On the classification of exotic patterns

As exotic patterns are not associated with families of rules of increasing size that could be used as a scale for any rating system, the question arises as to where one should classify them in my hierarchy.

Remember that this hierarchy is based on:

- chain length for patterns in the bivalue-chains, whips, g-whips, braids, g-braids,... family

- size of the base set for patterns in the Subsets, g-Subsets family

and that these two quantities are equal for patterns that can be considered from both points of view.

1) Classification versus priority

One of the problems with exotic patterns is, they are brittle: if other rules are applied before them, they may disappear and, depending on the other rules used, the eliminations they allowed may no longer be available.

Said otherwise, exotic patterns, when added to a resolution theory with the confluence property may destroy this property if they are used too late.

There is thus a dilemma between using an exotic pattern as soon as available and classifying it (and the puzzles) according to its real logical complexity.

In Sudoku Explainer, rating and order of rule application are confused.

But there's no reason to do the same in any resolution strategy.

When a pattern is brittle, one can decide to apply it as soon as available (or immediately after another set of rules, e.g. SSTS) and nevertheless assign it a high rating.

In this post, I won't discuss when an exotic pattern should be applied (this is a matter of personal taste), but only which rating it should be assigned.

For the hardest puzzles, whether it should be applied before or after some basic rules is somehow irrelevant as, usually, no basic rule can be applied at the start. But for puzzles with lower ratings, this may drastically change statistics for the pattern.

There are currently not so many exotic patterns.

For the sk-loop pattern, it is easy to see that it can be considered as a g-Subset of size 16. This immediately defines its classification.

So, I'll now concentrate on the J-Exocet cases. I first need precise definitions.

2) Standard Jk-Exocet

For definiteness, let me recall what I consider as a standard Jk-Exocet, k = 2, 3, 4, 5, ...

I have previously written "basic" Jk-Exocet but considering that we usually speak of "standard Fish" and not of "basic Fish" (vs e.g. Franken or Mutant ones), I think "standard" is better.

The following definition is a refinement of both David's initial definition: (http://forum.enjoysudoku.com/jexocet-pattern-defintion-t31133.html) and of my re-writing of it (http://forum.enjoysudoku.com/jexocet-pattern-defintion-t31133-12.html), with the S-conditions broadened as introduced here (http://forum.enjoysudoku.com/jexocet-pattern-defintion-t31133-145.html)

I'll also make a distinction between a Jk-Exocet-in-columns and a Jk-Exocet-in-rows, based on the quasi-fish part in the S-lines.

standard Jk-Exocet-in-columns, for k = 2, 3, 4 , 5, ...:

Conditions on the Pattern of Cells

- Code: Select all
`*-------*-------*-------*`

| B B . | . . . | . . . | B = Base Cell Pair

| . . . | Q . . | R . . |

| . . . | Q . . | R . . | Q = 1st Object Cell Pair

*-------*-------*-------* R = 2nd Object Cell Pair

| . . S | S . . | S . . |

| . . S | S . . | S . . | S = S-Cells, included in three S-columns

| . . S | S . . | S . . |

*-------*-------*-------* . = Any candidates (including base digits)

| . . S | S . . | S . . |

| . . S | S . . | S . . |

| . . S | S . . | S . . |

*-------*-------*-------*

The "Base Cell Pair" consists of two cells in the same "Base Row" and the same "Base Block"

Each of the two "Object Cell Pairs" (Q and R) is made of two cells in the same horizontal band as the Base Cell Pair (B) (the Base Band) but not in the same block/box as B; the two Object Cell Pairs may be in the same block.

The first "S-column" intersects the Base Block but doesn't contain any of the Base Cells.

The 2nd and 3rd "S-columns" intersect the Base Band by passing respectively through the Q and R Object Cell Pairs.

Conditions on candidates

1) No base cell is decided.

Each base cell may have up to k candidates, but together, they contain exactly k digits (called the "Base Digits").

2) Each Object Cell Pair is made of two cells: the "Target Cell" and the "Companion Cell"

Each Target Cell contains at least one Base Digit (plus possibly non-Base Bigits)

Companion Cells contain no Base Digit

3) For each Base Digit, there are two rows containing all its occurrences in the S cells (be it as a candidate or as a decided value).

