## Clues

Advanced methods and approaches for solving Sudoku puzzles

### Clues

What is "a clue" - how is that different from T&E :?:
insolvent

Posts: 4
Joined: 09 March 2005

A clue is one of the numbers that is already present when you start the game. Trial and error is a system whereby your next move isn't determined by logic, but at random and is often incorrect.
Chris

Posts: 12
Joined: 06 March 2005

### Clues

I accept that, in a simple puzzle, one can derive the answers from the "clues" of number relationships. Are you asserting as a fact that the very difficult/fiendish puzzles can be solved similarly and without trialling? Having tried one or two I have run into the problem of all blank squares [empty cells] having multiple possible numbers without any obvious way forward except trialling.
insolvent

Posts: 4
Joined: 09 March 2005

### Re: Clues

insolvent wrote:I accept that, in a simple puzzle, one can derive the answers from the "clues" of number relationships. Are you asserting as a fact that the very difficult/fiendish puzzles can be solved similarly and without trialling? Having tried one or two I have run into the problem of all blank squares [empty cells] having multiple possible numbers without any obvious way forward except trialling.

Yes, thats what I'm saying. All the Pappocom puzzles (not those that appear in the Torygraph [Telegraph]) for example can be solved with just logic. Even the difficult or fiendish puzzles can be solved without trial and error.
Chris

Posts: 12
Joined: 06 March 2005

### Re: Clues

Chris wrote:Yes, thats what I'm saying. All the Pappocom puzzles (not those that appear in the Torygraph [Telegraph]) for example can be solved with just logic. Even the difficult or fiendish puzzles can be solved without trial and error.

Very interesting ... what you are saying is, of course, an assertion not a proof. Some people, like yourself, say these puzzles can be solved with just logic. Others say they can only solve them with trialling to an unspecified degree. The solutions are shown as just a set of completed cells with no logical explanation as to how the solution is arrived at. The 'tips' section finishes at a trivial level and there are no worked examples of a very difficult or fiendish problem. An acceptable hypothesis would be that the solutions are generated by a computerised trialling program and the problem solvers either trialled or guessed. I have not yet found a way forward as I have been unable to find any proof that these complex problems ARE capable of being solved through just logic.
insolvent

Posts: 4
Joined: 09 March 2005

### Re: clues

I agree with 'insolvent'. I have seen many quotes saying that logic is all you need. Certainly, applying the basic rule (each number appears once in each unit) solves some but not all puzzles. Looking deeper solves more puzzles, but I have not been able to eliminate trialling completely.
I was a teacher for long enough to know to never give a student marks for the answer, only for how they got to the answer. So what about a Su Doku solver program with a log file of WHY it chose to put each number in each cell, and in which order it solved the cells? As my program develops I will certainly put that into it. If there is no logical process that will solve all puzzles then on a reasonably fast PC you could certainly make a solver program with a trial-and-error algrothim look like it really knows what its doing, when its really just guessing.
In fact, if logic really is enough then the puzzle is trivial and rather boring. It becomes like Rubik's Cube, anybody can solve it because it requires only the ability to blindly follow rules. Solving the puzzle then becomes an indication of the persistance and accuracy of the solver, not their flair, inspriation or "intellegence" (whatever that is :) ).
howshaw

Posts: 9
Joined: 13 March 2005

I believe the assertion that all of the puzzles (even difficult and fiendish) can be solved using logic alone.

I have managed to complete all such puzzles since I started Su-Doku'ing a fortnight ago (with many hours of effort for some of them!)

