Example of an order-3 Sudoku puzzle. This particular grid is logic-solvable. 

... future work, it is interesting to note that Sudoku can also be modelled as a graph colouring problem. This is done by considering each of the n 4 cells in a grid as a node, and then adding edges between any two nodes corresponding to a pair of cells in the same row, column, and/or box (meaning that the n 2 nodes occurring in each row/column/box will form a clique of size n 2 .) Further edges can then also be added due to the pre-filled cells that are supplied with the puzzle -for example in fig. 1 it is clear that nodes (cells) 9 (top right) and 10 (first on second row) should never be the same colour, and so we can add an extra edge between these in order to ensure that they will never be allocated the same colour in a feasible solution. Given such a graph, the task is to then colour the nodes using exactly n 2 colours. Graph colouring has, of course, been widely studied in the past (see [14], for one example) and in the future it is likely that various techniques from this field could show applicability to Sudoku and, indeed, vice-versa. 4 Finally, it is worth stressing that although Sudoku itself might not seem to have great practical implications in a real-world/industrial context, to its credit it is a problem that is very easy to understand, and it is certainly the case that it has encouraged many people to take an interest in constraint satisfaction problems. Perhaps more importantly though, it is noticeable that Sudoku features various similarities with other important combinatorial optimisation problems, and so its study should allow us to gain deeper insights into these as well. As an example, consider a typical timetabling problem where the aim is to assign a number of events to a limited number of timeslots and rooms in accordance with a set of constraints. In these problems it is common, among other things, to encounter pre-assignment constraints (e.g. "event 3 must be scheduled into room 6 in timeslot 8", etc.). In the past, various stochastic search techniques have been applied to handle these sorts of constraints in timetabling (see, for example, some of the works in [15]). However, it is noticable that this sort of constraint is actually very similar to the constraints introduced by the fixed cells in Sudoku. This suggests that it should also be useful to consider hybrid algorithms (of the sort described here) for these sorts of problems as well. Here, we refer the reader to papers by Merlot et al. [15] and Duong and Lam [16], where some preliminary work on this matter has been ...

... its simplest form, Sudoku can be defined as follows. Given an n 2 × n 2 grid divided into n 2 distinct n × n boxes (denoted by the bold lines in fig. 1), the aim is to fill the grid so that three separate criteria are ...

... some of the cells in a Sudoku grid will have been pre-filled by the puzzle master (see fig. 1). The player will then use these to logically determine the values for other cells in the grid, eventually allowing him-or-her to complete the puzzle. As can be imagined, how many and which cells the puzzle-master chooses to fill will therefore be particularly important if the puzzle is to be enjoyable for the player. Generally speaking, a "good" puzzle (from the player's perspective) should be configured in such a way so that is logic-solvable -that is, the player should be able to complete the puzzle in a logical sequence of steps using forward- chaining logic only (obviously the deductive abilities of different players will vary). In particular, a player should not usually be required to make random choices, especially when the grid is still quite empty, because if this guess turns out to be wrong, he-or-she will then have to go through the unsatisfying process of backtracking and re-guessing. For these reasons "good" Sudoku puzzles tend to have just one possible solution in each ...