## The Riddle of Sho

Advanced methods and approaches for solving Sudoku puzzles

### The Riddle of Sho

Consider the following puzzle,

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

(which was posted by Tso in the "Pyrpolising" thread).
Now, let's suppose that a Sudoku enthusiast named Sho have recently read some threads in this forum and have decided to try his skills on this puzzle. He feels absolutely no fear though he have red that this is a very hard puzzle. In fact, Sho managed to make some logical deductions and then he arrives to the following grid:

Code: Select all
`*-------------------------------------------------------*| 14   29    3   | 1278   17      1289 | 6     48    5  || 14   256   256 | 3      16      1268 | 48    9     7  || 67   8     79  | 59     569     4    | 2     3     1  ||----------------+---------------------+----------------|| 68   4     56  | 15     2       1356 | 9     7     38 || 28   569   79  | 457    34567   356  | 13    15    28 || 27   3     1   | 79     8       59   | 45    6     24 ||----------------+---------------------+----------------|| 9    7     8   | 6      145     15   | 13    2     34 || 3    1     26  | 2489   49      7    | 458   458   69 || 5    26    4   | 1289   139     1238 | 7     18    69 |*-------------------------------------------------------*`

However, Sho could no longer continue to solve this grid because he noted something bizarre in it. So, after spending some time scratching his head and thinking "What's going on here?!", Sho concluded that this puzzle should be an invalid one and so decided to try his lucky with another puzzle.
Nevertheless, this puzzle in perfectly valid, having just one solution. So, Sho must have done something wrong when solving it, arriving to an invalid grid. Now, if you want to help Sho, can you tell him why is the pencilmarked-grid above invalid, and then show him what he have done wrong?

Remember, Sho only like to solve puzzles by hand, so try to do the same when analysing the grid above and don't use a computer. (Try also to not look at the grid solution of this puzzle, otherwise things would be easy). Also, before starting to solve this riddle, read carefully the notes that follow.
---------------------------------------------------------------------------------------------

Note 1. If you have discovered the answer to Sho's riddle, then please don't post it here in order to not spoil the fun for the others: you can send it to me in a private message or email. Thanks.

Note 2. In order to prevent some doubts that may occur, here is the purpose of the riddle explained:

Step 1. Show that the grid is invalid just by analysing it, and not by trying to figure out what candidate or candidates should not have been eliminated (leave this for the next steps). You don't need to do any logical deductions (eliminations or inclusions), just try to find what is wrong with the grid.

Step 2. With the answer to step 1 in hand, try to figure out how could Sho arrive to his invalid grid (hint: solve the puzzle from the beggining using only basic logic).

Step 3. Now you have your chance to show what you are really made of: put your brain and logical Sudoku reasoning into work and try to make the deductions that Sho have done, according to your conclusions from step 2. A small advice: keep a jug of coffee right beside you, just in case...

Step 4. If you have done the previous steps correctly, you will necessarily discover what have Sho done wrong, which is the same to say what deductions have Sho done wrong. After all this work, you will be recompensated not only with an elegant solution for this puzzle but also with an interesting solution grid.

Be clever and have fun.

Good luck!

Carcul
Last edited by Carcul on Tue Jul 18, 2006 8:17 am, edited 2 times in total.
Carcul

Posts: 724
Joined: 04 November 2005

Just by curiosity, have someone managed to solve this riddle?
Carcul

Posts: 724
Joined: 04 November 2005

### Re: The Riddle of Sho

Carcul wrote:
Code: Select all
`*-------------------------------------------------------*| 14   29    3   | 1278   17      1289 | 6     48    5  || 14   256   256 | 3      16      1268 | 48    9     7  || 67   8     79  | 59     569     4    | 2     3     1  ||----------------+---------------------+----------------|| 68   4     56  | 15     2       1356 | 9     7     38 || 28   569   79  | 457    34567   356  | 13    15    28 || 27   3     1   | 79     8       59   | 45    6     24 ||----------------+---------------------+----------------|| 9    7     8   | 6      145     15   | 13    2     34 || 3    1     26  | 2489   49      7    | 458   458   69 || 5    26    4   | 1289   139     1238 | 7     18    69 |*-------------------------------------------------------*`

What were the logical steps used to...
... remove the 9 from r1c4?
... remove the 9 from r1c5?
... remove the 4 and 5 from r6c4?
... remove the 4 from r5c8?
... remove the 9 from r9c6?
tso

Posts: 798
Joined: 22 June 2005

Tso wrote:What were the logical steps used to...

