checker board 5T E P 2- \' \\. s rer ---3 N o w wh;c. h \JJ e., co.n be covere.d pte ces (o n-e... w? f-h dom, f'1o s (:e uld hqve... lerh 8W., o B'y.j o)

7 Example 6: Trevor is trying to fatten up to fit into his tuxedo before an important day. Over a 3 day period, he pledges to eat bacon at least once per day, and 45 times in all. Show there will be a period of consecutive days where he eats bacon exactly 14 times. Le+ s, (. be.. -1-he.. SUM of b Q C.Oh -1-e... on J"Y. (;. -=/ < 5, ( 5 <.. - < 5 == lf-5 2._ 3o ==;> 4- <. 5 t /lf- < S t llf < < 2.. 3D 5 t- }Lf ==- 5CJ- (,O 5. js.rllf (.. c. rany e.-,.--_ r s, s1.. 2 & s3o S,-rl'f St.+- llf ll.f- (> 59 /et1sf 2 of S r a d '-l+e r(!. 1'1 + a,.,.j ;) ; t. ;... ).:>.3 o arc.. \ we ha.ve.-: 5 1.,./LfJ S2.+1lf;. ; 5 3 t-l'f \l ' ',.. from 5-==S +tf for.scj(v).e (.....;. J l e.2e.ct. Tre'"' r c.+ 'r 1 + /ne...s', if

8 Example 7: Show that by placing 5 points anywhere within a square of side length 2, there will surely be a pair of points that are distance J2 or less from each other. OiS'ec+ tbe.. Sguare_ inf.o by Sgt(te.S : - 5 poin + 5 \ 5gua.re5? s.. One... Sf, u Cl( e... leas+- 2. poin-l-s: