About Colouring

Advanced methods and approaches for solving Sudoku puzzles

Postby stuartn » Wed Aug 10, 2005 6:08 pm

The box/line intersection chain I have introduced above is something new which I had never seen before and which doesn't fall in the above categories.


For which you deserve full credit Nick70.:D

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Postby tso » Wed Aug 10, 2005 6:29 pm

Wolfgang wrote:So forcing chains really can use 'box elimination' with contradicting chains, i was mistaken.


Sorry for scrambling my last post; thanks for pointing it out.

Nishio is not really forcing chains or 'contradicting chains', though my post may have made it appear that way to show the similarity to the original post. In Nishio, you place a digit and see if you are able to place all other like digits. If not, you know your original placement is wrong. The advantage is it is often very easy to follow it completely to the end in your head.

A more "traditional" way to describe the same Nishio is this:

1) Place a 6 in r1c7.
2) Now, the only place in column 6 that a 6 can go is row 8. Place a 6 in r8c6.
3) Now, the only place in box 7 for a 6 is r9c3. Place a 6 there.
4) Now, there is no place for a 6 in box 1. Backtrack, removing all the 6s placed and r1c7 is NOT a 6.
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Postby tso » Wed Aug 10, 2005 7:23 pm

Nick70 wrote:After my thoughts have eaten, I propose this notation for discussion:

[x&y] is the intersection of units x and y.

[R9&C3]=6 => [B1&C3]<>6 => [B1&R1]=6 => [B3&R1]<>6 => [R3&C9]=6
[R8&C1]=6 => [B9&R8]<>6 => [B9&C7]=6 => [B3&C7]<>6 => [R3&C9]=6


Though either [R1&B1]<> 3, r1b1<>3 or r1c123<>3 is clear, [R1&B1]= 3, r1b1=3 or r1c123=3 may not be without the definition included, as you are using "=" to mean something different than "equals".
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Postby Nick70 » Wed Aug 10, 2005 7:51 pm

tso wrote:Though either [R1&B1]<> 3, r1b1<>3 or r1c123<>3 is clear, [R1&B1]= 3, r1b1=3 or r1c123=3 may not be without the definition included, as you are using "=" to mean something different than "equals".

Of course you are right, actually they are both wrong mathematically, since R1B1 is a set we should use the element-of and not-element-of symbols, which aren't par tof the standard font unfortunately.
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Postby stuartn » Wed Aug 10, 2005 8:19 pm

For goodness sake tso - stop being so pedantic - its clearly obvious that in this context (and with this font) that "=>" means 'implies that'. What would you suggest otherwise?

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rant over. fridge open
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Postby tso » Wed Aug 10, 2005 8:28 pm

How about ε?

Others that may be useful:

∩≡≈±
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Postby stuartn » Wed Aug 10, 2005 8:34 pm

and look - they're all there on my keyboard - why haven't I spotted them before?:D
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Postby tso » Wed Aug 10, 2005 8:50 pm

stuartn wrote:For goodness sake tso - stop being so pedantic - its clearly obvious that in this context (and with this font) that "=>" means 'implies that'. What would you suggest otherwise?

stuartn

rant over. fridge open


Sorry, I may not have been clear. The symbol "=>" is not in question. That's common usage -- I use it constantly. What is in question is the use of the equal sign in [R1&B1]=1. Anyone coming in late to the conversation or reading a post that is linked to out of context should be able to figure out [R1&B1]<>1, but they may very well misinterpret [R1&B1]=1, as it seems to say that r1c123 are all equal to '1'.

I don't know if something like r1∩b1 ε 1 is acceptable.
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Postby tso » Wed Aug 10, 2005 9:01 pm

stuartn wrote:and look - they're all there on my keyboard - why haven't I spotted them before?:D


I had to turn my keyboard upside down to find them.


If you don't have a character map ap handy, you can either copy and paste the symbols or type these in, leaving out the spaces:


ε
& # 9 4 9 ;


& # 8 7 4 5 ;


& # 8 8 0 1 ;


& # 8 7 7 6 ;

Anyway, it was just a suggestion. Sheesh.
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Postby Nick70 » Wed Aug 10, 2005 9:52 pm

I didn't know you could use HTML entities in the posts.

