New Problems

Chapter 1

  1. 1.1 How can the latin subsquares which occur in the Cayley tables of Bruck, Bol and other classes of loops be characterized? (page 28).
  2. 1.2 What is the largest number of s × s subsquares for fixed s (1 < s < n) which can be avoided by a given n × n latin square L? Does this number depend on the structure of L: for example, whether L is group-based? Also, the same question when s is allowed to take all values (1 < s < n). (page 36)

Chapter 2

  1. 2.1 Which classes of groups other than those discussed on pages 75 to 78 of [DK2] are super P-groups? (page 67)
  2. 2.2 Is it true that if L is the multiplication table of a non-soluble group then, not only does L have at least one, and possibly many, square roots L si1_e , but at least one of these square roots is a latin square. (page 70).

Chapter 3

  1. 3.1 Does there exist an odd integer n such that a weakly completable critical set of size less than ⌊n 2/4⌋ exists for the cyclic latin square of order n? (page 95)
  2. 3.2 What is the largest size (in terms of n) for a critical set in a cyclic latin square of order n? (page 95)
  3. 3.3 What are the sizes of minimal critical sets for the various main classes of non-cyclic latin squares? (page 95)
  4. 3.4 What is the size of a minimal critical set for a Sudoku latin square L whether group-based or not? (page 96)
  5. 3.5 What is the largest size which a critical set in a latin square may have? (page 98)
  6. 3.6 How can latin bitrades of genus 1 (or any other genus greater than zero) be characterized? (page 103)
  7. 3.7 'What is the minimal order of a latin square into which a given separated (or non-separated) latin trade T of size h may be embedded? (page 104)
  8. 3.8 Are there combinatorial structures other than latin bitrades which can be represented topologically and hence assigned a genus? (page 105)
  9. 3.9 Is it true that every 1 4 ε si2_e -dense partial latin square is completable? (page 106)
  10. 3.10 Is Rodney's conjecture that every latin square contains a duplex true? (page 121)

Chapter 4

  1. 4.1 Is it true that, for odd n ≥ 7, every transversal of Ln can be extended in at least two distinct ways to a decomposition of Ln into transversals? (page 157)
  2. 4.2 Do there exist latin squares of orders n ≠ 3 h which contain as many as 1 18 n 2 ( n − 1 ) 3 × 3 si3_e subsquares? (page 158)

Chapter 5

  1. 5.1 What are the conditions for two latin directed triple systems to be orthogonal? (page 188)
  2. 5.2 Can the concept of orthogonality be modified to cover the case of nonlatin directed triple systems? (page 188)
  3. 5.3 Do pairs of maximal orthogonal r × n latin rectangles exist for all integers r such that n/3 < r < n when n is sufficiently large? (page 194)

Chapter 6

  1. 6.1 What is the largest number of totally diagonal latin squares that can exist in a pairwise orthogonal set? (page 219)
  2. 6.2 Can a 25 × 25 bimagic square be constructed by a modification of the method described in Keedwell(2011c)? (page 222)
  3. 6.3 What is the size of a minimal critical set for a 4 × 4 magic square? Is this the same for all such squares in the same class? Does it vary according to which of the twelve classes a particular square belongs? (page 224)
  4. 6.4 Is it possible to construct an n 2 × n 2 Sudoku latin square (n ≥ 4) all of whose n × n subsquares are magic squares of which no two are equivalent? (page 224)
  5. 6.5 How many non-isomorphic and non-equivalent Room designs of order 2n exist? (page 233)

Chapter 8

  1. 8.1 Can a complete directed graph with a prime number p of vertices be separated into p disjoint Hamiltonian paths? (page 277)

Chapter 9

  1. 9.1 For which orders n not divisible by 8, if any, do 4-homogeneous latin squares exist? (page 285)
  2. 9.2 Find the complete spectum of integers n−k for which (n−k)-homogeneous latin squares exist. (page 285)
  3. 9.3 For which integers h do orthogonal h-homogeneous latin squares exist or, alternatively, can it be proved that they do not exist for some values of h? (page 285)
  4. 9.4 Do atomic latin squares of composite but non-prime-power orders exist? (page 294)
  5. 9.5 For which even orders do N ∞-squares exist? (page 294)

Chapter 10

  1. 10.1 Can the bounds t ≤ m + 1 when m is odd and t ≤ m when m is even for the number t of latin squares of order 2m in a mutually nearly orthogonal set be attained when m ≠ 3? (page 296)
  2. 10.2  What is the upper bound Nq (n) on the number of squares in a set of MQOLS of order n? (page 301)
  3. 10.3 Is it true that, for some n, n > 6, Nq (n) > n − 1. (page 302)
  4. 10.4 For which orders n > 10 do D-type and/or C-type latin power sets of at least two members exist? (page 304)

Chapter 11

  1. 11.1 Do latin triangles exist of all odd orders and of all even orders except 4, 6 and 10? (page 325)
  2. 11.2 Alternatively, is there an integer n 0 such that, for all n > n 0, an LT(n) exists? (page 325)
  3. 11.3 For a given order n, how many different LT(n)'s exist? (page 325)
  4. 11.4 In this context, how should we define “different”? (page 325)
  5. 11.5 What is the maximum number of mutually orthogonal LT(n)'s that can exist? (page 325)