3) Standard Advanced Jk-Exocet

The following definition is the extension of the above, proposed by Blue while I was preparing this post: (http://forum.enjoysudoku.com/jexocet-pattern-defintion-t31133-152.html). Blue's idea is, there's no reason to restrict the Target Cells to the Base Band.

I keep both definitions here, as I don't know which will prevail and it isn't at all the topic of this thread to discuss this, but it's clear for me that Blue's is more elegant.

Standard Advanced Jk-Exocet-in-columns, for k = 2, 3, 4 , 5, ...:

Conditions on the Pattern of Cells

- Code: Select all
`*-------*-------*-------*`

| B B . | . . . | . . . | B = Base Cell Pair

| . . . | S . . | S . . |

| . . . | S . . | S . . |

*-------*-------*-------*

| . . S | S . . | S . . |

| . . S | S . . | S . . | S = S-Cells, included in three S-columns

| . . S | S . . | S . . |

*-------*-------*-------* . = Any candidates (including base digits)

| . . S | S . . | S . . |

| . . S | S . . | S . . |

| . . S | S . . | S . . |

*-------*-------*-------*

The "Base Cell Pair" consists of two cells in the same "Base Row" and the same "Base Block".

The first "S-column" intersects the Base Block but doesn't contain any of the Base Cells.

The 2nd and 3rd "S-columns" do not intersect the Base Block (but they may be in the same tower).

In each of the two S-columns not crossing the Base Block, one of the S-cells is designated as a Target Cell.

Conditions on candidates

1) No Base Cell is decided.

Each Base Cell may have up to k candidates, but together, they contain exactly k digits (called the "Base Digits").

2) Each Target Cell contains at least one Base Digit (plus possibly non-Base Bigits)

3) For each Base Digit, there are two rows such that all the occurrences of this digit in the S cells (be it as a candidate or as a decided value) are either in the Target Cells or in these two rows.

4) Proof of the Standard and the Standard Advanced Jk-Exocet

The proof I gave for the standard Jk-Exocet works with almost no change for both versions.

Let a and b be the (unknown) true values of the two base cells. By the rules of Sudoku, each of a and b must be the value of a cell in each of the three S-columns.

But condition 3 entails that each of a and b can be the value of at most two S cells that are not Targets. Therefore (as neither a nor b can be in the Base Row or the Base Block anywhere outside the S Cells) each of a and b must also be the value of one of the Target Cells.

We have thus proven that the values of the two target cells are the same as those of the two base cells.

We don't know these values, but this is enough to exclude any non base digit from the target cells.

5) Classification of the standard Jk-Exocet

At first sight, formulating the whole set of conditions for the standard Jk-Exocet requires 6 + 3*k CSP-variables / 2D-cells, associated with:

- the 2 Base Cells,

- the 2 Target Cells,

- the 2 Companion Cells,

- the 3k cinj cells, one i for each S column and one k for each base digit

(see details here http://forum.enjoysudoku.com/pattern-based-classification-of-hard-puzzles-t30493-69.html)

However, contrary to the patterns in the main two families of rules mentioned at the start of this post, all these 2D-cells don't play similar roles.

In my view of the Jk-Exocet, the S-columns play the major role and, for each Base Digit, the covering conditions for the S-cells are Fish-like. The complexity of this part of the pattern is analogous to the complexity of k combined Swordfish, i.e. 3k.

Once this part of the pattern is found, the remaining Q, R and B cells are very easy to determine and the relevant conditions are very easy to check.

So, I think the standard Jk-Exocet should be classified at level 3k (taking only its Fish part into account).

6) Classification of the standard Advanced Jk-Exocet

Classifying the Advanced version is only a little more subtle:

At first sight, there are only 2 + 3k CSP-variables / 2D-cells, associated with:

- the 2 Base Cells,

- the 3k cinj cells, one i for each S column and one k for each base digit, as in the standard case.

But listing all the constraints remains complex.

Writing the covering conditions of the S-cells require the choice of two target cells as in the standard case, so it can't be considered as simpler. Globally, it even looks a little more complex; but the difference may be small.

Finally, I think the standard Advanced Jk-Exocet should also be classified at level 3k.