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Solving twins (separating sides). User created Dec 16 and 22 https://www.calcudoku.org/forum/viewtopic.php?f=3&t=1007 
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Author:  clm [ Sun Dec 24, 2017 1:18 pm ] 
Post subject:  Solving twins (separating sides). User created Dec 16 and 22 
The “user created” puzzle on Dec 16, 2017 has been solved by 83 puzzlers till the moment, and the “user created” puzzle on Dec 22, 2017 has been solved by 123 puzzlers. But I have not build these two puzzles, they are the top (left) and bottom (right) sides of a 7x7 twin puzzle already published by Patrick Min in one of his books. I can not reveal in which one, since the solution is “eternal” and other puzzlers could still try both. First of all I asked Patrick his approval to do this “experiment” (and his cooperation in order to insert both puzzles consecutively and separated by a small number of days). The 7x7 twin puzzle did not score originally any points in the book, so no problem was apparently arising for assigning now some points (in my particular case I am not going to score those 19 points, since for me it is enough satisfaction to have the opportunity of exposing, in this Section “Solving strategies and tips”, my “theory”). So, what you have done, my good friends, Calcudokers, is solving independently both sides of a twin puzzle. Those who have solved both sides, can verify this simply by checking that all numbers occupy the same positions (it seems that nobody had noticed this) (BTW, you have the solution of the other side, so those who did not solve both can now quickly solve the other side, I apologize for that). Since a long time ago I have been “worried” and “obsessed” with the idea (I have not a formal demonstration) that the twin puzzles could have an independent (and unique solution) for both sides, so making unnecessary the “cooperation” between both sides. Please refer to this other post (Oct 06, 2011) "The essential info in a puzzle": viewtopic.php?f=3&t=77 . That’s why I have tried this “experiment” to "prove" it. If this becomes finally true (I wait debate and surprises in the future), and of course the difficulty increases a lot when you solve both sides independently, as you have seen, any time you find a twin puzzle you can try it, making many “trivial” twins (specially in the range of the 5x5’s and the 6x6’s) much more challenging (this could be interesting for the advanced Calcudokers), while, if you are in a hurry, use both sides simultaneously. As an example, I have solved all the twin puzzles in one of the Patrick’s books following this process. However, it must be said that there is an exception to this “rule”, that’s the case of the twins puzzles “fm 0”, this has been found by frederick (I had with him different PM’s / mails on the subject). I believe the reason is because the “fm 0” puzzles have “less information” since the multiplication rule, for instance, can not be applied (it produces a zero for any line), so having more “uncertainties” (in some way similarly to the case of the puzzles with negative numbers where, for instance, in the “symmetric” puzzles with negative numbers, you can not either apply the addition rule since it also produces a zero). The parity rule, in the other hand, can be applied (being more useful in puzzles “fm 0”). Merry Christmas. 
Author:  beaker [ Sun Dec 24, 2017 9:39 pm ] 
Post subject:  Re: Solving twins (separating sides). User created Dec 16 an 
Merry Chistmas, clm and also a festive wish to all the other users during this time of the year ( picture a smiling emoticon with a Santa hat on) 
Author:  jaek [ Tue Dec 26, 2017 5:47 pm ] 
Post subject:  Re: Solving twins (separating sides). User created Dec 16 an 
This is really cool. I often try to solve the twins using just one side or the other, but far from formally proving any sort of independent solvability, I always end up noticing some aspect of the other puzzle. 
Author:  pnm [ Tue Dec 26, 2017 6:24 pm ] 
Post subject:  Re: Solving twins (separating sides). User created Dec 16 an 
beaker wrote: Merry Chistmas, clm and also a festive wish to all the other users during this time of the year ( picture a smiling emoticon with a Santa hat on) 
Author:  michaele [ Wed Dec 27, 2017 1:14 pm ] 
Post subject:  Re: Solving twins (separating sides). User created Dec 16 an 
It is interesting to know that the puzzle from Dec. 16 was one side of a twin puzzle, I have been away for Christmas and had not seen the Dec. 22 puzzle. I had to use trial and error to solve the puzzle, after I had solved it I tried a few more times to solve it using analysis only, but could not solve it without resorting to trial and error. When I have created twin puzzles (see my number swap puzzles) I try to create them in a way that both sides of the puzzle can not be solved without reference to the other side, but I am not sure if that is how they are by the time I make changes to ensure the puzzle is solvable. If I create more twin puzzles I will try to ensure that they can only be solved by using both sides of the puzzle. 
Author:  clm [ Thu Dec 28, 2017 9:51 pm ] 
Post subject:  Re: Solving twins (separating sides). User created Dec 16 an 
michaele wrote: It is interesting to know that the puzzle from Dec. 16 was one side of a twin puzzle, I have been away for Christmas and had not seen the Dec. 22 puzzle. I had to use trial and error to solve the puzzle, after I had solved it I tried a few more times to solve it using analysis only, but could not solve it without resorting to trial and error. When I have created twin puzzles (see my number swap puzzles) I try to create them in a way that both sides of the puzzle can not be solved without reference to the other side, but I am not sure if that is how they are by the time I make changes to ensure the puzzle is solvable. If I create more twin puzzles I will try to ensure that they can only be solved by using both sides of the puzzle. The puzzles can be solved only with analysis, using the parity rule, the additon rule, and being patient to eliminate improper combinations. In fact, to ensure the unicity of the solution, the analysis is mandatory. I have solved several of your very interesting proposed puzzles and IIRC, in the swap puzzles, it was necessary the cooperation of both sides . I did not mention this interesting situation in the post, not because they are not "official" yet (this could come in a near future) but because your "circular replacement" or "swap puzzles" do not properly enter into the "twin puzzles" category, since the numbers do not occupy the same positions. I am convinced that twin puzzles in which the "cooperation" between both sides is required are perfectly possible to design, of course, being this curious condition actually observed simply the consequence of how the Patrick's software is building and verifying these puzzles. Thank you very much, michaele, for your comments. 
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