A useful trick is not just to look at what possible numbers can go into a cell but looking at groups of numbers instead. Say there were 5 cells available in a box and you know of two numbers one or the other must appear in one of two boxes and vice versa. Logically these teo numbers could not be used in any of the other 3 cells in that box which will reduce the possibilities for these and possibly if you're lucky cascade through some other boxes/rows/columns!
steveb

Posts: 2
Joined: 14 March 2005

### Algrothims and guesses

So you come to a point where there are two cells, each could have digits x or y, but is it x,y or y,x ? I think most people consider that a perfectly valid approach is to try x,y and see what happens. If it leads to an impossible position then you back track and try y,x. Indeed it is a valid approach and has been used for thousands of years to prove very famous assertions (that sqrt(2) is irrational is one example). Guessing is the basic principle behind division algrothims (if long division is still taught the same as I learnt it at school), and for solving 2nd order differential equations and factorising equations in general cases, guessing is all there is. Newton "invented" iteration and calculus but its all guessing really.
What we are asserting (I still see no proof) is that any puzzle can be solved by a single general linear sequence of operations that must always lead to a single solution. That there is some function F(clues) = solution. We can think of F() as a mapping in 81 dimensional space, or a set of 81 linear equations, or loads of other visualations BUT... by saying that there is a logical solution to the puzzle we are removing from our "armoury of techniques" the one method that solves the vast majority of problems in maths, physics and the natural sciences... guessing. We should be very sure before we "tie our hands" quite so tight.
howshaw

Posts: 9
Joined: 13 March 2005

### Re: Clues

Chris wrote:Yes, thats what I'm saying. All the Pappocom puzzles (not those that appear in the Torygraph [Telegraph]) for example can be solved with just logic. Even the difficult or fiendish puzzles can be solved without trial and error.

OK Chris, I accept what you say. After a few days has passed a number of things have happened:
1. I read an article by Wayne Gould in the Times Online where he stated that all Pappocom puzzles can be solved with just logic. An assertion by the author has a credibility like no other so I was prepared to accept that as a working hypothesis.
2. Being a bear of exceedingly small brain, it has taken me a while to get to grips with the Difficult and Fiendish puzzles. I have now gained an understanding of the different techniques required to solve these puzzles and have solved a Difficult and two Fiendish puzzles from the Times .... and a few minutes is all that is now required to solve an Easy puzzle.
I am now a happy bunny :D :!:
insolvent

Posts: 4
Joined: 09 March 2005

### Re: Algrothims and guesses

howshaw wrote:So you come to a point where there are two cells, each could have digits x or y, but is it x,y or y,x ? I think most people consider that a perfectly valid approach is to try x,y and see what happens. If it leads to an impossible position then you back track and try y,x. Indeed it is a valid approach and has been used for thousands of years to prove very famous assertions (that sqrt(2) is irrational is one example). Guessing is the basic principle behind division algrothims (if long division is still taught the same as I learnt it at school), and for solving 2nd order differential equations and factorising equations in general cases, guessing is all there is. Newton "invented" iteration and calculus but its all guessing really.
What we are asserting (I still see no proof) is that any puzzle can be solved by a single general linear sequence of operations that must always lead to a single solution. That there is some function F(clues) = solution. We can think of F() as a mapping in 81 dimensional space, or a set of 81 linear equations, or loads of other visualations BUT... by saying that there is a logical solution to the puzzle we are removing from our "armoury of techniques" the one method that solves the vast majority of problems in maths, physics and the natural sciences... guessing. We should be very sure before we "tie our hands" quite so tight.

Guessing is indeed a valid approach, and you should play the game whichever way you want to of course, thats up to you. However I find it more satisfying to have solved the puzzle through my own logical thought as opposed to trying lots of different combinations and elmiminating all those that end in error. Trial and error will certainly get the puzzle complete, but for me it is not just about finishing but about having some fun on the way. If trial and error is fun for you, then great, keep at it. All I say is that it isn't a necessary weapon.

I am now a happy bunny :D :!:

I'm glad you now agree insolvent. :D

(I think Pappocom should really activate the smileys from the phpbb admin panel). ;)
[Administrator/Pappocom: smileys now activated for a trial period]
Chris

Posts: 12
Joined: 06 March 2005

### Solving techniques

I'm new to this forum , so hello!

