**Optimum solution based on constraint problems:**

Whether depending on what solver you are using so there constraints are often expressed as relationships between variables as e.g., x1 + x2 < x3. Moreover,, just to be able to discuss constraints more formally that we use the following notation regularly:

So here a constraint C_{ijk } justified that tuples of values variables x_{i}, x_{j } and x_{k} all are allowed to take simultaneously. However in plain English there a constraint normally talks regaids things that can't happen so but in our formalism then we are looking at tuples as v_{i}, v_{j}, v_{k} that x_{i}, x_{j} and x_{k} can take simultaneously. Moreover as a simple example, now assume we have a CSP with two variables x and y, and in which x can take values {1,2,3}, when y can take values {2,3}. After then the constraint that x=y would be written like:

C_{xy}={(2,2), (3,3)},

So there the constraint that x

C_{xy} = {(1,2),(1,3),(2,3)}

Conversely a solution to a CSP is an assignment of values that one to each variable in such a way that no constraint is broken. So here it depends on the problem at hand so the user might want to know that there is a solution such i.e., they will take the first answer given.

However they may utilize all the solutions to the problem and they might want to know that no solution exists. In fact sometimes there the point of the exercise is to find the optimum solution based on some measure of worth. So sometimes, it's possible to do this without enumerating all the solutions thus other times there it will be necessary to find all solutions or then work out which is the optimum. However in the high-IQ problem there a solution is simply a set of lengths, one per square. Mostly the shaded one is the 17th biggest that answers the IQ question.