## The continuum of Sudoku logic.

Advanced methods and approaches for solving Sudoku puzzles
Personally, I've no problem with conditional logic. But, as you said, I like to be able to recognise the patterns, rather than begin at {condition A} and see what that implies for {condition B} until somewhere between C and Z I hit an absurdity.

If I can't see it all, I'm not really interested.
Karyobin

Posts: 396
Joined: 18 June 2005

### Re: sorry to Quibble...

Doyle wrote:... but in Statement 3 of Example E (of the original post), r4c4 = 4, not 1

As to the broader issue, I'm more with Karyobin, if only because resorting (my bias shows) to T/E inhibits finding more direct processes. Would the fish be in our aquarium otherwise?

What I tried to get across was that resorting to T/E -- however it is defined -- once all other methods at your disposal have been exhausted is not the same as doing the end run around the proper solution as it is often characterized. Also, many of the things she did -- and you do -- while solving -- some people will think of as T&E for shifting reasons.

My argument to a large degree was aimed at those who *create* puzzles to engourage them not to censor the hardest ones that might flummux us.

The thing is, in actual solving terms, we are actually in agreement on the T&E issue!
tso

Posts: 798
Joined: 22 June 2005

PaulIQ164 wrote:So far as I can tell, the first three deductions can be rephrased in a way that doesn't involve any conditional statements (e.g., in situation B, the logic is that r1c1, r1c2 and r1c3 are three cells with three candidates between them, therefore must contain some permutation of 123, and therefore r1c4 cannot conatin 1 2 or 3, so must be 4). Once you get to D and beyond, I can see no way to make the deductions without using conditional statements (if-thens or '==>'s). Doesn't this offer a point at which you could be justified in saying there's a break in the continuum, and that perhaps logic at the other side of it is not valid (beautiful fun, whatever)?

A bowling ball is point of squat, pyramid shaped building:

1) if the ball rolls forward, then it will roll down the front, if it rolls down the front, then it will reach the ground.
2) if the ball rolls back, then it will roll down the back, if it rolls down the back, then it will reach the ground.
3) if the ball rolls left, then it will roll down the left side, if it rolls down the left, then it will reach the ground.
4) if the ball rolls right, then it will roll down the right side, if it rolls down the right, then it will reach the ground.
Therefore, the ball will end up on the ground.

Or:

1) if r1c1 "rolls forward", that is, equals 1, then it will cause r1c4 to "roll down the front", that is, equal 3, which will lead to r4c4 "being on the ground", that is, equal 1.
2) if r1c1 "rolls back", that is, equals 2, then it will cause r4c1 to "roll down the back", that is, equal 3, which will lead to r4c4 "being on the ground", that is, equal 1.

Therefore, r4c4 "is on the ground", that is, equals 1
tso

Posts: 798
Joined: 22 June 2005

Karyobin wrote:Has the tipping-point moved - absolutely, and it's been fun to watch. Still, for the time being Forcing Chains are about as far down the T & I road as I'd like to go.

Me too I suppose. Supercoloring and Magic are indistinguishable to me.

Karyobin wrote:
As the eye of the beholder becomes trained, what was ugly may become beautiful.

Well my eye's going to have to become pretty trained in order for me to appreciate the kind of "Oh well, it's one of two options so off we go..." T & I approach that I envisaged when I wrote that!

However, bearing in mind the invention and undeniable practical use of Forcing Chains and Turbot Fish, amongst others, I do wonder whether the problem with genuine T & I lies in the fact that 'normal' people can't hold all the steps in their minds at any single point (possibly because it's impossible to know how many steps will/won't lead to a result). This may lead to a feeling that one has somehow lost touch with the puzzle in a way that does not occur with the other more defined /refined strategies.

Jelly Beans in a Glass Jar -- if there are 5, you can count them, if there are 1000 you must estimate them. I maintain that it is impossible to draw a line that separates the two situations. Maybe I can count to 20, you can count to 25.

Also, not much has been posted on tactics of *finding* forcing chains. Though you *could* just search blindly, there are various ways of narrowing the possibilies so that you won't feel like it could be any number of steps. I’m trying to put together a post that describes how I look for chains in *some* limited situations -- not neccessarily to supply a tool for doing so, but to show that there is logic behind the proof.

