# Sudoko theory
**URL:** https://boards.straightdope.com/t/sudoko-theory/475342
**Category:** The Game Room
**Created:** [December 2, 2008, 3:23pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342 "2008-12-02T15:23:40Z")
**Posts on this page:** 20
**Page:** 1
### Author:  [@twickster](https://boards.straightdope.com/u/twickster)
#### Post date: [December 2, 2008, 3:23pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/1 "2008-12-02T15:23:40Z")
(Not sure if this should go here or GQ – if not enough math people hang out here, I guess we can move it.)
If not a single 9 is among the given numbers that you start with, will it be possible to pin down the 9’s later on in the game? Or are you doomed to be stuck where I am, which is a whole slew of “this can be this or this” (or, more discouragingly, “this, this, this, or this”) squares and no way to start the domino train that will sort all of them out?
---
### Author:  [@Sitnam](https://boards.straightdope.com/u/Sitnam)
#### Post date: [December 2, 2008, 3:36pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/2 "2008-12-02T15:36:29Z")
I did my final undergrad Mathematics project on this, the short answer is ‘no, there is not always a domino effect’. The long answer comes with a beautiful Power Point presentation and numerous mentions of Latin Squares.
---
### Author:  [@Mosier](https://boards.straightdope.com/u/Mosier)
#### Post date: [December 2, 2008, 3:37pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/3 "2008-12-02T15:37:58Z")
Unless there is more than 1 possible solution for the numbers you’re looking at, you can eventually arrive at the solution.
---
### Author:  [@chrisk](https://boards.straightdope.com/u/chrisk)
#### Post date: [December 2, 2008, 3:43pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/4 "2008-12-02T15:43:12Z")
Yes, I would agree that there are certainly solvable positions that wouldn’t start with any 9. Somewhere along the way, you’d be able to fill in all other 8 numbers in a row, column, or square, and thus get a 9 by the process of elimination.
On the other hand, obviously you cannot have a solvable sudoku with no 8’s OR nines, because whenever you got all the other 7 numbers filled in, there’s still no conceivable way of telling that one blank space should be an 8 and the other a 9, or vice versa. If you came up with one legal solution, you’d be able to generate another by swapping all the 8s with 9s.
---
### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [December 2, 2008, 4:17pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/5 "2008-12-02T16:17:29Z")
> [@chrisk](#):
>
> On the other hand, obviously you cannot have a solvable sudoku with no 8’s OR nines, because whenever you got all the other 7 numbers filled in, there’s still no conceivable way of telling that one blank space should be an 8 and the other a 9, or vice versa. If you came up with one legal solution, you’d be able to generate another by swapping all the 8s with 9s.
A solution exists. That sounds pretty solvable to me.
---
### Author:  [@twickster](https://boards.straightdope.com/u/twickster)
#### Post date: [December 2, 2008, 4:31pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/6 "2008-12-02T16:31:20Z")
> [@ultrafilter](#):
>
> A solution exists. That sounds pretty solvable to me.
Well, yeah, to _you_ 😛
---
### Author:  [@ShibbOleth](https://boards.straightdope.com/u/ShibbOleth)
#### Post date: [December 2, 2008, 4:35pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/7 "2008-12-02T16:35:06Z")
> [@twickster](#):
>
> Well, yeah, to _you_ 😛
Post the puzzle you’re stuck on.
---
### Author:  [@twickster](https://boards.straightdope.com/u/twickster)
#### Post date: [December 2, 2008, 5:16pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/8 "2008-12-02T17:16:49Z")
**6 \_ 7 / 2** 4 \_ / \_ \_ **3**
4 \_ \_ / 1 \*\*5 6 \*\* / \_ 7 \_
\_ **1 \_ / \_ 3** 7 / **6 4** \_
**2** 4 \_ / \_ 1 5 / **7 6 8**
\_ \_ \_ / **7** 6 \*\*4 \*\* / \_ 2 \_
\*\*7 6 1 \*\* / \_ \_ 2 / \_ \_ **4**
\_ \*\*2 4 \*\* / 6 **7** \_ / \_ \*\*8 \*\*\_
\_ 7 \_ / **5 2** \_ / 4 \_ 6
**5** \_ 6 / 4 \_ \*\*1 / 2 \_ 7 \*\*
Bold are the given numbers; not bold is where I think I am with this, which may or may not be correct.
After preview: dang, my extra spaces disappeared. Hope the slashes help.
---
### Author:  [@Claptree](https://boards.straightdope.com/u/Claptree)
#### Post date: [December 2, 2008, 5:56pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/9 "2008-12-02T17:56:45Z")
I had a go, and got to your current situation, where I hit the proverbial wall. It rarely takes me that long to figure out a sudoku, so I went to [sudokusolver.co.uk](http://sudokusolver.co.uk), and ran the puzzle there.