Tso, obviously that is part of the riddle. Anyway, you don't need to know that in order to start by helping Sho and tell him why is the grid invalid.
However, you will need to know what were those logical steps in order to show Sho what he have done wrong.

Good Luck!
Last edited by Carcul on Mon Jul 10, 2006 10:56 am, edited 1 time in total.
Carcul

Posts: 724
Joined: 04 November 2005

I'm assuming we can pitch in with guesses/comments now?

It's an extremely rotationally symetrical puzzle with opposite digits, pairs etc. adding up to 10 and with the same techniques used to solve at approprate symetrical points. I'm doing it by hand now and looking for where the pattern breaks. Is this anyhing to do with the answer?
Last edited by underquark on Mon Jul 10, 2006 11:11 am, edited 2 times in total.
underquark

Posts: 299
Joined: 06 September 2005

Underquark wrote:I'm assuming we can pitch in with guesses/comments now?

Yes, but I will give no additional clues. All you need to know is in the original post.

Underquark wrote:Is this anyhing to do with the answer?

No. To solve the riddle, you "only" have to answer two questions:

1. Why is the pencilmarked grid above invalid?
2. What have Sho done wrong?

Good luck to you to!
Carcul

Posts: 724
Joined: 04 November 2005

The 9 in r9c5 shouldn't be there. If the rotational symmetry hypothesis holds true then it would seem that 5 is logical for r5c5.
underquark

Posts: 299
Joined: 06 September 2005

Carcul wrote:
To solve the riddle, you "only" have to answer two questions:

1. Why is the pencilmarked grid above invalid?
2. What have Sho done wrong?

Either I'm missing the point or I'm assuming that this is more complex that it is.

1. The grid is invalid because Sho removed a candidate with no logical reason to do so -- and that candidate is actually the correct number for that cell.

2. I don' t see that we have anyway of infering what Sho did wrong. She could have accidently erased more than one digit, had a momentary mental lapse, misread her writing, assumed some fact not in evidence, i dunno.

Wait -- is it possible that Sho was under the impression that the theory put forth in the thread where she found this puzzle was valid? Does applying that false theory somehow lead to these erroneous exclusions? (I'm just guessing because I make an effort to forget any theories that are proved false -- not enough room in my head for the true stuff.)
tso

Posts: 798
Joined: 22 June 2005

Tso wrote:Wait -- is it possible that Sho was under the impression that the theory put forth in the thread where she found this puzzle was valid?

Do you really think that I were capable of making a riddle based on such a strange theory? Come on Tso, that would be ridiculous. This a riddle intended to do real thinking, not for joking. And btw, Sho is a boy, not a girl.

Tso wrote:The grid is invalid because Sho removed a candidate with no logical reason to do so -- and that candidate is actually the correct number for that cell.

That's true, but how do you know that? How did you discovered that the candidate is actually the correct one for his cell, without knowing the solution grid, and without writing in the implications?
That was precisely the purpose of the riddle: to show that the grid is invalid without proving first that there is one (or several) particular candidate that should not have been eliminated.
So I see that you discovered the answer to the second question (btw, I have written in my post that any solution should not be posted here...), but not to the first.

Ok, in order to prevent further confusion, here is the purpose of the riddle:

1. Show that the grid is invalid just by analysing it, and not by trying to figure out what candidate or candidates should not have been eliminated (leave this for the next step). You don't need to do any logical deductions (eliminations or inclusions), just try to find what is wrong with the grid.