My favorite Sudoku end up with a large number of 2-candidate cells from which I’m able to produce a forcing chain while completely ignoring the cells with more than 2 candidates. It may very well be possible to intentionally create puzzles of this type that require longer and longer chains that are nevertheless, easy to find once you know how to look.

When I first heard of Swordfish, I tried to solve several problems using the tactic -- it doesn't work unless there is a Swordfish to be found! I did a lot of searching for nothing. T&E? I dunno. I had to learn when and how to look for them. How I do it is probably NOT how you do it.

It's all a matter of degree, a sliding scale. That's my point. Any line drawn will be arbitrary and personal -- and will change over time.

-- 1 inch sided square
-- whole number sided square
-- square
-- rectangle
-- parallelogram
-- convex polygon
-- polygon

Shapes as metaphorical Sudoku patterns -- proofs being closed lines.

Towards the top of the list, the patterns are very specific. But the number of separate descriptions required to cover everything we see is very high. They are very useful if we find them, very easy to create a (computer or human) search that will identify them -- but we cannot find them if they aren’t there, and any one specific shape will probably NOT be there. (We can also easily create a search that tells us that as well.)

Towards the bottom, the patterns become more and more general, requiring fewer and fewer rules to encompass the all we see. They are much more likely to be found, though they may become more difficult to identify. Searches of any kind are more difficult to construct, either to find or exclude the existence of the patterns. Imagine 100 sided non-convex polygon drawn on a large piece of paper with many other open ended segmented lines -- hard to see, but undeniable once found.

As a human solver, I want to work as far down the list as is comfortable AND appropriate.

Switching metaphors, it seems that my argument that elephant guns are fair is often met with the reply that it makes hunting ducks too easy and no fun. But I’m clearly not suggesting that you use an elephant gun for ducks! Only that birdshot will not bring down and elephant, sometimes there *are* elephants, elephant hunting is a qualitatively different experience from duck hunting, I like elephant meat -- and hope we don't run out of elephants!

(No animals were harmed in this post, though by dogs need to be taken for a walk badly.)
tso

Posts: 798
Joined: 22 June 2005

a facetious, off-topic comment to this excellent discussion...

tso wrote:elephant guns ... makes hunting ducks too easy and no fun.
A nice analogy... but, I think it'd be far harder to hit a small, fast-moving target, such as a duck, with a single large bullet than with a hail of shot.

Simes
simes

Posts: 324
Joined: 11 March 2005
Location: UK

... but, I think it'd be far harder to hit a small, fast-moving target, such as a duck, with a single large bullet than with a hail of shot.

An' I once shot an arrow from a longbow through a hoop that'd been thrown in the air. Really! (I teach archery an' all, y'know).

Jelly Beans in a Glass Jar -- if there are 5, you can count them, if there are 1000 you must estimate them. I maintain that it is impossible to draw a line that separates the two situations.

Odd that. I once read that 7 is about the limit for the average human to count visually in one glance, but there you go.

My favorite Sudoku end up with a large number of 2-candidate cells from which I’m able to produce a forcing chain while completely ignoring the cells with more than 2 candidates.

Me too! Whoooaaahhhh!!!

Luuuurrrrvve the whole quadrilateral thing too, tso - exactly the way I'd have explained it, had I thought about it properly. It doesn't matter how complex or irregular the shape - if there's rules, they can be used.

The elephant gun-thing I think I've seen from you once before, and I agreed with it whole-heartedly then as well. Though of course I'm disgusted that you eat elephants. I mean, I once had horse meat, but I was in France at the time and everything tastes of garlic anyway.

As a Parthian Shot, this thread brings to mind a sadly rather surreal post I lodged on the "I'm wicked and I'm lazy..." thread the other day, about future generations spotting patterns in candidate distributions that we would class as, well, hard to spot. I really did mean it, apart from the bit about town-planning, that is... It's all just a question of more complex polygons...Frank Herbert really did have some things wired.
Karyobin

Posts: 396
Joined: 18 June 2005

simes wrote:a facetious, off-topic comment to this excellent discussion...

tso wrote:elephant guns ... makes hunting ducks too easy and no fun.
A nice analogy... but, I think it'd be far harder to hit a small, fast-moving target, such as a duck, with a single large bullet than with a hail of shot.

