We do not pay enough attention to the definition of the problems given to our solvers. Firstly, by selecting a tailored definition to our system, we can give it an unreasonable help. More importantly, ambiguities in the definition may lead to a different problem although we believe that they are always identical.

This is often due to the ambiguities of natural language. However, we will see that different problems come from the same definition because some people add constraints that they feel evident; nevertheless, we have the right to reject these constraints.

The definition of Sudoku does not seem to pose difficulties, it is very straightforward: one must complete a 9×9 grid with numbers from 1 to 9 so that each column, each row, and each of the nine 3×3 subregions contain all the digits from 1 to 9. This definition appears in the magazines that publish these problems. Moreover, a few add another constraint: there is only one solution for this puzzle.

The question is whether one can use this uniqueness constraint for solving Sudoku puzzles. This constraint is respected in all the published puzzles: they always have exactly one solution. However, this can be considered as a constraint only for those who create new puzzles. The requirement of a unique solution exist for many problems; for instance, a chess problem is cooked if there are several solutions. Even then, very difficult problems, such as the magic cube, described in a preceding post, has millions of solutions; as each one is very hard to find, the problem is interesting.

If this constraint must be respected by the author of a Sudoku puzzle, does a human or machine solver have the right to use it? I have never seen a chess problem where the solution uses this uniqueness constraint. Some constraints are necessary for creating a beautiful problem; this does not mean that they can be used for solving it. However, the first question that needs to be asked is: can this constraint be useful for solving Sudoku puzzles?

Of course, knowing the number of solutions of a problem may be very helpful: as soon as all the solutions have been found, one can stop the process. Indeed, no more solutions can be found if the author has not made a mistake. It is very easy to implement this possibility. For instance, a parameter indicates to CAIA the maximum number of solutions that it can find, so that it avoids wasting a lot of time when there are millions of solutions. The default value for this parameter is 50, but one has just to instantiate it to 1: CAIA will stop as soon as it has found one solution.

It is less obvious to use this restriction even before finding a solution. I will illustrate this point in a problem published in the *Figaro Magazine.* Far too often, after a newspaper has published a Sudoku problem, it only gives the solution: it does not indicate the steps that were taken for finding this solution. Fortunately, the *Figaro Magazine* also gives an excellent description of the main steps; therefore, it is possible to see which rules have been used. Usually, there are only the constraints given with the definition of the problem, but not the one stating the uniqueness of the solution. However, a problem stated in the 17 July 2015 issue uses it. The following diagram describes the situation after finding the value of three squares. The lines are numbered 1 to 9 from the top.

It is easy to see that the possible values for B2 and C2 are 7 and 9. In the same way: 1, 7, and 9 are the possible values for B5 and C5. (5 cannot be in C5 because in column C, 5 can must be in C7 or C9, the only squares where 5 can be put in the lower left subregion). If 1 were neither in B5, nor in C5, each solution with B5=7 and C5=9 would give a symmetrical solution where 7 and 9 are switched in squares B2, C2, B5, and C5. If there are N solutions with B5=7 in this hypothesis, there would be also N solutions with B5=9. If K is the number of solutions when 1 is in B5 or C5, the total number of solutions would be K+2*N. As the uniqueness constraint states that it is equal to 1, this means that K=1, and N=0: there is no solution without 1 in B5 or C5. This is very useful, that means that D5 is not 1. As in the center subregion 1 is either in D5 or F6, we have F6=1, and it is easy to complete the Sudoku with this information.

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CAIA does not make this deduction: it has not received the uniqueness constraint. For this problem, it finds the same possible values for squares B2, C2, B5, and C5; unfortunately, it cannot deduce the following steps of the Figaro solution. In its solution, it sees that the only values for E1 are 2 and 5. 2 leads to a contradiction, and it directly finds the solution with E1=5. Incidentally, this proves that the value of N is actually zero. However, contrary to the solution given in the Figaro, it has to backtrack once.

I understand that many people want to use the uniqueness constraint, although I do not agree with them for several reasons:

*Real problems may have none or many solutions, and it is interesting to solve them. Why are they giving a special status to artificial problems?

*Even when there is certainly one solution, why transform the search for a solution into a game of chance. The first to find the solution is the lucky one who considers firstly the good value for the unknowns.

*For other problems that have only one solution, as Chess problems, one never uses this uniqueness for finding this solution.

*In many cases, including the Figaro, the given Sudoku definition does not include this constraint.

*If the problem is cooked, nobody will see it.

*The author of the problem must check that it has just one solution: the uniqueness constraint is certainly necessary for creating a puzzle. Naturally, for checking it, he cannot use the fact that the problem has just one solution! Therefore, one must also have a solver that does not use this constraint. Is it necessary to have two solvers for the same problem?

All this shows that, even when the definition of a problem looks evident at first sight, it may be difficult to agree on this definition: the solvers cannot help but interpret this formulation. In this case, the solutions and the solving methods may not be the same as those given in the initial formulation: they do not solve the same problem. Therefore, if it is possible to define a problem, it is certainly not easy. Unfortunately, AI researchers have to give unambiguous definitions to their problem solvers.

The Wikipedia page on chess retrograde analysis has a mind bending example under “The a posteriori convection” ( https://en.wikipedia.org/wiki/Retrograde_analysis#The_a_posteriori_.28AP.29_convention ).

In the opening position, it’s possible that en passant capture is a legal move, but to actually play the move, the rule in chess problems is that you must be able to *prove* that it is legal. It turns out that it can be proven in the case that white can legally castle. It turns out that with the rules of chess problems, en passant capture on the first move is legal only if white castles at some point in the solution — *and black plays moves that try to prevent white’s castling at all costs, because then white wouldn’t have proven the legality of castling, which would make white’s first move illegal after the fact*.

Good luck trying to implement this in AI

An excellent example. Many years ago, after reading Smullyan’s books, I considered to realize a system solving retrograde analysis problems. Luckily, I preferred to develop CAIA, this example shows that such problems may be very difficult to solve!