2) Complete the following statement (correctly) : Let and n be positive integers. Then there is a -fold TS(n) if and only if .

2) Complete the following statement (correctly) : 

    Let λ and n be positive integers. Then there is a λ-fold TS(n) if and only if     .

3) There is only one idempotent quasigroup of order 3 on symbols {1, 2, 3}; find its table, a LS(3).

  If it is used in the 3n + 1 construction of a 2-fold TS(10), find the two triples containing each of the following pairs:

  a) (2, 1), (3, 3)

  b) (3, 1), (3, 3)

  c)  ∞ , (2, 2)

4) Wilson's Construction is to be used to construct a STS(13). Find the appropriate deficiency graph, and a 1-factorization of it.

5) Here is a solution to Heffter's difference problem for a cyclic STS(13) on the integers (mod 13): the two difference triples are {1, 3, 4} and {2, 5, 6}. 

  a) Find the two corresponding base blocks.

  b) Find the third element in the triple containing each of the following pairs: 

      i) 2, 12     ii) 0, 3     iii) 5, 7

 Bonus Problem: (10 extra points)

  Let V be a finite, non-empty set. Prove that there is a PBD on V with every block having size 2.