Search Methods and Heuristics - Artificial intelligence:

Here now we come to the question of how constraint solvers search for solutions -constraint preserving coursework of values to variables - for the CSPs they are given. In fact the most obvious approach is to utilize a depth first search as: assign a value there to the first variable and in which check whether this assignment doesn't break any constraints. After then, move at to the next variable and then assign it a value and check that this doesn't break any constraints so then move at to the next variable and so on. Where an assignment does break a constraint after then choose a different value for the assignment until one is found that satisfies the constraints. Whether one cannot be found then this is where the search must backtrack. However in such a situation there the previous variable is looked at again and so the next value for it is tried. Furthermore in this way, all possible sets of assignments that, will be tried for a solution be found. So therefore the following search diagram - taken from Smith's tutorial paper - shows that how the search for a solution to the 4-queens problem progresses until it finds a solution as: 

Conversely we see that the first time it backtracks is after the failure to put a queen in row three given queens in positions as (1,1) and (2,3). Moreover in this case there it backtracked and move the queen in as (2,3) to (2,4). Ultimately, this didn't work out either then it had to backtrack further and then moved the queen in as (1,1) to (1,2). Alternatively this led fairly quickly to a solution.