Simes

That’s the point -- that the more powerful methods have their place but may be clumsy or useless if brought out too soo.

Karyobin wrote:
My favorite Sudoku end up with a large number of 2-candidate cells from which I’m able to produce a forcing chain while completely ignoring the cells with more than 2 candidates.

Me too! Whoooaaahhhh!!!

I think a great Sudoku Variant would have NO clues, but all cells would have a 2 candidates. Step one, start looking for wings, fish and chains. Similar to the Digit Place puzzle

Karyobin wrote:… future generations spotting patterns in candidate distributions that we would class as, well, hard to spot. I really did mean it, apart from the bit about town-planning, that is... It's all just a question of more complex polygons...Frank Herbert really did have some things wired.

Like most people, when ever I tried one of those peg solitaire puzzles, I *always* ended up with three left, maybe two, but never one and certainly not one in the center. It just seemed too complex, too opaque to see a useful number of jumps in advance. Then I read “Purging Pegs Properly” in Winning Ways by Conway, Guy and Berlekamp. In half an hour, I was the *master* of peg solitaire! What I had thought was nearly impossible became nearly trivial in just a few pages. It seemed so obvious to me afterwards -- why *hadn’t* I thought of it? If those pages existed for Sudoku …
tso

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Joined: 22 June 2005

Well that's what I was hoping to encourage, y'see. Maybe it'll get picked up on later, or something.
Karyobin

Posts: 396
Joined: 18 June 2005

The thing is, though, if it becomes too simple to solve even the hardest puzzle then there is no challenge and the fun goes out of it, thus rendering it a waste of time doing the puzzle at all.

It's a game, a past time, a form of entertainment. It's also a puzzle that requires solving, not a piece of art that requires admiring. Take the challenge away and you are left with a grid containing numbers that doesn't mean anything.

Luna
lunababy_moonchild

Posts: 659
Joined: 23 March 2005

Take the challenge away and you are left with a grid containing numbers that doesn't mean anything.

Not at all Luna, it's a question of altering the challenge. On the face of it, yes, it is merely a grid with some clues and off-you-go. But becoming good at cryptic crosswords doesn't take any of the challenge away, the challenge then becomes pattern-spotting and trying to cut down your solution time - exactly like with sudoku.

Games have always been used to teach higher-level thinking skills, some of the simplest ones ('Go' for example) having the most mind-boggling complexities. Games are how children develop, why shouldn't adults follow suit?

Where one person sees a meaningless grid of numbers, another sees a chance for growth.
Karyobin

Posts: 396
Joined: 18 June 2005

Karyobin

Actually, that happened to me. I was doing the Daily Mail puzzles until I realised that all I was doing was filling in the blanks. They then became completely uninteresting to me and I stopped doing them altogether. As it turns out I found the Times Book 1, this site and things changed (thus, as you said, the challenge changes) but the Daily Mail's puzzles, although they have changed, still don't interest me. And I'm not the most sophisticated solver out there.

With the other logic puzzles on the market, once I figured out how to do them - and it was very quickly so they couldn't have been that sophisticated because I'm not that good - they all became the same and thus meaningless to me, so I stopped doing them.

Yes, humans need to develop and grow and be challenged but the point I was making is ......... once you've masterd every single solving technique open to you and the Very Hard (to use Pappocom's definition since we are on this site) becomes as perfunctory to do as the DM's ones became to me, then it seems, to me, they will get meaningless. The challenge only exists when you 'have somewhere to go' as the Amercians would say i.e. progress can be made. What I'm saying is, once you've got there (i.e. mastered everything there is to master) there is nowhere else to go i.e. progress cannot be made.

