## Some thoughts on Eureka notation

Advanced methods and approaches for solving Sudoku puzzles

### Some thoughts on Eureka notation

true = value is contained in realm
false = value is not in realm

x=y Strong inference both x and y cannot be false

x-y weak inference both x and y cannot be true

xrealm, yrealm xy are values if not present they continue from the last value of the previous inference

1 to 9 values could be multiple but must be unique

rncn realm of a cell row and column b stands for block n could be multiple but must be unique

rncn,rncn cells not in the same row or column

(*inference)#realm inference * is true for # realm

inferences are enclosed with parenthesis if values are common to a realm

\$\$:realm function or method to further define a realm

=> therefore precedes a solution

-valuerealm solution is value is false for the realm

+valuerealm or valuerealm value is true

; end of step

stte singles to the end

lclste locked cells and sets to end

any thoughts, comments or suggestions are welcome
Last edited by ArkieTech on Mon Jan 07, 2013 1:41 pm, edited 3 times in total.
dan

ArkieTech

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Joined: 29 May 2006
Location: NW Arkansas USA

### Re: Some thoughts on Eureka notation

x=y Strong inference x or/and y must be true

x-y weak inference both x or/and y must be false

Don't understand the wording. I thought a strong inference was that both can't be false (or one or the other must be true) and weak meant both can't be true. Why the "or/and"?

stte singles to the end

lclste locked cells and sets to end

I have no idea what value this has to others, but I don't find it useful. The absence of such a statement implies no more advanced moves needed, i.e., basics to the end, and that's more than enough information for me.
Marty R.

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Joined: 23 October 2012
Location: Rochester, New York, USA

### Re: Some thoughts on Eureka notation

Marty R. wrote:
x=y Strong inference x or/and y must be true

x-y weak inference both x or/and y must be false

Don't understand the wording. I thought a strong inference was that both can't be false (or one or the other must be true) and weak meant both can't be true. Why the "or/and"?

stte singles to the end

lclste locked cells and sets to end

I have no idea what value this has to others, but I don't find it useful. The absence of such a statement implies no more advanced moves needed, i.e., basics to the end, and that's more than enough information for me.

Thanks for the input Marty.

I am going to remove the word "both" it is misleading.

My understanding both can be true in a strong inference (and) an at least one must be (or).

In a weak inference if one is true the other can't be. ie both cannot be true but both could be false.

or

strong both cannot be false
weak both cannot be true

!A => B (if not A, then B) OFF implies ON -- A is not defined

A => !B (if A, then not B) ON implies OFF -- !A is not defined

Might be better--what do you think?

"Basics to end" means different things to different folks. Some consider all of the simple sudoku set as basic. Others consider quads as advanced.

stte and lclste are not open to opinion.
dan

ArkieTech

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Location: NW Arkansas USA

### Re: Some thoughts on Eureka notation

Example from Here

Code: Select all
`qnp = quantum naked pair. *--------------------------------------------------------------------* | 2      4      5      | 6      9      8      | 7      1      3      | | 7      8      1      |g25     4      3      | 9      6     e25     | | 6      9      3      | 125    7     h12     | 248    24     2458   | |----------------------+----------------------+----------------------| | 8      1      27     | 237    5      9      | 234    247    6      | | 5      3      9      | 27     6      4      | 1      8      27     | | 4     f27     6      |j1237-8 28    i127    | 23     5      9      | |----------------------+----------------------+----------------------| | 1      5     d248    |a278    3      27     | 6      9     d248    | | 3      6     d248    | 9      1      5      | 248    247   d2478   | | 9     c27    c278    | 4     b28     6      | 5      3      1      | *--------------------------------------------------------------------*8r7c4=(8-2)r9c5=2r9c23-(2r78c3=qnp:27r578c9)aur:48r78c39-r2c9=r2c4-(2=1)r3c6-r6c6=(1)r6c4 => -8r6c4`
dan

ArkieTech

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Joined: 29 May 2006
Location: NW Arkansas USA

### Re: Some thoughts on Eureka notation

!A => B (if not A, then B) OFF implies ON -- A is not defined

A => !B (if A, then not B) ON implies OFF -- !A is not defined

Might be better--what do you think?

Sorry, can't help. I don't understand much beyond the definitions and don't think about them because a lot of weak inferences seem strong to me. I just try my best by alternating the "=" and "-" signs.
Marty R.

Posts: 1508
Joined: 23 October 2012
Location: Rochester, New York, USA

### Re: Some thoughts on Eureka notation

Personally I find "strong both cannot be false, weak both cannot be true" to be the clearest. But in terms of a specification it is probably best to say it several different ways.

JasonLion
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Location: Silver Spring, MD, USA

### Re: Some thoughts on Eureka notation

JasonLion wrote:Personally I find "strong both cannot be false, weak both cannot be true" to be the clearest. But in terms of a specification it is probably best to say it several different ways.

Thanks I agree.
dan

ArkieTech

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Joined: 29 May 2006
Location: NW Arkansas USA

### Re: Some thoughts on Eureka notation

Hi,

As foreigner, I am reluctant to enter that discussion, but seing how difficult it is to adjust the wording for "strong inference" and "weak inference", I would put the discussion one step above.

