Sudoku, Logic and Proof

As you probably know, a Sudoku puzzle is a $9\times 9$ grid divided into nine $3\times 3$ subgrids.  Some of the cells in the grid contain a symbol: usually the symbols are the numbers $1,2,\ldots, 9$. The idea is to put a symbol into each empty cell so that every symbol appears exactly once in each row, exactly once in each column and exactly once in each of the nine $3\times 3$ subgrids. An example of a Sudoku puzzle is given below: it was the daily puzzle on [3] on 9 June 2011, when I wrote this article.