R1∩B1∋1

and its negation is what I'd like to write. It seems only the affirmative version is possible? I cann't find the crossed out ∋ symbol.
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Postby stuartn » Wed Aug 10, 2005 10:30 pm

had to turn my keyboard upside down to find them.



crikey - you've keys UNDER there aswell??
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special symbols

Postby Pat » Thu Aug 11, 2005 9:48 am

tso (2005.Aug.10) wrote:How about ε ?

Others that may be useful:

∩ ≡ ≈ ±


many thanks for the special symbols!

i'd like to use something like 4 ε r7 ∩ BR to indicate a 4 must be in the r7 slice of box Bottom-Right, just as r3c4 ε { 1,6 } indicates the cell r3c4 must be either 1 or 6.

the may also come in handy.

the crossed-out version of ε is sorely needed!

- Pat
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Re: About Colouring

Postby Addlan » Thu Aug 11, 2005 1:20 pm

Bunnybuck wrote:One example would be Addlan's puzzle:
Code: Select all
 .  .  2 | .  9  . | 1  .  7
 .  3  8 | 6  .  . | .  .  .
 4  .  . | .  .  . | .  .  .
---------+---------+---------
 .  .  . | .  .  5 | .  .  .
 .  .  9 | .  1  . | 3  .  .
 .  .  . | 4  .  . | .  .  .
---------+---------+---------
 .  .  . | .  .  . | .  .  4
 .  .  . | .  .  7 | 9  2  .
 8  .  6 | .  3  . | 7  .  .



I am sorry, but I copy this puzzle from Nick.:)

It is a very good article. The above puzzle can be solved by forcing chain, but the chain is very long so that the effort will be more than "try and error".

Try 4 at r8c3 and have the following result
{5} {6} {2} {8} {9} {3} {1} {4} {7}
{7} {3} {8} {6} {4} {1} {5} {9} {2}
{4} {9} {1} {7} {5} {2} {6} {8} {3}
{2} {8} {7} {3} {6} {5} {4} {1} {9}
{6} {4} {9} {2} {1} {8} {3} {7} {5}
{1} {5} {3} {4} {7} {9} {2} {6} {8}
{9} {7} {5} {1} {2} {6} {8} {3} {4}
{3} {1} {4} {5} {8} {7} {9} {2} {6}
{8} {2} {6} {9} {3} {4} {7} {5} {1}

Try 4 at r4c3 and get the following result
{56} {56} {2} {3} {9} {4} {1} {8} {7}
{7} {3} {8} {6} {2} {1} {4} {9} {}
{4} {9} {1} {5} {78} {28} {26} {6} {3}
{3} {12678} {4} {9} {68} {5} {26} {167} {28}
{26} {268} {9} {7} {1} {28} {3} {4} {5}
{126} {12678} {5} {4} {68} {3} {26} {167} {289}
{29} {29} {7} {1} {5} {6} {8} {3} {4}
{15} {15} {3} {8} {4} {7} {9} {2} {6}
{8} {4} {6} {2} {3} {9} {7} {5} {1}

These two are the extreme cases of chains and combine the two possible tables above and you will get the results. Apparently the above processing is not worthy and also someone doesn't know when to stop the chain.

My idea is to restrict forcing chain only for pairs or a digit. If a pair (a,b), both a and b will lead to a digit c, then try to find the forcing chain. For example a pair (1,2) in a gid, if '1' can lead to digit '2' somewhere, then try to find a forcing chain for digit '2'.

Another drawback for forcing chain is it may waste effort if it cann't find anything, while "try and error" will never. So make sure forcing chain or forcing colour or whatever is worthy.
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Re: special symbols

Postby tso » Thu Aug 11, 2005 5:44 pm

Pat:

The negation sign is & # 1 7 2 ;, so you could write ¬∋.

All possibilities are listed here.

BUT, some symbols didn't display on my Internet Explorer, but did in Firefox.

Embedded gifs are possible:

3ImageR1ImageB1

These gifs are here. I didn't see a "does not contain the element" symbol" but they have the "is not an element of" symbol, which could be reversed with a paint program or simply write the equation backwards as above.
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