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Step by step analytical solution of a 9x9 “very difficult” https://www.calcudoku.org/forum/viewtopic.php?f=3&t=196 
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Author:  arjen [ Sat Apr 21, 2012 10:03 am ] 
Post subject:  Re: Step by step analytical solution of a 9x9 “very difficul 
Excellent, I restarted 2 times. I'm curious to see statistics. A graph of the last month with the lines started / finished / solver rating. @Patrick, when you are bored .... :) 
Author:  pnm [ Sat Apr 21, 2012 10:30 am ] 
Post subject:  Re: Step by step analytical solution of a 9x9 “very difficul 
arjen wrote: I'm curious to see statistics. A graph of the last month with the lines started / finished / solver rating. @Patrick, when you are bored .... :) Interesting though. But at the moment I'm not logging when a puzzle is started (I can see of course when it's loaded, but not if/when someone starts entering numbers)(so I maybe I should log that fact :) 
Author:  jomapil [ Sat Apr 21, 2012 1:58 pm ] 
Post subject:  Re: Step by step analytical solution of a 9x9 “very difficul 
Clm, a pretty and clean analytical solution. In short, we must " transform " the products in sums and calculate a,b,c,d values. Thanks for your solution. 
Author:  jomapil [ Sat May 12, 2012 12:31 pm ] 
Post subject:  Re: Step by step analytical solution of a 9x9 “very difficul 
At the beginning it looked like I must do many "trial and error" to solve the 9x9 of today (12May2012). But, after all, as far as I remember it was the first time I solved a 9x9 puzzle 100 % analytically. Has anyone any explanation about some characteristic of this puzzle that permitted that? ( It's only a curiosity of mine ). 
Author:  jomapil [ Wed May 16, 2012 12:33 pm ] 
Post subject:  Re: Step by step analytical solution of a 9x9 “very difficul 
Clm: For the second time in less than a week I solved the 9x9 puzzle of today, exclusively by analytic and logical means ( in part due to your advices ). But the 9x9 of yesterday ( 15May2012 ) I used many trials and errors and it took many hours to complete ( with many restarts due to the mistakes ). So, it looks like many diagrams are not possible to use only analytical methods, isn't it? 
Author:  clm [ Wed May 16, 2012 6:09 pm ] 
Post subject:  Re: Step by step analytical solution of a 9x9 “very difficul 
jomapil wrote: Clm: For the second time in less than a week I solved the 9x9 puzzle of today, exclusively by analytic and logical means ( in part due to your advices ). But the 9x9 of yesterday ( 15May2012 ) I used many trials and errors and it took many hours to complete ( with many restarts due to the mistakes ). So, it looks like many diagrams are not possible to use only analytical methods, isn't it? First of all congratulations for the two analytical solutions, on May 12 and May 16, we are in the good way. But I think that probably all diagrams, though difficult, can be solved with analytical methods, according to my previous assertion of: "If a calcudoku has a unique solution, it can be solved using only analytical means" (thread "A 6x6: Is the analysis enough to solve a calcudoku?"). This is not a theorem (it must be demonstrated and this demonstration is still pending). The yesterday's case, May 15, 2012, is not an exception, in my opinion. It's more difficult than usual, as every tuesday, of course. As it's a large 9x9 it will require several graphics and some time. I will try to send the step by step solution in the following days, opening a separate thread (in this same section "Solving strategies and tips"), in order to separate the discussion from this thread, which is more related to the 9x9 on tuesday Apr 17, 2012. 
Author:  jotempe [ Thu May 17, 2012 12:03 am ] 
Post subject:  Re: Step by step analytical solution of a 9x9 “very difficul 
clm, as I have already written elsewhere, the validity of your assertion depends on your definition of "analytical solution". Look e.g. at "Note 1" and "Note 2" in the solution that you started the thread with. They essentally say someting like: 'it could be either x or y. However if this were x, then we would have a, and ths would lead to b, which is contradictory, because of c. So it must be y'. Now, a purist will tell you, that this is a reductio ad absurdum proof, or, in plain English, a trial and error method, only that you performed the entire trial "in your mind", without putting any numbers in the diagram  as you have good enough memory to "see" the diagram with them in. So, the question is "how many numbers can be put in the diagram when considering a hypothesis, before you stop calling it anaytical solution?" 
Author:  jomapil [ Thu May 17, 2012 8:56 am ] 
Post subject:  Re: Step by step analytical solution of a 9x9 “very difficul 
Jotempe, in the end it looks like there will be ALWAYS " trial and error " ( TAE ). But what Clm intends to say is that we can reduce this TAE at a minimum of times. And the remaining few times we use TAE we can use them in a smarter way so as to mask that TAE component. Will it be so? 
Author:  clm [ Thu May 17, 2012 4:24 pm ] 
Post subject:  Re: Step by step analytical solution of a 9x9 “very difficul 
jotempe wrote: clm, as I have already written elsewhere, the validity of your assertion depends on your definition of "analytical solution". Look e.g. at "Note 1" and "Note 2" in the solution that you started the thread with. They essentally say someting like: 'it could be either x or y. However if this were x, then we would have a, and ths would lead to b, which is contradictory, because of c. So it must be y'. Now, a purist will tell you, that this is a reductio ad absurdum proof, or, in plain English, a trial and error method, only that you performed the entire trial "in your mind", without putting any numbers in the diagram  as you have good enough memory to "see" the diagram with them in. So, the question is "how many numbers can be put in the diagram when considering a hypothesis, before you stop calling it anaytical solution?" I understand by "analytical" solution when the known "analytical" tools are being used once "studied" and "understood". Clearly it is not possible to enterely "read" the puzzle (i.e., a 9x9) from the beginning so you must use methods to define the correct things and advance step by step, in this context there is no big difference if you use your memory (keeping the possibilities in mind) until some point (like in chess) or if you write the numbers in the paper or the screen. And if you make a "failure" in a written exposition, something that looks like a "guessing", that is mainly your fault, but it does not invalidate the original assertion. I think that as a puzzler uses more and more analytical tools probably will use less TAE. I am not against TAE, of course, by definition the "reductio ad absurdum" (or the TAE) is part of the "scientifical method" (in exact or empiric sciences ) and it's very good if it permits to find the solution since finding the solution is the target and it's always better than not arriving to any conclusion. What I say (in the line of the jomapil's explanation) is that it is possible (because the calcudoku, if it has a unique solution, should be like an "exact science", with a finite number of equations or relationships) to arrive to the solution using "only" analytical means; but this "only", here in the Forum, has always opened the debate. What is the meaning of "only"?. Logically, we must take decisions all the time among several possibilities, but this is analysis, if we base them in the logic, no matter the number of branches or possibilities (let's say within some reasonable limit, of course) being considered. For the practical situations, with the calcudokus, I understand by pure TAE when, let's say, we just proceed in this way: "OK, let's put a 5 here and see what happens; then if after 20 or 25 numbers entered everything crashes, we start again", we can loose hours with this procedure and we will never be sure where we are. I insist that it is better not using numbers "randomly" and without a base, but instead using the known tools and building logic hypothesis "from the beginning" of the solution process (and during the full process as well) and get accustomed to this method. As the puzzler is more skilled in these analytical tools they will become more clear and useful, I think. 
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