2. With the answer to 1. in hand, try to figure out how could Sho arrive to his invalid grid (hint: solve the puzzle from the beggining).

3. Now you have your chance to show what you are really made of: put your brain and logical Sudoku reasoning into work and try to make the deductions that Sho have done, according to your conclusions from step 2.
A small advice: keep a jug of coffee right beside you, just in case...

4. If you have done the previous steps correctly, you will necessarily discover what have Sho done wrong, which is the same to say what deductions have Sho done wrong. After all this work, you will be recompensated with an elegant solution for this puzzle.

Good luck!
Carcul

Posts: 724
Joined: 04 November 2005

Hm, i am out of competition here, because i have studied this puzzle using solver programs. It is rather hard and there are hundreds of ways, how it can be solved, more or less elegant. But for sure i would not make 4 of the 6 eliminations tso mentioned, because easier ones (for me) are possible.
And i am still far from seeing your point. So i cannot imagine, that one who tries to do it on paper, would solve your riddle. I would be happy to be able to finish it my way (after hours), so the additional challenge to detect order and reason, what Sho has done and done wrong, is more than difficult. I wouldn't be surprised, when you are the only one, who would take such a challenge and bring it to an end.
ravel

Posts: 998
Joined: 21 February 2006

Ravel wrote:So i cannot imagine, that one who tries to do it on paper, would solve your riddle.

Just keep trying, because it is possible. You just need to be very smart.
Carcul

Posts: 724
Joined: 04 November 2005

In the "Sudoku Discussions Forum", Gurth Bruins have posted a solution for this riddle that, unfortunately, is wrong. You may want to have a look at it here.
However, in face of it, I have rewritten "Step 2" in the original post as an additional hint. I have also transferred to the original post the explanations on the various steps.

Carcul
Carcul

Posts: 724
Joined: 04 November 2005

The fact that you haven't gotten the answer you are looking for though you've posted the riddle in multiple forums should imply to you what is obvious to me -- that riddle is not sufficiently defined to be solved without guessing.

I'm sorry, but your terms are hopelessly muddled. No one knows -- or *can* know from what you have given -- exactly what answer you have in mind. Your puzzle is only slightly more specific than "Guess what number I'm thinking of."

1) We have no way of guessing what type of situation Sho would think "bizarre". It is undefined and subjective. We have no way of knowing why SHO assumed the puzzle was invalid rather than the much more logical assumption that he made a mistake. I'm forced to infer that SHO is not too bright -- a possiblity that makes guessing his error-riddled logical deductions even more unlikely

2) What he did wrong was make eliminations without logical basis. You will not accept this -- why? I have no idea.
But this IS the fact of the matter. He made a mistake. We can only PRETEND to know why he did so.

3) Your question was "can you tell him why is the pencilmarked-grid above invalid, and then show him what he have done wrong?"
Here is a mathematical statement: 3=4. Can you tell me why it is invalid? I could say "because 4 is greater than 3" or "because 3 is less than 4" or "3x5=15 AND 4x15=20 AND 15<>20, therefore, 3<>4" -- but the truth is, "3=4" is invalid because IT ISN'T TRUE. The posted grid is invalid. There are candidates that have been removed without valid reasons. You are asking us to find the INVALID REASONS leading to these deductions! The technical terms for that are "guessing" and "telepathy". It is ONLY possible to reconstruct a series of VALID deductions. Why did I write 3=4? Typo? Stupididty? Insanity? You cannot know -- no amount of figuring will help

4) Your computer/by hand requirement is quite meaningless. We rarely can tell for sure about many of the solutions posted if they were done by hand, by computer or something in between.
5) "Step 1. Show that the grid is invalid just by analysing it, and not by trying to figure out what candidate or candidates should not have been eliminated (leave this for the next steps). You don't need to do any logical deductions (eliminations or inclusions), just try to find what is wrong with the grid." You want us to analyse it, but NOT analyse it. No logical deductions, just an undefined "figure it out". What on earth are you talking about? I am taller than my brother, but he is the tallest man in town. How can this be? Is it (a) I am not a man? (b) I am not in town? (c) I am a liar? You CANNOT know. There is nothing to figure out, only to guess. There is no satisfaction knowing the answer, as it could have been another answer just as well.