It is possible - because I've done it - to solve the Fiendish puzzles by logic alone. For me, the time to do so can vary from all weekend to an hour and a half but I can assure you all that there is no trial and error involved in my solving. Currently I'm solving the Fiendish puzzles in the book (no I don't peek at the answers!), before that I was solving the puzzles which appear in the Daily Mail - that have nothing to do with Pappacom - with increasing boredom because they were so easy. There was no rating on these puzzles either and they don't appear on the Daily Mail website.

I reckon that the reason that there isn't an explanation for the more advanced techniques of solving is because it's so difficult to put into words. That and everybody's techniques are different. I know I changed mine in order to have half a chance of solving the Fiendish puzzles and I'm not entirely sure that my techniques don't change, even slightly, for each new puzzle. Anyway, it would get too boring to read a long list of solving techniques, in my opinion. It's a puzzle that is set to be solved. Writing all of the possible solving techniques would be akin to publishing the answer with the puzzle instead of after it.

Actually I never look at the answer. Once I've solved the puzzle - and I always do, whether it takes me hours or days - I check each row and each column (I solve mine on paper) then I don't need the answer. This is simply my personal preference and for my own personal enjoyment, sadist that I am!

I thoroughly enjoy solving and am lucky enough to have the time to fit it into my life, it's rapidly becoming and obsession and is extremely good for exercising my grey matter, determination and downright patience.
Happy solving!

Luna
lunababy_moonchild

Posts: 659
Joined: 23 March 2005

Hi All,

Happy solving to you too Luna
I too am getting abit obsessive and now my challenge is to do the puzzles quicker. This is in part prompted by a mention of the Rubiks cube (a few posts previous):

howshaw wrote:In fact, if logic really is enough then the puzzle is trivial and rather boring. It becomes like Rubik's Cube, anybody can solve it because it requires only the ability to blindly follow rules. Solving the puzzle then becomes an indication of the persistance and accuracy of the solver, not their flair, inspriation or "intellegence" (whatever that is ).

Years ago, (when they first came out and I was still in primary school) I learnt a solution to the Rubiks cube. All it involves is knowing a few standard moves, which I can still remember now, so I can still do the cube. However, I am not a particularly quick cube solver (~5mins) and this is because I do the moves more or less blindly and don't tune them to the starting configuration much.

It strikes me that there are also a few 'moves' in Su Doku (e.g. look for easy numbers by considering the 3 3x3 boxes, check the intersecting lines, place the exclusive pairs / multiples, and repeat until done - sorry not expressed very well, but I'm sure you all know what I mean). Because there are only a few 'moves', I think it would be relatively easy to write a program to do it (I haven't done it yet because I havent found the time yet....). But like solving the rubiks cube, just doing the 'moves' might not get you to the solution very quickly, and some degree of flair (or whatever it is called) is required to know where to begin and thus to get to the solution with the minimum number of steps. So, when publishing a solution to the problem, it would be interesting to know what the minimum number of steps (or maybe minimum number of pencil marks?) are as well. Perhaps this is something to consider for the contests ?

Anyway, must refine my techniques !
Sparrow

Posts: 5
Joined: 19 March 2005

### Solving techniques

Luna, you mention placing exclusive pairs/multiples, but how do you know what goes where.

these puzzles are driving me mad...in the past I managed all types of the Times Su doku, from easy to fiendish - no problem, but it seems to me they have got so much harder!! i have a pile of unsolved puzzles now, and I will never give in and just look at the answers, so i feel i'm losing it slightly. Would you mind trying to explain how you place exclusive pairs and multiples in case I'm missing something. I feel frustrated when I look across 3 boxes and it is possible to put multiple numbers in 2 out of 3 rows and columns...i don't like guessing, but just feel stumped!

Thanks
Guest

### Re: Solving techniques

Sue C wrote:Luna, you mention placing exclusive pairs/multiples, but how do you know what goes where.

these puzzles are driving me mad...in the past I managed all types of the Times Su doku, from easy to fiendish - no problem, but it seems to me they have got so much harder!! i have a pile of unsolved puzzles now, and I will never give in and just look at the answers, so i feel i'm losing it slightly. Would you mind trying to explain how you place exclusive pairs and multiples in case I'm missing something. I feel frustrated when I look across 3 boxes and it is possible to put multiple numbers in 2 out of 3 rows and columns...i don't like guessing, but just feel stumped!

