SpAce wrote:http://sudoku.com.au/sudokutips.aspx?Go=U3-1-1992

Is that the authoritative source on this stuff?

To me, yes it is.

SpAce wrote:JC also provided great examples here:

continuous-loops-with-als-nodes-t34306.html#p261523

You are right to associate JC van Hay to that topic. He made me discover Steve K's writings about matrices and Easter Monster, and I have learnt a lot from him...

You have borrowed Denis's definition. I recall Steve K's one

- Size is nxn

- Each row contains at least one truth (a strong inference set)

- Each column, except the first column, has the following quality:

The top non-empty entry is in conflict with each entry below it

Note there is a subtle difference here from the pigeonhole matrix.

- The following cells are always blank:

For each row i, rowicolumnj for all j>i+1

Note this is a significant difference

- The first column, the result column, is proven to contain at least one truth - thus a derived strong inference set.

I reformulate in a less "mathematical" language

- A TM is a square matrix (n rows AND n columns)

- Rows contain SIS (native or derived i.e. bivalue, bilocation, 3-value, 3-location, ..., ALS or any other derived SIS)

- The matrix is empty above its first upper diagonal (FUD), (this why it is called "triangular")

- All non empty cell below FUD has a weak link to the FUD candidate(s) above, (but not required between each other)

- Matrix inference is : candidate(s) in the first column are in a strong inference (one must be true)

In your subsequent post, you ask:

...how would people write the linked Triangular Matrix example in chain format?

The puzzle of Steve K's example (Easter Monster, after several eliminations):

- Code: Select all
` +-----------------------+----------------------+----------------------+`

| 1 478 3458 | 3567 389 5678 | 3489 369 2 |

| 238 9 378 | 4 16 27 | 138 5 368 |

| 3458 248 6 | 1235 389 1258 | 7 139 3489 |

+-----------------------+----------------------+----------------------+

| 2468 5 1478 | 9 126 3 | 128 167 678 |

| 389 126 139 | 126 7 4 | 3589 126 3589 |

| 2369 1267 1379 | 8 5 126 | 1239 4 3679 |

+-----------------------+----------------------+----------------------+

| 7 148 4589 | 1235 348 1258 | 6 239 3459 |

| 456 3 145 | 1267 126 9 | 245 8 457 |

| 4589 468 2 | 3567 348 5678 | 3459 379 1 |

+-----------------------+----------------------+----------------------+

The TM written "Eureka-like"

- Code: Select all
`1r2c7 1r2c5`

1r4c78 1r4c5 1r4c3

1r8c5 1r8c3 1r8c4

1r6c6 1r3c6 1r7c6

=> -1 r6c7

Note that all candidates are digit 1. This is probably recognised as a mutant jellyfish r248c6\b268c3 in the UFG ?

Fish experts can confirm... (I am not sure of my choice of cover sets).

So, this example is at the same time much interesting, as matrices are able to explain how complex fishes work, but also not relevant as a training to translate a matrix into an AIC. To me, some "full" n-fishes (n>=3) are impossible to write as AIC's, if they are not natively reduced to an AIC (with "full" I mean fishes with no node or only few nodes left void of the fish digit). This is the case right now. You have to use them as patterns.

Steve K himself proposed an AIC presentation:

As an AIC, I could write:

(1): r2c7=c5-[c5r48,r3c6]=[r4c78=c3-r8c3=c4-c6r7=r6] => r6c7 <>1

Note that I need to do two things:

- recognize the multiple forbiddings caused by (1)r2c5

- break out a sub-chain - contained above by the brackets [].

I would reformulate a bit his writing to:

(1)r2c7 = r2c5-[r4c5*,r8c

5**,r3c6***]=[r4c78*=r4c3-r8c3**=r8c4-r7c6***=r6c6] => r6c7 <>1

with stars to show the strong inference connections. Typo corrected, Thanks, SpAce for spotting.

EDIT Feb. 01: in his sequent post, SpAce draw attention on the meaning of commas in the first square bracket. "OR" must be meant un-ambiguously. So, the following is proposed:

(1)r2c7 = r2c5-[r4c5*|r8c5**|r3c6***]=[r4c78*=r4c3-r8c3**=r8c4-r7c6***=r6c6] => r6c7 <>1

A simpler chain is also proposed by SpAce:

(1)r2c7 = r2c5 - (r4c5 | r3c6) = [ (1)r4c78 = r4c3 - r8c3 = r8c45 - r7c6 = (1)r6c6 ] => -1 r6c7

I agree with this chain, just noting that it is the translation of another matrix than the above:

- Code: Select all
`1r2c7 1r2c5`

1r4c78 1r4c5 1r4c3

1r8c3 1r8c45

1r6c6 1r3c6 1r7c6

=> -1 r6c7

Otherwise, one may notice a finned X-wing r24c578 in rows 1&2 of the TM, and an almost-almost kite r8,c6 in rows 3&4 of the TM.

Therefore a first possible presentation is (kind of kraken):

[Kite(1)r8c5=r8c4-r7c6=r3c6]-r2c5=r2c7

(1)r6c6

(1)r8c3 - r4c3 = [FXW(1)r4c78*=r4c5-r2c5=r2c7]

=> -1 r6c7

So, on one line you could get:

(1)r6c6*=[r2c7=r2c5-(Kite

r3c6=r7c6-r8c4=r8c5) = r8c3-r4c3=(FXW r4c78=r4c5-r2c5=r2c7)] => -1 r6c7

"*" and highlighted blue "=" to focus the strong link of 1r6c6 to the kite.

or with shortened patterns (1)r6c6*=[r2c7=r2c5-(Kite

r37c6-r8c45) = r8c3-r4c3=(FXW r4c578-r2c57)] => -1 r6c7

Two other examples of a TM :

http://forum.enjoysudoku.com/october-3-2017-t34109.htmland a net :

http://forum.enjoysudoku.com/extreme-9-9-17-t34038.htmlPlease, don't ask me the equivalent AIC !

Just FWIW...