What means "true" and "False".

For a candidate, we can for sure say "is part of the solution" for "true" and "is not part of the solution" for "false".

But the writing rules becomes more complex as soon as we handle objects other than a candidate.

Usually, in ALS AHS a group of candidates is written for example 1r1c1r2c3

then "true" means one of 1r1c;1r2c3 is part of the solution
"false" means "none of them is part of the solution.

But if the group is a possible solution for 2 cells (or any AAHS) say 12r1c1r2c3, then
"true" means that both are part of the solution
"False" means that at least one of them (may be one of 2 sub groups) is false.

It seems difficult to skip in that exercice
the list of objects that can be handled
the exact meaning of "false" and "true" for each of them
the writing rules to apply

Hope this is not off topic
champagne
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Location: France Brittany

### Re: Some thoughts on Eureka notation

Champagne, You've caused me to break a promise I made to myself to stay out of this. I thought DonM had taken up the baton but it appears he's dropped it.

While I accept that for beginners the understanding weak and strong links is a hurdle, I have little sympathy for those with years of experience still have problems digesting the difference between them. It seems that they have been gulping their food rather than chewing it first!

That said, most strong links used in basic chains are conjugate and can also be used as weak links which I admit is confusing at first. But look at this chain: (1=2)r1c1 – (2=3)r1c9
This proves that there's a strong link (1)r1c1 = (3)r1c9 but that doesn't mean if one is true the other is false, and that is what AICs are all about.

Combining these cells into (123)ANS:r1c19, (1=3)r1c19 is a strong link as 1 and 3 can't both be false, however both could still be true so there is no alternative weak link.

Changing subject, here's the existing convention for group nodes (nodes that consist of more than one cell)

(1)r1c123 true when any of these cells holds 1, false otherwise
(12)r1c123 true when two of these hold 1 AND 2, false otherwise
(123)r1c123 true when together these cells hold 1 AND 2 AND 3, false otherwise

Therefore there are implied logical AND operations between the listed candidates which saves having to write (1&2&3)r1c123

(1|2)r1c123 uses a logical OR symbol to mean true when these cells hold either candidate, false when they hold when neither of them.

So when the candidates in a cell are 123 we can write (1=2|3)r1c1 and the link to the next node then becomes 2 OR 3.

This can be used in group nodes too eg (12=3|4)r1c12

Myth Jellies suggested using (12=34#1)r1c12 where #1 signifies the minimum number of truths to be held by 3 & 4 to make the second term true.
Checking this out, 12 will be false when one or both of these digits is false, so 34#1 will be true when one or both of them are true.

I like this because it stands out better and (123#2) is awkward to express using logical operators. I've been using this for several years now and it handles the AAHS situations that concern you.

I have never yet found a situation where a logical XOR (exclusive or) has been needed, but perhaps that's because I haven't been actively searching, but simply waiting for one to hit me. [Edit: What was I thinking? That's wrong as I often use conjugate chains which is what a XOR interence represents. It's just that they don't mix with AICs, and have to be translated into weak and strong links.]

Regarding SK loops, in my view these aren't pure AICs because they represent three rather that two divisions of the truths

Code: Select all
`*-------------*            (12=34)r2c13 –  (34=56)r13c2    | .   345  .  |   Truths    0  2             0  2| 134 .   124 |             1  1             1  1| .   456  .  |             2  0             2  0  *-------------*           (12=34#1)r2c13 – (34=56#1)r13c2                              (12#1=34)r2c13 – (34#1=56)r13c2`

As usually notated in the top line, the inferences used aren't logically valid under Eureka conventions. To express these loops as AICs would need the two chains shown in the bottom lines. (I'm not suggesting we do that, simply that we shouldn't think of these loops as regular AICs).

As to whether I intend to contribute further, I've been there, done that, and got the broken nose and cauliflower ears to prove it, so please include me out – at least until there is a wider interest than is shown at present.
Last edited by David P Bird on Mon Jan 07, 2013 5:05 pm, edited 1 time in total.
David P Bird
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### Re: Some thoughts on Eureka notation

Thanks champagne and David P Bird for your input. Today's Vanhegan extreme Here has a good example of David's discussion.

Code: Select all
`*--------------------------------------------------------------------------------*| 346     136     2        | 8      b159     15-3     | 7       459     469      || 467-3   5       8        |a39      2      c37       | 6-3     1       469      || 9       137     137      | 6      b157     4        | 35      2       8        ||--------------------------+--------------------------+--------------------------|| 1       8       4        | 7       3       9        | 2       6       5        || 367     3679    379      | 15     b15      2        | 8       347     47       || 2       37      5        | 4       8       6        | 9       37      1        ||--------------------------+--------------------------+--------------------------|| 5       4       79       | 2       6       8        | 1       79      3        || 37      2       1379     | 1359    4       1357     | 56      8       679      || 8       1379    6        | 1359    1579    1357     | 4       579     2        |*--------------------------------------------------------------------------------*als xy-loop(3=9)r2c4-(9=7)r135c5-(7=3)r2c6 => -3r1c17,r2c7; stte`
dan

ArkieTech

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Joined: 29 May 2006
Location: NW Arkansas USA