The answer (not to the puzzle, but from the solver):
Solved it, but had to guess, and there might be more than one solution.
So, IMO, not a great puzzle.
---
### Author:  [@Hal\_Briston](https://boards.straightdope.com/u/Hal_Briston)
#### Post date: [December 2, 2008, 5:59pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/10 "2008-12-02T17:59:47Z")
If anyone wants to give it a go, I’ve recreated the original puzzle in [Excel](http://www.sirblah.com/misc/sudoko.xls) and [HTML](http://www.sirblah.com/misc/sudoko.htm). I’ll pass this one over to my wife, who spends a good hour or two each day on sudoko.
---
### Author:  [@ShibbOleth](https://boards.straightdope.com/u/ShibbOleth)
#### Post date: [December 2, 2008, 7:01pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/11 "2008-12-02T19:01:37Z")
I think I have the answer, although I may have made a mistake\* that helped me solve it.
## 6 5 7 | 2 4 9 | 1 5 3
## 4 3 8 | 1 5 6 | 8 7 9
# 9 1 2 | 8 3 7 | 6 4 2
## 2 4 3 | 9 1 5 | 7 6 8
## 8 9 5 | 7 6 4 | 3 2 1
# 7 6 1 | 3 8 2 | 5 9 4
## 1 2 4 | 6 7 3 | 9 8 5
## 3 7 9 | 5 2 8 | 4 1 6
5 8 6 | 4 9 1 | 2 3 7
\*I think I made an invalid assumption on placing the first nine (positioning was correct, but it was lucky versus logical.
---
### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [December 2, 2008, 8:02pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/12 "2008-12-02T20:02:07Z")
Just for the record, if you type out the puzzle like this:
[noparse]
```auto
**6 _ 7 / 2** 4 _ / _ _ **3**
4 _ _ / 1 **5 6** / _ 7 _
_ **1 _ / _ 3** 7 / **6 4** _
**2** 4 _ / _ 1 5 / **7 6 8**
_ _ _ / **7** 6 **4** / _ 2 _
**7 6 1** / _ _ 2 / _ _ **4**
_ **2 4** / 6 **7** _ / _ **8** _
_ 7 _ / **5 2** _ / 4 _ 6
**5** _ 6 / 4 _ **1 / 2 _ 7**
```
[/noparse]
you’ll get this as the output:
```auto
**6 _ 7 / 2** 4 _ / _ _ **3**
4 _ _ / 1 **5 6** / _ 7 _
_ **1 _ / _ 3** 7 / **6 4** _
**2** 4 _ / _ 1 5 / **7 6 8**
_ _ _ / **7** 6 **4** / _ 2 _
**7 6 1** / _ _ 2 / _ _ **4**
_ **2 4** / 6 **7** _ / _ **8** _
_ 7 _ / **5 2** _ / 4 _ 6
**5** _ 6 / 4 _ **1 / 2 _ 7**
```
Unfortunately, the spaces in the editor don’t look the same as the spaces in the code box, so you’ll have to preview and muck around a bit, but that’s your best bet for representing something with multiple spaces.
ETA: If you want to spoiler your sudoku, you’ll need to put the spoiler box inside the code box.
---
### Author:  [@lieu](https://boards.straightdope.com/u/lieu)
#### Post date: [December 2, 2008, 8:14pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/13 "2008-12-02T20:14:03Z")
**Shibb** , you’re close but you’ve got two 8s in your second row from the top and two 2s in your third row from the top. I think if you invert the 8 and 2 in the upper left box you’re good.
This was a toughie.
---
### Author:  [@Eureka](https://boards.straightdope.com/u/Eureka)
#### Post date: [December 2, 2008, 8:18pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/14 "2008-12-02T20:18:35Z")
**lieu** ,
You are correct about there being at least one mistake in **Shibb’s** answer, but since there is already an 8 in the third row from the top, your solution won’t work. Incidentally, the first row from the top has two 5s.
---
### Author:  [@CatInASuit](https://boards.straightdope.com/u/CatInASuit)
#### Post date: [December 2, 2008, 9:11pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/15 "2008-12-02T21:11:52Z")
As far as I can tell there is only one solution, but to get it required a guess from the stage twickster was at to begin with in order to complete it.
I can provide the link in the chain, if required 😉
---
### Author:  [@panamajack](https://boards.straightdope.com/u/panamajack)
#### Post date: [December 2, 2008, 10:04pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/16 "2008-12-02T22:04:06Z")
The solution is unique (confirmable with any computer sudoku solver, like [this one](http://sudoku.goodcamel.com/resolver).
I’m still wondering where the nest is. Any theories on that?
