Dividing Ranks into Regiments using Latin Squares

Transcription

1 Dividing Ranks into Regiments using Latin Squares James Hammer Department of Mathematics and Statistics Auburn University August 2, / 22

2 1 Introduction Fun Problem Definition Theory Rewording the Problem Examples Yet Another Problem Rephrasing the Question Known Results 2 Mutually Orthogonal Latin Squares History Another Conjecture 3 Outside Mathematics Sudoku Puzzles 2 / 22

3 Fun Problem Setup In the Star Trek universe, there are many different Affiliations; however, surprisingly enough, they all share a very similar command structure. 3 / 22

4 Fun Problem Affiliations & Ranks Empires United Federation of Planets (U) Klingon Empire (K) Romulan Empire (R) Cardassian Empire (C) Rank Captain (O6) Commander (O5) Lieutenant (O3) Ensign (O1) Question: Each empire sends one of each rank to a peace hearing. The hearing would like to promote communication by seating them in such a way that every row and every column has exactly one representative from each affiliation and from each rank. Can this be done with the sixteen officers? 4 / 22

5 Fun Problem Affiliations & Ranks Empires United Federation of Planets (U) Klingon Empire (K) Romulan Empire (R) Cardassian Empire (C) Rank Captain (O6) Commander (O5) Lieutenant (O3) Ensign (O1) Question: Each empire sends one of each rank to a peace hearing. The hearing would like to promote communication by seating them in such a way that every row and every column has exactly one representative from each affiliation and from each rank. Can this be done with the sixteen officers? 4 / 22

6 Fun Problem Affiliations & Ranks Empires United Federation of Planets (U) Klingon Empire (K) Romulan Empire (R) Cardassian Empire (C) Rank Captain (O6) Commander (O5) Lieutenant (O3) Ensign (O1) Question: Each empire sends one of each rank to a peace hearing. The hearing would like to promote communication by seating them in such a way that every row and every column has exactly one representative from each affiliation and from each rank. Can this be done with the sixteen officers? 4 / 22

7 Fun Problem Affiliations & Ranks Empires United Federation of Planets (U) Klingon Empire (K) Romulan Empire (R) Cardassian Empire (C) Rank Captain (O6) Commander (O5) Lieutenant (O3) Ensign (O1) Question: Each empire sends one of each rank to a peace hearing. The hearing would like to promote communication by seating them in such a way that every row and every column has exactly one representative from each affiliation and from each rank. Can this be done with the sixteen officers? 4 / 22

8 Definition Theory Definition A latin square of order n is an n n array, each cell of which contains exactly one symbol in {1, 2,..., n}, such that each row and each column of the array contains each of the symbols in {1, 2,..., n} exactly once. Example: / 22

9 Definition Theory Definition A latin square of order n is an n n array, each cell of which contains exactly one symbol in {1, 2,..., n}, such that each row and each column of the array contains each of the symbols in {1, 2,..., n} exactly once. Example: / 22

10 Definition Theory Definition Two latin squares L 1 and L 2 of order n are said to be orthogonal if for each (x, y) {1, 2,... n} {1, 2,... n} there is exactly one ordered pair (i, j) such that cells (i, j) of L 1 contains the symbol x and cell (i, j) of L 2 contains the symbol y. Para Engles If L 1 and L 2 are superimposed, the resulting set of n 2 ordered pairs are distinct! L 1,..., L k are mutually orthogonal Latin squares if each L i and L j are orthogonal for all 1 i < j k. 6 / 22

11 Definition Theory Definition Two latin squares L 1 and L 2 of order n are said to be orthogonal if for each (x, y) {1, 2,... n} {1, 2,... n} there is exactly one ordered pair (i, j) such that cells (i, j) of L 1 contains the symbol x and cell (i, j) of L 2 contains the symbol y. Para Engles If L 1 and L 2 are superimposed, the resulting set of n 2 ordered pairs are distinct! L 1,..., L k are mutually orthogonal Latin squares if each L i and L j are orthogonal for all 1 i < j k. 6 / 22