Luna
lunababy_moonchild

Posts: 659
Joined: 23 March 2005

tso wrote:A bowling ball is point of squat, pyramid shaped building:

1) if the ball rolls forward, then it will roll down the front, if it rolls down the front, then it will reach the ground.
2) if the ball rolls back, then it will roll down the back, if it rolls down the back, then it will reach the ground.
3) if the ball rolls left, then it will roll down the left side, if it rolls down the left, then it will reach the ground.
4) if the ball rolls right, then it will roll down the right side, if it rolls down the right, then it will reach the ground.
Therefore, the ball will end up on the ground.

Or:

1) if r1c1 "rolls forward", that is, equals 1, then it will cause r1c4 to "roll down the front", that is, equal 3, which will lead to r4c4 "being on the ground", that is, equal 1.
2) if r1c1 "rolls back", that is, equals 2, then it will cause r4c1 to "roll down the back", that is, equal 3, which will lead to r4c4 "being on the ground", that is, equal 1.

Therefore, r4c4 "is on the ground", that is, equals 1

But is it not more elegant to simply say that since there is no flat point on the pyramid, the bowling ball will not be balanced anywhere on its surface, and so gravity will bring the ball to the ground?
PaulIQ164

Posts: 533
Joined: 16 July 2005

Yyyeeeeesssss.....succint certainly, but I don't think that really convey's tso's conditional point quite as thoroughly.

And to be obscenely pedantic, I do believe you will find no fewer than five flat points on a pyramid. Now horizontal's another matter, unless you spin it very fast first...
Karyobin

Posts: 396
Joined: 18 June 2005

Well exactly. I was trying to say that the conditional explanation of the pyramid situation, while perfectly logical, is not as elegant as a non-conditional one, just as I find conditional arguments in sudoku not as elegant.
PaulIQ164

Posts: 533
Joined: 16 July 2005

PaulIQ164 wrote:
tso wrote:A bowling ball is point of squat, pyramid shaped building:

But is it not more elegant to simply say that since there is no flat point on the pyramid, the bowling ball will not be balanced anywhere on its surface, and so gravity will bring the ball to the ground?

PaulIQ164 wrote:Well exactly. I was trying to say that the conditional explanation of the pyramid situation, while perfectly logical, is not as elegant as a non-conditional one, just as I find conditional arguments in sudoku not as elegant.

Sure, but you can't say that if it's a building in the real world rather than a mathematical construct. You have no *idea* if the ball will reach the ground each of the four directions unless you inspect the roof, the tiles, the gutter, the slope -- you will probably have to run imperial tests. ("Gravity" here is analogous to the "rules" of Sudoku.) Now, once we've tested this oddly shaped pyramidal building, and have assured ourselves that a ball placed on top will roll any of the four sides and reach the ground, the next time we see *an exact duplicate* of the building, we can make the same conclusion. Otherwise, for the thousands of buildings in between, we'll have to inelegantly climb up on the roof and check again. However, we well be able to "eyeball" the roofs from the parking lot, and with experience, will need to get a ladder less and less often.

My original post was never meant to suggest that we should use inelegant means in place of elegant ones, but rather to question if it follows that we should avoid puzzles that might *force* us to use the inelegant means? Should those who create them or software that creates them avoid and even condemn puzzles that require inelegant means?

I think we really agree: -- a solver who uses the most subjectively elegant means will most likely enjoy the experience the most. The type of T&E that a beginner might use -- get stuck, take a guess, see where it goes, erase and start again if needed -- very few enjoy, most will try to avoid. They type of T&E an expert might use in the course of a puzzle like the one below, not to solve it completely, but to leap across gaps in current Sudoku science -- or possibly advance it -- is a last resort only when convinced there is no other way.

Code: Select all
` 6 . . | . 4 . | . . 3  . 1 . | . . . | . 7 .  . . 5 | . . . | 8 . . -------+-------+------ . . . | 5 . 2 | . . .  3 . . | . 9 . | . . 2  . . . | 1 . 3 | . . . -------+-------+------ . . 8 | . . . | 9 . .  . 7 . | . . . | . 5 .  2 . . | . 3 . | . . 4 `

Conditional arguments that are tacked on *after* finding the puzzle by more round-about T&Eish means, are less satisfying that then those found by a more concrete method.

The areas we actually do disagree on, I think, are much smaller.
Last edited by tso on Sun Aug 07, 2005 8:40 pm, edited 1 time in total.
tso

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