Kirkman’s Schoolgirl Problem Charlie, Law Ka Kui Billy, Lai Ka Hin

PRESENTATION OUTLINE Kirkman's schoolgirl problem Round-robin Tournament Algorithm to find the solution (Frost method) Algorithm to find the solution for special cases (n ppl in a group, n days, n prime)

KIRKMAN’S SCHOOLGIRL PROBLEM: Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast. Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 

FIRST WE CONSIDER THE CASE WHEN EACH GROUP CONSISTS OF 2 PEOPLE

RULES (2 PEOPLE IN EACH GROUP) Consider a Big Group of n participants, where n is even Each day, we divide them into several small groups Each small group consists of 2 participants Each participant joins exactly 1 small group each day No two participants join the same group more than once

2 PARTICIPANTS EACH GROUP No of Small Groups formed: No of Groups formed Each Day: No of Days to exhaust all possibilities:

2 PARTICIPANTS EACH GROUP Round-robin Tournament Two participants (or groups) compete against each other once Examples: Football Leagues Chess Tournament Go Tournament

2010 FIFA WORLD CUP GROUP A Four teams: Uruguay Mexico South Africa France Number of Matches: Number of Matches Each Day: Number of Days:

FIXTURE Date Team Result 11/6(Match1) v.s. 1:1 11/6(Match2) 0:3 17/6(Match4) 0:2 22/6(Match5) 0:1 22/6 (Match6) 1:2 0:0

QUESTION 8 people, Andy, Benjamin, Chris, Dorothy, Ewan, Francisca, Greg, Hillary are in a meeting of AA (Alcoholics Anonymous) The coordinator wants to arrange them into groups of two so that they can share their experience with every other member Assume that each member meets each other member only once and all of them participate only once each day

QUESTIONS (CONT’D) What is the number of small groups formed each day? Ans: 4 How many days do they need to complete the session? Ans: 7 What is the number of small groups that can be formed? Ans: 28 Homework (1): Draw the timetable of the meetings

RULES (3 PEOPLE IN EACH GROUP) Consider a Big Group of n participants, where n is divisible by 3 Each day, we divide them into several small groups Each small group consists of 3 participants Each participant joins exactly 1 small group each day No two participants join the same group more than once

3 PARTICIPANTS EACH GROUP Question: What is the total number of small groups?

3 PARTICIPANTS EACH GROUP Number of small Groups Each Day: Number of Days: Total number of small groups:

CHINESE POKER (鬥地主) LEAGUE 9 participants (named by 1,2,3,4,5,6,7,8,9) 3 participants in each game 3 games each day The league lasts for 4 days Each participant only plays once a day No two participants meet more than once Question: How can we construct the fixture?

FROST’S METHOD First, we consider the fixture for player1, WLOG Each cell in the first row is filled with (1,a1,a2), (1,b1,b2),(1,c1,c2), (1,d1,d2) respectively Day 1 Day 2 Day 3 Day 4 1, a1, a2 1, b1, b2 1, c1, c2 1, d1, d2

FROST’S METHOD CONT’D Then, we can think about if it is a, then it is a1 or a2. 1, a1, a2 1, b1, b2 1, c1, c2 1, d1, d2 b, c, d a, c, d a, b, d a, b, c

FROST’S METHOD CONT’D Then, we can think about if it is ‘a’, then it is a1 or a2. 1, a1, a2 1, b1, b2 1, c1, c2 1, d1, d2 b1, c1, d1 a1, c1, d2 a2, b1, d2 a2, b2, c1 b2, c2, d2 a2, c2, d1 a1, b2, d1 a1, b1, c1

FROST’S METHOD CONT’D If a1=2, a2=3, b1=4…. Solution: 1, 2, 3 1, 4, 5 1, 6, 7 1, 8, 9 4, 6, 8 2, 6, 9 2, 5, 8 2, 4, 6 5, 7, 9 3, 7, 8 3, 4, 9 3, 5, 7

KIRKMAN’S SCHOOLGIRL PROBLEM: Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast. Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 

KIRKMAN’S SCHOOLGIRL PROBLEM(2) 15 young ladies (1,2,3,4…15) 3 participants a group 7 days Each lady only walks once a day No two ladies walk abreast more than once

{abc, ade, afg, bdf, beg, cdg, cef} Solution(1) 15 elements {1, a1,a2, b1, b2, c1, c2, d1, d2, e1, e2, f1, f2, g1, g2} Using the seven letters a,b, c, d, e, f and g, we form groups of triplets in which each pair of letters occurs exactly once: {abc, ade, afg, bdf, beg, cdg, cef}

{abc, ade, afg, bdf, beg, cdg, cef} Solution(2) {abc, ade, afg, bdf, beg, cdg, cef} Sun Mon Tue Wed Thu Fri Sat 1,a1,a2 1,b1,b2 1,c1,c2 1,d1,d2 1,e1,e2 1,f1,f2 1,g1,g2 b, d, f a, d, e a, b, c b, e, g a, f, g c, d, g c, e, f

SOLUTION(3)----HW (FINISH THE TABLE) Sun Mon Tue Wed Thu Fri Sat 1,a1,a2 1,b1,b2 1,c1,c2 1,d1,d2 1,e1,e2 1,f1,f2 1,g1,g2 b1, d1, f1 a1, d, e a2, b, c a1, b, c b2, e1, g1 a2, f, g a1, f, g a1, f1, g a2, d, e c1, d2, g2 c1, d, g b1, d, f b1, e, g b2, d, f c2, e2, f2 c2, e, f b2, e, g c1, e, f c2, d, g

Solution(4) a1,a2, b1, b2, c1, c2, d1, d2, e1, e2, f1, f2, g1, g2 Try to fill out a1,a2, b1, b2, c1, c2, d1, d2, e1, e2, f1, f2, g1, g2 into the box Substitute 2,3,4…15 into a1,a2….g2 Then, get the solution!!!!

ANOTHER ALGORITHM FOR CHINESE POKER(鬥地主) LEAGUE(1) This method can be used if n2 participants n participants a group n days *** n must be a prime number *** Example: 25 participants,5 a group,5 gorups each day 49 participants,7 a group,7 groups each day 121 participants, 11 a group, 11 groups each day

ANOTHER ALGORITHM FOR CHINESE POKER(鬥地主) LEAGUE(2) D1 G1 G2 G3 1 2 3 4 5 6 7 8 9 D2 G1 G2 G3 1 2 3 ←1 5 6 4 ←2 8 9 7 D4 G1 G2 G3 1 4 7 2 5 8 3 6 9 D3 G1 G2 G3 1 2 3 ←1 6 4 5 ←2 8 9 7

ANOTHER ALGORITHM FOR CHINESE POKER(鬥地主) LEAGUE(3) 1 2 3 . n n+1 n+2 2n 2n+1 2n+2 3n n2-n+1 n2 ←0 1 2 3 . n ←1 n+2 n+3 n+1 ←2 2n+3 2n+4 2n+2 ←(n-1) n2 n2-1

HOMEWORK 1. Draw the timetable of the meetings in slide 11. 2. Finish the table on slide 24 Extra Credit: Explain why the last algorithm fails when n is not a prime number

Online Discussion Would you suggest some daily applications? Which of the following is a better way to organize a contest, a round-robin or knock-off tournament? Why?