12 Definition Theory Definition Two latin squares L 1 and L 2 of order n are said to be orthogonal if for each (x, y) {1, 2,... n} {1, 2,... n} there is exactly one ordered pair (i, j) such that cells (i, j) of L 1 contains the symbol x and cell (i, j) of L 2 contains the symbol y. Para Engles If L 1 and L 2 are superimposed, the resulting set of n 2 ordered pairs are distinct! L 1,..., L k are mutually orthogonal Latin squares if each L i and L j are orthogonal for all 1 i < j k. 6 / 22

13 Definition Theory Example (4,4)(1,1)(2,2)(3,3) (1,2)(4,3)(3,4)(2,1) (2,3)(3,2)(4,1)(1,4) (3,1)(2,4)(1,3)(4,2) 7 / 22

14 Definition Theory Example (4,4)(1,1)(2,2)(3,3) (1,2)(4,3)(3,4)(2,1) (2,3)(3,2)(4,1)(1,4) (3,1)(2,4)(1,3)(4,2) 7 / 22

15 Definition Theory Example (4,4)(1,1)(2,2)(3,3) (1,2)(4,3)(3,4)(2,1) (2,3)(3,2)(4,1)(1,4) (3,1)(2,4)(1,3)(4,2) 7 / 22

16 Rewording the Problem Affiliations & Ranks Empires United Federation of Planets (U) Klingon Empire (K) Romulan Empire (R) Cardassian Empire (C) Rank Captain (O6) Commander (O5) Lieutenant (O3) Ensign (O1) Question: Each empire sends one of each rank to a peace hearing. The hearing would like to promote communication by seating them in such a way that every row and every column has exactly one representative from each affiliation and from each rank. Can this be done with the sixteen officers? 8 / 22

17 Rewording the Problem Observation Each of the sixteen officers can be represented as an ordered pair: (affiliation, rank). That is (U, O6), (U, O5),..., (C, O1). Orthogonal Latin Squares So, this means that we just need to create two orthogonal latin squares of order 4! 9 / 22

18 Rewording the Problem Observation Each of the sixteen officers can be represented as an ordered pair: (affiliation, rank). That is (U, O6), (U, O5),..., (C, O1). Orthogonal Latin Squares So, this means that we just need to create two orthogonal latin squares of order 4! 9 / 22

19 Rewording the Problem Observation Each of the sixteen officers can be represented as an ordered pair: (affiliation, rank). That is (U, O6), (U, O5),..., (C, O1). Orthogonal Latin Squares So, this means that we just need to create two orthogonal latin squares of order 4! 9 / 22

20 Rewording the Problem Example C U K R U C R K K R C U R K U C O1 O6 O5 O3 O5 O3 O1 O6 O3 O5 O6 O1 O6 O1 O3 O5 (C,O1) (U,O6) (K,O5) (R,O3) (U,O5) (C,O3) (R,O1) (K,O6) (K,O3) (R,O5) (C,O6) (U,O1) (R,O6) (K,O1) (U,O3) (C,O5) 10 / 22

21 Rewording the Problem Example C U K R U C R K K R C U R K U C O1 O6 O5 O3 O5 O3 O1 O6 O3 O5 O6 O1 O6 O1 O3 O5 (C,O1) (U,O6) (K,O5) (R,O3) (U,O5) (C,O3) (R,O1) (K,O6) (K,O3) (R,O5) (C,O6) (U,O1) (R,O6) (K,O1) (U,O3) (C,O5) 10 / 22

22 Rewording the Problem Example C U K R U C R K K R C U R K U C O1 O6 O5 O3 O5 O3 O1 O6 O3 O5 O6 O1 O6 O1 O3 O5 (C,O1) (U,O6) (K,O5) (R,O3) (U,O5) (C,O3) (R,O1) (K,O6) (K,O3) (R,O5) (C,O6) (U,O1) (R,O6) (K,O1) (U,O3) (C,O5) 10 / 22

23 Examples Natural Question Does there always exist two orthogonal latin squares of order n? Question Are there two mutually orthogonal Latin squares of order 2? Don t be Silly Of course not! There is only one latin square of order / 22

