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 A 6x6: Is the analysis enough to solve a Calcudoku? 
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Posted on: Thu Mar 08, 2012 12:56 am




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Post A 6x6: Is the analysis enough to solve a Calcudoku?
A 6x6: Is the analysis enough to solve a calcudoku?.

(this topic has been opened in this section with the idea of continuing here the possible debate relative to the analysis or to the “solver rating”, following the discussions maintained these recent days in the thread “what to do with the data?”, considering that this thread (opened by Patrick) should mainly be oriented to the Patrick’s paper “On the Choice of Browser and Numerical Intelligence”).

Below I solve the 6x6 puzzle referred to by Patrick in his paper (this puzzle is of the type discussed in the starling’s topic “Something I’ve always been curious about”)

First of all, I think I would like to stablish this assertion:

“If a calcudoku has a unique solution, it can be solved using only analytical means.”

I cann’t prove this assertion (for the moment [smile]) (otherwise it should be a theorem [biggrin]). With respect to the “solver rating”: I think a puzzle is really difficult when the 100% of the information initially given is essential (I talked a little about this in a previous topic “The essential info in a puzzle”). Some questions arise:

Why assigning a 0 (like in the case of the 4x4 puzzle of Figure 1 in the Patrick’s paper) to a puzzle that has at least a few variations?

If a puzzle can be easily solved why assigning to it a high “rating” (like 62 in the case of the 6x6 puzzle of Figure 2 in the Patrick’s paper? (on march 07, 2012, the 6x6’s easy, medium and difficult are rated 3.5, 5.4 and 43.4, respectively)

Considering the use of the natural log in the formula (what drives to the idea of some “exponential” increase) how can we really appreciate the level of difficulty of a puzzle? (in other words, a puzzle with 124 how “more difficult” is with respect to another with 62, 4 times i.e, … ?

Should this “rating” be “size” dependant?

I will use, as an example, the same 6x6 “difficult” puzzle from the Patrick’s paper (the puzzle also appeared in the site on Feb 08, 2012, Puzzle id: 417948). This time I will use conventional language just in case some beginners (though I think the puzzle is very easy) want to follow the full solution of this 6x6.

Do you think this puzzle

Image

is more difficult than this other one?

Image

(which corresponds to the Figure 2 of the referred Patrick’s paper). Let’s quickly solve the first drawing (with the blank cages) (I will use three steps).

Image

In this first step only very elementary analysis has been necessary.

Image

The subset of 9 numbers in the “intersection” white area (3x3 box) is the “complement” to three lines (the three upper rows, with a total of three 6’s, three 5’s, … , three 1’s) of the blue area and whatever the numbers are in the white area this area is also the “complement” (to three lines, now the three rightmost columns) of the yellow area so the blue set and the yellow set are identical. We could see this also reasoning in a different way: If a number N is present 0, 1, 2 or 3 times in the blue area it must be present 3, 2, 1 or 0 times, respectively, in the white area (three rows in total) so it would be present 0, 1, 2 or 3 times in the yellow area (to complete the three columns).

Because a 3, a 6 and a 2 are in cells d4, e5 and f6, we have only left the numbers 6, 5, 4, 3, 2, 1 (the “hidden numbers”) so some of the usual combinations like 5-5-1, 4-4-3 and 3-3-4 for an L-shape “11+” cage in the 6x6’s are not possible now, leaving only the 6-4-1 and the 6-3-2 (the 5-4-2 suppressed because 6-3-1 do not produce a “0-“). So a 6 must be present in “11+” and once a 6 goes to f4 then a6 = 6 >>> a5 = 3 thus “erasing” the combination 6-3-2 for the cage “11+”.

Image

Now we see the reason for the blank cages, the result (and operator) on those cages is irrelevant (or superfluous) though necessary to the software for “consistency”, that is, “any” result/operator (consistent with the content of the cage) can be assigned (as noted in the graphic, “several” different puzzles could be designed with the same solution). Only during the step 2 a more detailed analysis has been performed, the step 3 is straight forward, then: Why a rating of 62?.

Finally, I am permuting the 1’s, 2’s and 6’s in rows 2 and 3 obtaining now a “totally” different puzzle (also with blank cages and a unique solution) which is once more against the x-y-x hypothesis previously commented in the thread “Something I’ve always been curious about”.

Image


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Posted on: Thu Mar 08, 2012 3:27 am




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Post Re: A 6x6: Is the analysis enough to solve a Calcudoku?
clm wrote:
“If a calcudoku has a unique solution, it can be solved using only analytical means.”

As the number of permutations increases, you are likely to employ strategies that approach guessing. For example, sometimes, I write down beside the puzzle (I only solve puzzles with pencil after printing them out) all possible permutations of adjacent cages, and eliminate them one by one by checking whether they lead to contradictions. How far can I go with it and still consider the solution analytical? In fact, by checking whether a permutation in a cage leads to a contradiction, I am just trying it out without completely admitting it. The difference with what I call "trial and error" is subtle, and becomes more subtle as the my checks involve more and more cages and cells. I go only as far as I can while still keeping all possibilities open. Trial and error, for me, means writing a solution in a cell or a cage and see whether it works. But the distinction is purely subjective.

In other words, the distinction between "trial and error" and "analysis" is not an absolute distinction. It depends both on the solver and on how far the solver is prepared to go to eliminate alternatives before writing numbers in the grid.

I agree with you that, as the solution is unique, it is certainly possible to solve the puzzle by eliminating the wrong alternatives. But, if in order to do so I have to deal with half a dozen "1- " and "2-" cages and/or with more than one large cage containing sums (which result in too many (whatever "too many" means to me) alternatives), I am going to consider the puzzle, for all practical purposes, only solvable by trial and error.