Thanks

Hi Sue

I can understand the frustration - if you look in the other Topics, Particularly The Times (all about the very Hard Puzzle), and Solver Programs, you will find plenty of Tips/Rules etc that should help you. For me the challange has been to create a Program (in EXCEL using VBasic) to solve the little blighters. Like others, it has now reached the stage where it can solve the V Hrad Puzzle on the Times web site + all Fiendish Puzzles (76 -100) in the SuDoku book + all the Daily Puzzles.

You may regard this as cheating, but it has been a real cahllange to get it to work and fun . If you send me the Puzzle(s) you are stuck on, I can return the Log file of all the steps needed to solve it - but try the other topics on this site as well - there is lots of info there.

Good luuck
Tony Williams

Posts: 18
Joined: 02 April 2005

### Solving Techniques

Sue

"Luna, you mention placing exclusive pairs/multiples, but how do you know what goes where. "

I re-read my post and I'm not sure where you get this from - or what you mean! I do understand about them driving you mad though, I have, literally, taken all weekend to solve a Fiendish one from the book and my techniques of doing so vary from puzzle to puzzle.

However, what I do is this: when I start a new puzzle, I go through all of the numbers, starting with the 1's and working my way up to the 9's to see if there are any obvious placements. Then, I check each row and each column to see what's missing from that particular row/column. Then, I check each square of 9 cells to see what's missing and where I can fit it in.

As I mentioned, I do mine on paper with a pencil (and eraser!), so I write the missing numbers for each row at the side of the row and those from the columns under each column. As for the squares of cells, I draw a mini grid at the bottom of the page and list all the missing numbers from each square in the appropriate section - kind of like a giant noughts and crosses grid.

Other than that, I do exactly what it says in the book and on the website.
Generally, if I stare at the puzzle long enough I spot what's missing and how, logically (i.e. it can only be that number in that cell), it can be fitted in. It does eventually come to me - and is usually staring me in the face!

Obviously, given that it can take me a whole weekend to solve the one puzzle, it's not a fool-proof method - I'm sure that there will be other, more efficient ways of solving them - and I certainly don't claim to be an expert on the subject. This I can say, the more tired I am, or for that matter angry (won't go into that!) the longer it takes me to solve, because I'm distracted and not concentrating fully on the puzzle.

I'd just like to state that I do not - in spite of the fact that I use pencil and paper - write in a number on the off-chance that it could be that and then rub it out in the event that it's wrong and try something else (which is trial and error). I don't place any number unless I've got a logical reason for doing so - or so I think at the time! - and in the event that I rub it out it's because I got the logic wrong not because I'm trying it out.

As it turns out I haven't done one in over a week. I've never done one from the Times so can't comment on them, I do the ones from the book. I genuinely don't look at the answers, even when finished. I check rows and columns and assume that if they are correct the puzzle is solved. I'm concerned only with my own entertainment and gain immense satisfaction when I do eventually solve one that has been holding me back for the whole weekend (which is why I understand your frustration!).

I do hope that helps you out. However, if I may suggest, it might be a useful learning exercise if you look at the answer to one of the puzzles that remain unsolved and then try to work out, from the answer, the logic behind it. I did that with one of the Daily Mail puzzles that was set over Christmas and was driving me mad. The Daily Mail has nothing to do with Pappocom but I thought at the time that the DM ones were the only puzzles available. The exercise was of limited use to me though (because I still couldn't figure out where they got the answer from), but it may be worth a try. The Pappocom puzzles are logical - believe me, I felt the difference straight away - and there is only the one answer, so that might work for you.

It's really a personal thing. I'd rather use pencil and paper and work away at the puzzle until I figure it out - a whole weekend is the longest - a lot of people, as you can see, like to write programs to solve them which is a different challenge and probably gives a unique insight to them.

Let me know how you get on!

Luna
lunababy_moonchild

Posts: 659
Joined: 23 March 2005

Next