HP01 Solving “hardest puzzles” continued   

 

P2. The EXOCET Pattern

 

This is surely the key property, widely spread and often combined to itself, but also to the SK loop. 

 

The EXOCET property has been defined based on an Allan Barker study off 2 very special puzzles “Fata Morgana” and “Trompe l’oeil” found by tarek.

 

020400000000009200098000004000600007500010000040003800300000060000700001004008500 Fata_Morgana

000400000007109000090020040000000800300060005070002010005000000800070003010200070 Trompe_l'oeil

 

These 2 puzzles staid unsolved with the set of rules of my solver

(one more example up to now, the puzzle from Mauricio shown in P5 symmetry).

 

Allan Barker built a big SLG clearing partially two cells See The Illusion of Fata Morgana.

 

I named the underlying property EXOCET which appeared to be the most frequent property in hardest puzzles.

 

Here after the definition of an EXOCET on the puzzle Fata Morgana  and some clues to find it .

 

the start in fata morgana, after one single, is the following

 

 

1||167__ 2____ 13567 |4___ 35678_ 1567_ |13679 135789 35689

2||4____ 13567 13567 |1358 35678_ 9____ |2____ 13578_ 3568_

3||167__ 9____ 8____ |1235 23567_ 12567 |1367_ 1357__ 4____

 

4||1289_ 138__ 1239_ |6___ 24589_ 245__ |1349_ 123459 7____

5||5____ 3678_ 23679 |289_ 1_____ 247__ |3469_ 2349__ 2369_

6||12679 4____ 12679 |259_ 2579__ 3____ |8____ 1259__ 2569_

 

7||3____ 1578_ 12579 |1259 2459__ 1245_ |479__ 6_____ 289__

8||2689_ 568__ 2569_ |7___ 234569 2456_ |349__ 23489_ 1____

9||12679 167__ 4____ |1239 2369__ 8____ |5____ 2379__ 239__

 

An EXOCET has a base of 2 cells, in row or column. Here the base is r13c1.

 

The base contains 3 or 4 digits (3 here, digits 1;6;7). As far as I can see, in the direction of the base (here stack 1) none of the digits of the base is given, but this is not part of the rule.

 

We have an EXOCET if we can combine the base with a 2 cells target in such a way that any digit belonging to the base belongs to at least one of the cells of the target.

 

Here, the target is r6c3 ; r9c6

 

As a matter of fact, any 2 digits solution for the base will have the same 2 digits solution in the target.

 

Working on the “2 digit super candidates” possible solutions for the base gives then much faster ways to crack the puzzle.

 

One question comes immediately: How to detect an EXOCET.

 

1) If we combine the above given rules for a base

 2 cells in row or column

 3 or 4 digits in total

 None of the digits given in the stack/band

 

 The number of possibilities is very low

 

2)        To easily detect the target and work on “super candidates”, I personally prepare a reduced PM where all givens and cells containing none of the digits of the base are ignored.

 

 

In other cells, the candidates not belonging to the base are marked on block ‘+’, giving that reduced map

 

167  X    167+ |X   67+  167+ |167+ 17+  6+ 

X    167+ 167+ |1+  67+  X    |X    17+  6+ 

167  X    X    |1+  67+  167+ |167+ 17+  X   

 

1+   1+   1+   |X   X    X    |1+   1+   X   

X    67+  67+  |X   X    7+   |6+   X    6+

167+ X    167+ |X   7+   X    |X    1+   6+

 

X    17+  17+  |1+  X    1+   |7+   X    X   

6+   6+   6+   |X   6+   6+   |X    X    X   

167+ 167  X    |1+  6+   X    |X    7+   X   

 

 

The target has always one cell in each block of the band/stack out of the block containing the base and not in the row/column of the base. Again this is a practical rule and not part of the ‘principles’.

 

A couple of cells here and only one could work: r6c3;r9c2.

 

We have just to verify at that point in each floor/rookery that no solution exists with a digit in the base and not that digit in the target. I do it here for digit 1 starting with a partial solution after    1 r13c1 true  (and I re introduce to clarify the givens ‘1’)

 

1 . . |. . . |. . .

. . . |1 . . |. 1 .

1 . . |1 . 1 |1 1 .   

 

. 1 1 |. . . |1 1 .   

. . . |. 1 . |. . .

. . 1 |. . . |. 1 .

 

. 1 1 |1 . 1 |. . .   

. . . |. . . |. . 1   

. 1 . |1 . . |. . .  

 

If we assume now 1 r6c3 false and 1r9c2 false we come there

 

1 . . |. . . |. . .

. . . |1 . . |. . .

. . . |. . . |1 . .   

 

. 1 1 |. . . |. . .   

. . . |. 1 . |. . .

. . . |. . . |. 1 .

 

. 1 1 |. . 1 |. . .   

. . . |. . . |. . 1   

. . . |. . . |. . .    no possibility in row 9 ; box 9

 

The EXOCET pattern has been studied  with 2  different bases

 

_ 2 cells having 3 digitd (and 3 super candidates) as in Fata Morgana.

_ 2 cells having 4 digits (and 6 super candidates) as in “Golden Nugget” and “Silver Plate”.

 

I have identified more than 50 000 puzzles having the 3 digits pattern.

We will see later more details on how to use the property to crack puzzles, but some examples can be found in these threads

 

bi bi pattern in hardest puzzles

Great Monster Loops

More Monster Loops, Structure and Symmetry

 

interesting “double EXOCET” patterns  have been found with the 4 digits base.

Here is an example

 

000406700007000000000070015290000000006040100000008002030000000800300090005010600 coly007  

 

1359  1258  12389 |4      23589 6     |7     238    389  

3459  2458  7     |12589  23589 12359 |23489 23468  34689

3469  2468  23489 |289    7     239   |23489 1      5    

-------------------------------------------------------

2     9     1348  |1567   356   1357  |3458  345678 34678

357   578   6     |2579   4     23579 |1     3578   3789 

13457 1457  134   |15679  3569  8     |3459  34567  2    

-------------------------------------------------------

14679 3     1249  |256789 25689 24579 |2458  24578  1478 

8     12467 124   |3      256   2457  |245   9      147  

479   247   5     |2789   1     2479  |6     23478  3478 

 

In red and blue the 2 EXOCET patterns with the same digits 2;3,8;9.

 

All targets are reduced to 2389. Blue and red can not have the same digits. R1c5=5

 

I have more than 300 puzzles having that double EXOCET pattern;. Here after 2 more:

 

070100006000000009003090500008060005020000300000004800006080050200001000040700000

col0906 3        

 

010200008700000009003090500008060005000002300100004600006030050400000000070000000

col0906 11       

 

The entire file will be available for downloading on my website later