---
### Author:  [@twickster](https://boards.straightdope.com/u/twickster)
#### Post date: [December 2, 2008, 11:44pm UTC](https://boards.straightdope.com/t/sudoko-theory/475342/17 "2008-12-02T23:44:32Z")
Thanks all. I’ve moved on from that one. The one I did this afternoon had a similar situation – not a single 8 in the given numbers – but turned out to be solvable\*, answering my original question in the non-theoretical fashion that works best with me on these things.
\*I had all but three numbers in one column. Two of the blanks were each “either a 2 or a 5,” so the 8 was in the third blank.
---
### Author:  [@Wilson](https://boards.straightdope.com/u/Wilson)
#### Post date: [December 3, 2008, 12:12am UTC](https://boards.straightdope.com/t/sudoko-theory/475342/18 "2008-12-03T00:12:27Z")
> [@CatInASuit](#):
>
> As far as I can tell there is only one solution, but to get it required a guess from the stage twickster was at to begin with in order to complete it.
>
> I can provide the link in the chain, if required 😉
It’s solveable. I’m not sure what you mean by a guess - agreed that at the point **twickster** is at, there’s nothing else easily obvious. However, at that point, I’ve found at least one square that only allows two choices, and one of the choices leads to later impossibilities. Finding that out required going pretty far down the “domino train” with the wrong answer however. Once you fill in that square with the other number, the train is easy.
---
### Author:  [@lieu](https://boards.straightdope.com/u/lieu)
#### Post date: [December 3, 2008, 2:58am UTC](https://boards.straightdope.com/t/sudoko-theory/475342/19 "2008-12-03T02:58:29Z")
Thanks, **Eureka** , I kept working between meetings and finally drew **Wilson’s** conclusion.
**Panamajack** , what’s a nest?
---
### Author:  [@Oslo\_Ostragoth](https://boards.straightdope.com/u/Oslo_Ostragoth)
#### Post date: [December 3, 2008, 6:05am UTC](https://boards.straightdope.com/t/sudoko-theory/475342/20 "2008-12-03T06:05:12Z")
twickster: I got as far as you did. I plugged it into Simple Sudoku, which gave me a hint: Exclude based on “colors”. The help file offers this:
> [@Simple Sudoku](#):
>
> Solving with Colors:
> Of interest are individual candidates that are in only two cells in a given group (a row, column or box). Those two cells have a ‘conjugate’ relationship where one must be the value (true) while the other must not (false). Because we don’t yet know which is which, a recognised strategy is to visualise this relationship using two colors. (I’ve arbitrarily chosen blue and bright green for the following examples but don’t get confused by the pale green shaded cells which simply indicate “filtered” cells – ie cells containing a specific candidate or candidates). Typically there are a number of ‘conjugate pairs’ present at any given time. Sometimes these conjugate pairs link with other conjugate pairs forming a chain of alternate true/false cell states, and these chains may expose candidates which can be excluded safely.
>
> Whenever two cells in a conjugate chain have the same color and also share the same group, that color must be the ‘false’ color since each group can only have one of any value.
>
> In the following example, filtering has been added so only candidate 9’s are visible. (Cells which contain the filtered candidate are automatically highlighted pale green in Simple Sudoku unless subsequently recolored - which is the case for all but one cell in this example). It is easy to see that there are a number of conjugate relationships allowing alternate coloring in a chain. Starting with the cell R2C2, which I’ve arbitrarily colored green, it has 2 conjugates - cells R6C2 (since there are only 2 candidate 9’s in Column 2) and cell R2C3 (since there are only 2 candidate 9’s in row 2) which I’ve colored blue. Both these 2 blue cells (R6C2 & R2C3) have other conjugate cells which I’ve colored green. Repeating this technique around the grid we soon discover that there are blue cells sharing the same group. Since any group can only have one of a given value, the blue cells must represent the false values so all bright green cells can safely be assigned the value 9. (In the following examples, don’t be confused by any pale green cells which simply indicate filtering on a specific candidate.)
> Also, whenever a candidate outside the chain relates by column, row or box to two alternately colored cells in a conjugate chain, that ‘non-chain’ candidate can be excluded.
>
> This is best understood by looking at the example below. Filtering has been applied so only candidate 5’s are visible. The cells labeled A & B form a conjugate pair as they are the only candidate 5’s in column eight. Cells labeled B & C also form a conjugate pair as they are the only candidate 5’s in the bottom right box. Finally, cells C & D form a conjugate pair as they are the only candidate 5’s in row eight. As these three conjugate pairs all link together by separate conjugate relationships, they form a chain and can be colored alternately as shown. The yellow highlighted cell remotely relates to linked conjugate pairs of alternate colors (cells A & D). Since one of the two colors (blue & green) represents the true value 5, this remote candidate can safely be excluded.
I swear I am not stupid, but I don’t understand this at all. ☹ Can anyone help a brother?
[Next page](https://boards.straightdope.com/t/sudoko-theory/475342.md?page=2)