How odd that people should be so very against puzzles requiring "Trial and Error", as it is referred to by everybody here.

"Trial and Error" as the term is used here, appears to refer to the use of proof by contradiction, a.k.a. reductio ad absurdum, to verify an entry in a Su Doku puzzle. I do wonder why puzzles requiring this technique are looked down upon so by many users of this forum. Most feel, it seems, that for a puzzle to require this technique makes it in some way "inelegant". But why is this so? Used correctly, proof by contradiction is as logical as any other method of proof.

To me, the only Su Doku puzzles that are inelegant are the ones with more than one solution - they seem something of a faux pas (though with one of these, the solver might still take pleasure from demonstrating to himself that there exist multiple solutions, and if there are not so many that it tries his patience, identifying all of the solutions). If a Su Doku puzzle has a unique solution then I may take pleasure from proving to myself that it is so, by whatever means.

I now copy a statement that I saw made by another poster on this site (unfortunately I don't remember who made it or where I saw it). Many mathematical theorems can be proven using proof by contradiction, for example the fact that the square root of 2 is not the ratio of two integers. There are lots of others, many of them very elegant proofs (my favourite proof is the proof that the set of real numbers is uncountably infinite, which is a proof by contradiction). Say I have a Su Doku puzzle partially filled in and I see there are two squares within a box, row or column where some symbol could go. If I show that putting it in one box leads to a chain of logical deductions from which a contradiction follows, then the alternative must be correct. How elegant! Of course if one wishes to show that the Su Doku has only one solution, then if one makes the correct choice first time, one must go back and check that the alternative does lead to a contradiction; otherwise, we shuold be unsure of the uniqueness of the solution we have found...

Final point. The choice of "Trial and Error" to describe the technique is a bad one; it is a misnomer. Tral and Error would surely imply that the solver blindly enters numbers in the grid and goes on to see whether or not he finds a solution. This is not what proof by contradiction is about; it's about making a very specific hypothesis and showing that that hypothesis cannot be sustained as taken to its logical conclusion, it produces absurdities. It's clear-cut and above all logical, which actually attempting to solve by guesswork would never be.