Mathematics 502
Homework (due Jan 31)
A. Hulpke
1) Show that for n ≤ 4 any Latin square of order n can be obtained from the multiplication table of a group by permuting rows, columns and symbols. Show that this is not true
for n = 5.
2) a) Construct a pair of MOLS of order 4.
b) Construct a pair of MOLS of order 12.
3) Suppose that n is an order for which MOLS exist. Does every latin square of order n
have an orthogonal partner?
4) Using the fact that the polynomial x2 − x − 1 is irreducible over GF(3) construct the
addition and multiplication tables of a field with 9 elements.
5) Let n be an integer and m = n2 . Show that an m × m Sudoku problem can only have
a unique solution if all of the following conditions are fulfilled:
• It contains at least m − 1 numbers
• It contains numbers in n(n − 1) different rows
• It contains numbers in n(n − 1) different columns
(It is not known whether there is a setting of a 9 × 9 Sudoku with less than 17 numbers
that has a unique solution.)
6∗ ) (2 points) Write a program that solves Sudokus. (You may use a programming language of your choice but you may not use a preexisting “solver” library.)
Use it to solve the Sudoku
and to show that the Sudoku
7.8...3..
...2.1...
5........
.4.....26
3...8....
...1...9.
.9.6....4
....7.5..
.........
.......1.
..8.3....
.......4.
12.5.....
.....47..
.6.......
5.7...3..
...62....
...1.....
has no unique solution.