6) "Step 2. With the answer to step 1 in hand, try to figure out how could Sho arrive to his invalid grid (hint: solve the puzzle from the beggining using only basic logic). " Again, we must guess what you or SHO deem "basic logic". The words "figure out" imply logical methods -- but that is NOT what is required -- guesswork is. I think he arrived at his invalid grid *by making a mistake*. Am I wrong? Is what I'm suggesting untrue? I think that he made a deduction that was without justification. I have the wrong answer but what I'm saying is true? That's not a world I want to live in.

An that's the real problem. The puzzle or riddle is not sufficiently defined. Any solution you give -- when and if you post yours -- will not be satifying. I'll probably disagree with your solution. This is fine in a "A man is lying face down in the desert dead. There are no footprints in any direction. What happened?" type of puzzle, because it's fine to have many solutions. ("He's a dead parachutist" is no more valid than "He's a dead backpacker. The wind blew away his footprints" Neither is provably true from what is given.)
tso

Posts: 798
Joined: 22 June 2005

Hi Tso.

Thanks for being so detailed and direct in your critique. Mind you that, although the solution given by Gurth is wrong, it is however very close to the correct one: his proof for Step 1 is precisely the type of correct proof, and he managed to find it without further assistance. By basic logic I mean naked/hidden singles, locked candidates, pairs/triples/quads.
Maybe you are right when you say the riddle is not sufficiently defined. However, I feel that if I were to give more information then the solution would be rather obvious. Do you have a suggestion on how I could improve the riddle?

Tso wrote:1) We have no way of guessing what type of situation Sho would think "bizarre".

Yes you do, and in two ways: 1) by analysing the grid methodically, and 2) by paying attention to what happens in this forum.

Tso wrote:We have no way of knowing why SHO assumed the puzzle was invalid rather than the much more logical assumption that he made a mistake.

But, if Sho were to conclude that he made a mistake, then probably he would found what mistake was it and how to fix it in order to solve the puzzle. But, in that case, I wouldn't have a reason to make this riddle! Naturally, we must assume that Sho was naive (or pehaps too proud) by assuming the grid is invalid and not he made a mistake. However, to solve the riddle that issue is not important.

Tso wrote:What he did wrong was make eliminations without logical basis.
(...)I think he arrived at his invalid grid *by making a mistake*.

That is obvious Tso. In what other way could he arrive at an invalid grid?

Tso wrote:Your computer/by hand requirement is quite meaningless.

But, with a computer you would probably be able to confirm that the grid is invalid - however, the computer wouldn't be able to say why (unless the computer program is one that is much better than all the ones I have "seen" so far). Yes, we humans are smarter than computers! In the mean time, by hand you will be able to prove why, although you must be some kind of "methodic". ( I am giving so many clues in all these words that I cannot believe no one noticed yet what the riddle is all about).

Tso wrote:You want us to analyse it, but NOT analyse it.

I don't understand what you mean.

Tso wrote:The words "figure out" imply logical methods

If you want to go to the bathroom I think you also need logical methods. Everything we do makes use of logical methods. The words "figure out" imply thinking, make a conclusion, not guessing.

Tso wrote:The puzzle or riddle is not sufficiently defined.

When I wrote this riddle, I thought that a person with, for example, your knowledge of Sudoku, and that, more importantly, pays attention to what is going on in this forum, would conclude in short time what is involved in the riddle and that with some work would reach the solution (or, at least, go through steps 1 and 2). How wrong I was.
Well, never mind, perhaps you are right: this was just a stupid idea I have, and it is better to forget it.

Carcul
Carcul

Posts: 724
Joined: 04 November 2005

Please consult the Post in the Sudoku Discussions Forum (hyperlink above) for an ADDITIONAL HINT I have decided to give.
Carcul

Posts: 724
Joined: 04 November 2005

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