Last edited by giulio on Thu Mar 08, 2012 5:45 am, edited 1 time in total.



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Posted on: Thu Mar 08, 2012 5:38 am




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Post Re: A 6x6: Is the analysis enough to solve a Calcudoku?
That pretty much sums up the way I do it for the difficult 6's or greater.


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Posted on: Thu Mar 08, 2012 5:49 am




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Post Re: A 6x6: Is the analysis enough to solve a Calcudoku?
I know somewhere else in the forum there was a discussion about being analytical or doing things by trial and error, whether it is termed "guessing" or not. Someone else said that it is called "guessing" if you write anything down (even on the side I think?) to determine later that it in fact leads to a contradiction and that it doesn't count as "guessing" if you do it all in your head. I would "write things down" so I don't have to stress my brain too much by keeping it all only my head. But does guessing in one's head now count as "guessing" or something like "logic by process of elimination"? So perhaps giulio's assertion that "the distinction is purely subjective" has merit.


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Posted on: Thu Mar 08, 2012 6:41 am




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Post Re: A 6x6: Is the analysis enough to solve a Calcudoku?
IMO, "guessing" and "trial and error" are synonyms. After all, when you "try" a number, it means that you are making a "guess". Regardless of whether you write things down or not, when you write a single number in a cell, either you have a sequence of logical steps that leads you to the conclusion that it is the correct solution, or you are guessing.

Obviously, the better you are at keeping logical steps and numbers in your head, the less you need to write things down. But without writing things down, I wouldn't be able to solve the more difficult puzzles, and I do like to solve them. If for somebody that means that I am guessing, then so be it! [smile]


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Posted on: Thu Mar 08, 2012 7:20 pm




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Post Re: A 6x6: Is the analysis enough to solve a Calcudoku?
giulio wrote:
IMO, "guessing" and "trial and error" are synonyms. After all, when you "try" a number, it means that you are making a "guess". Regardless of whether you write things down or not, when you write a single number in a cell, either you have a sequence of logical steps that leads you to the conclusion that it is the correct solution, or you are guessing.

Obviously, the better you are at keeping logical steps and numbers in your head, the less you need to write things down. But without writing things down, I wouldn't be able to solve the more difficult puzzles, and I do like to solve them. If for somebody that means that I am guessing, then so be it! [smile]


Yes, but is guessing in your head "guessing" or "being analytical"?


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Posted on: Thu Mar 08, 2012 10:31 pm




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Post Re: A 6x6: Is the analysis enough to solve a Calcudoku?
LOL


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Posted on: Fri Mar 09, 2012 1:26 am




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Post Re: A 6x6: Is the analysis enough to solve a Calcudoku?
sneaklyfox wrote:
giulio wrote:
IMO, "guessing" and "trial and error" are synonyms. After all, when you "try" a number, it means that you are making a "guess". Regardless of whether you write things down or not, when you write a single number in a cell, either you have a sequence of logical steps that leads you to the conclusion that it is the correct solution, or you are guessing.

Obviously, the better you are at keeping logical steps and numbers in your head, the less you need to write things down. But without writing things down, I wouldn't be able to solve the more difficult puzzles, and I do like to solve them. If for somebody that means that I am guessing, then so be it! [smile]


Yes, but is guessing in your head "guessing" or "being analytical"?


In my opinion, it is different "guessing" as an "strategy", let's say, all the time, with many options or branches (and part of this discussion was maintained in the other thread "Method: Guessing" in this same section "Solving strategies and tips") than "guessing" when you have only two branches or, let's say, with the puzzle in a very advanced state, focusing to a very concrete problem. "Guessing" and "trial and error" have the same meaning, from a practical point of view, in the solution process (we can forget the "semantic" question, though guessing would mean only "imagining" while "trial and error" implies also "testing" and "proving").
But, when we do a "guessing" or "trial and error" with a determined number in a calcudoku and arrive to a contradiction, now, studying the reasons for the contradiction, we can always find and think about those analytical reasons and consequently define the analytical way for that step, that's what I mean. The process could be "mental" or "with pen and paper" (outside the brain [smile]), it does not matter. Now, if we "rewind" the time ("revisiting the past") going back to the same point we were before the "trial and error" (I would not see any objection for that), we may depart with a new analytical knowledge or conclusion to advance that step. And so until the end. Finally the full analytical process can be written.
A different thing is doing "guessings" arbitrarily (like in a lotery, this is valid for timed puzzles because they have a low number of possibilities and also we have little time), in this case we would probably never arrive to the solution (it will depend on the complexity of the puzzle) and certainly we would never learn about the "nature", the "essence", the "structure" of the different types of puzzles.


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Posted on: Fri Mar 09, 2012 3:05 am




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Post Re: A 6x6: Is the analysis enough to solve a Calcudoku?
As I see it, the only difference between "guessing" and "trial and error, is that "guessing" implies that we choose one of possible alternatives, while "trial and error" sounds completely neutral: perhaps meaning that we go through all possibilities in an order that doesn't depend on the status of the particular puzzle (e.g., from the smallest to the larger numbers, or even throwing a die). In that vein, it could be argued that "guessing" is then a better strategy, because choosing a number without following a pre-defined strategy inevitably relies on heuristics. If you have played enough puzzles and have learned the "puzzle language" (who was saying that? Gee, my apologies, but my memory is completely shot!), you are more likely to "guess" the right solution.

Don't you just love splitting hairs? :-)


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