24 Examples Natural Question Does there always exist two orthogonal latin squares of order n? Question Are there two mutually orthogonal Latin squares of order 2? Don t be Silly Of course not! There is only one latin square of order / 22

25 Examples Natural Question Does there always exist two orthogonal latin squares of order n? Question Are there two mutually orthogonal Latin squares of order 2? Don t be Silly Of course not! There is only one latin square of order / 22

26 Examples Natural Question Does there always exist two orthogonal latin squares of order n? Question Are there two mutually orthogonal Latin squares of order 2? Don t be Silly Of course not! There is only one latin square of order / 22

27 Elbow Grease With a little bit of playing around, one can create two orthogonal latin squares of orders 3 and 5 as well. 12 / 22

28 Yet Another Problem Euler s Officer Problem (reworded) We have six positions: Quarterback Running Back Full Back Wide Receiver Tight End Center Question: And six SEC teams: Auburn Orange War Eagle Blue Alabama Crimson Vanderbilt Gold LSU Purple Georgia Black Can a grid be set up so that each row and each column has exactly one representative from each position and exactly one representative from each team? 13 / 22

29 Yet Another Problem Euler s Officer Problem (reworded) We have six positions: Quarterback Running Back Full Back Wide Receiver Tight End Center Question: And six SEC teams: Auburn Orange War Eagle Blue Alabama Crimson Vanderbilt Gold LSU Purple Georgia Black Can a grid be set up so that each row and each column has exactly one representative from each position and exactly one representative from each team? 13 / 22

30 Yet Another Problem Euler s Officer Problem (reworded) We have six positions: Quarterback Running Back Full Back Wide Receiver Tight End Center Question: And six SEC teams: Auburn Orange War Eagle Blue Alabama Crimson Vanderbilt Gold LSU Purple Georgia Black Can a grid be set up so that each row and each column has exactly one representative from each position and exactly one representative from each team? 13 / 22

31 Rephrasing the Question Euler s Officer Problem Does there exist a pair of mutually orthogonal latin squares of order 6? Note: This is indeed equivalent as we are looking at one latin square which contains player positions and another that contains teams. No pair can be used twice, since there is only one type of player from each team. 14 / 22

32 Rephrasing the Question Euler s Officer Problem Does there exist a pair of mutually orthogonal latin squares of order 6? Note: This is indeed equivalent as we are looking at one latin square which contains player positions and another that contains teams. No pair can be used twice, since there is only one type of player from each team. 14 / 22

33 Known Results Euler s Officer Problem Does there exist a pair of mutually orthogonal latin squares of order 6? Note: The answer is no. There does not exist a pair of mutually orthogonal latin squares of order 6. This result is due to Gaston Tarry (an amateur French mathematician) in 1900 via brute-force methods (before computer searches). 15 / 22

34 History Euler s Observations Euler knew that there was not a pair of orthogonal latin squares of order 2, and he suspected that there did not exist a pair of orthogonal latin squares of order 6. This led him to conjecture the following. False Fact Euler s Conjecture There does not exist a pair of orthogonal latin squares if n 2 (mod 4). That is to say n {2, 6, 10, 14, 18, 22,...} 16 / 22

35 History Euler s Observations Euler knew that there was not a pair of orthogonal latin squares of order 2, and he suspected that there did not exist a pair of orthogonal latin squares of order 6. This led him to conjecture the following. False Fact Euler s Conjecture There does not exist a pair of orthogonal latin squares if n 2 (mod 4). That is to say n {2, 6, 10, 14, 18, 22,...} 16 / 22

36 History Euler s Observations Euler knew that there was not a pair of orthogonal latin squares of order 2, and he suspected that there did not exist a pair of orthogonal latin squares of order 6. This led him to conjecture the following. False Fact Euler s Conjecture There does not exist a pair of orthogonal latin squares if n 2 (mod 4). That is to say n {2, 6, 10, 14, 18, 22,...} 16 / 22

37 History Euler s Observations Euler knew that there was not a pair of orthogonal latin squares of order 2, and he suspected that there did not exist a pair of orthogonal latin squares of order 6. This led him to conjecture the following. False Fact Euler s Conjecture There does not exist a pair of orthogonal latin squares if n 2 (mod 4). That is to say n {2, 6, 10, 14, 18, 22,...} 16 / 22

38 History The Award Goes To R.C. Bose and S.S Shrikhande gave a case where this is not true (order 22). In fact, R.C. Bose, S.S. Shrikhande, and E.T. Parker showed the following: Falsity of Euler s conjecture There exist at least two orthogonal latin squares of order n {2, 6} 17 / 22

39 History The Award Goes To R.C. Bose and S.S Shrikhande gave a case where this is not true (order 22). In fact, R.C. Bose, S.S. Shrikhande, and E.T. Parker showed the following: Falsity of Euler s conjecture There exist at least two orthogonal latin squares of order n {2, 6} 17 / 22

40 History The Award Goes To R.C. Bose and S.S Shrikhande gave a case where this is not true (order 22). In fact, R.C. Bose, S.S. Shrikhande, and E.T. Parker showed the following: Falsity of Euler s conjecture There exist at least two orthogonal latin squares of order n {2, 6} 17 / 22

41 Another Conjecture False Fact MacNeish s Conjecture Let n = p r 1 1 pr 2 2 pr k k, where p r 1 1 pr 2 2 pr k k are all distinct (i.e. the prime factorization of n.) Then the maximum number of Mutually Orthogonal Latin Squares of order n is min{p r 1 1 pr 2 2 pr k k } 1. Observation Euler s Conjecture is a special case of MacNeish s Conjecture. 18 / 22

42 Another Conjecture False Fact MacNeish s Conjecture Let n = p r 1 1 pr 2 2 pr k k, where p r 1 1 pr 2 2 pr k k are all distinct (i.e. the prime factorization of n.) Then the maximum number of Mutually Orthogonal Latin Squares of order n is min{p r 1 1 pr 2 2 pr k k } 1. Observation Euler s Conjecture is a special case of MacNeish s Conjecture. 18 / 22

43 Another Conjecture Reasoning MacNeish had good reason for this conjecture. He discovered two constructions (finite fields and direct product) that could produce that many Mutually Orthogonal Latin Squares of order n. Falsity of MacNeish s Conjecture In 1959, E.T. Parker discovered a construction to produce three Mutually Orthogonal Latin Squares of order 21! Note 21 = 3 7; so, by MacNeish s Conjecture, there should be only two Orthogonal Latin Squares of order / 22

44 Another Conjecture Reasoning MacNeish had good reason for this conjecture. He discovered two constructions (finite fields and direct product) that could produce that many Mutually Orthogonal Latin Squares of order n. Falsity of MacNeish s Conjecture In 1959, E.T. Parker discovered a construction to produce three Mutually Orthogonal Latin Squares of order 21! Note 21 = 3 7; so, by MacNeish s Conjecture, there should be only two Orthogonal Latin Squares of order / 22

45 Another Conjecture Reasoning MacNeish had good reason for this conjecture. He discovered two constructions (finite fields and direct product) that could produce that many Mutually Orthogonal Latin Squares of order n. Falsity of MacNeish s Conjecture In 1959, E.T. Parker discovered a construction to produce three Mutually Orthogonal Latin Squares of order 21! Note 21 = 3 7; so, by MacNeish s Conjecture, there should be only two Orthogonal Latin Squares of order / 22

46 Sudoku Puzzles Popular Puzzles Latin squares exist outside the world of mathematics as well! Sudoku Puzzles are one example of this. Variations There are all kinds of variations of Sudoku Puzzles; however, the 9 9 square variety is the most common. 20 / 22

47 Sudoku Puzzles Popular Puzzles Latin squares exist outside the world of mathematics as well! Sudoku Puzzles are one example of this. Variations There are all kinds of variations of Sudoku Puzzles; however, the 9 9 square variety is the most common. 20 / 22

48 Sudoku Puzzles Example Stating the Obvious Sudoku problems are a special type of Latin square! 21 / 22

49 Sudoku Puzzles Example Stating the Obvious Sudoku problems are a special type of Latin square! 21 / 22

50 War Eagle! Thank You For Your Kind Attention! Figure: War Eagle! 22 / 22