3) Rating resolution rules and puzzles; classification results
3.1) Rating resolution rules and puzzles; ordering the rules
introduces several ratings of puzzles, studies their
correlation and justifies the choice of length as a measure
of complexity of a chain rule.
This page is not yet revised and it keeps a strong flavour
of its historical development on the Sudoku Player's Forum.
3.2) Classification results
gives various classification results, shows that different
kinds of generators of minimal puzzles lead to different
statitics (and thus proves that there is some bias),
introduces the notion of a controlled-bias generator, uses
it to produce unbiased statistics. As a collateral result,
it gives a close estimate of the real distribution of minimal
3.3) The nrczt-rating of nrczt-whips as a guide for defining the rating of any
chain or any pattern
How can one rate chains based on subsets (e.g. ALS-chains)
with different lengths and different maximal sizes for the
subsets they use, in a way consistent with the rating of
nrczt chains or whips?
Remember first that the complexity of a resolution rule is
in its condition pattern, not in its conclusion.
Remember also that the kind of complexity I am considering
is mainly complexity of resolution rules and that it is
extended to complexity of puzzles in the SER style (=
complexity of the most complex rule necessary to solve it).
The question then is mainly: how can the presence of Subsets
in a chain be taken into account in the definition of chain
length? For ALS-chains, e.g., there are two kinds of
parameters: chain length and size of each of the ALSs; how
should they be combined?
After all the time and work on the subject of ratings, the
answer now seems very clear to me: as the nrczt-whips can
solve almost all the puzzles (in more than 10 million
minimal puzzles randomy generated by various kinds of
generators - bottom-up, top-down, conrollled-bias - no
puzzle was found that couldn't be solved by nrczt-whips),
the NRCZT-rating constitutes an almost universal rating, to
which all the other ratings can be compared statistically.
It has very good properties, the first three of which are
not shared with the widely used SER:
- pure logic definition,
- full supersymmetry,
- very strong correlation (0.95) with log(number of chains),
which shows that it is statistically a logarithmic measure
- strong correlation (0.895) with the well established SER
for the human solvable puzzles, in spite of their a priori
very different definitions,
- almost confluence of the underlying resolution
theories (i.e. almost independence of the implementation);
if one wants strict confluence and indepence, just replace
erywwhere in this section whips by braids and NRCZT by
It is therefore justified to take it as a reference for
defining ratings based on different kinds of chains or
3.3.1) Consider first chains of ALS, AHS and A-Fish.
- define the length of such a chain to be the sum of the
sizes of all its defining Subsets, a Single being considered
as a subset of length 1 (see remarks below for precisions);
- define the LS-rating of a puzzle as the length of the
longest such chain necessary to solve it.
- the "restricted commons" don't count in the subset sizes,
they more or less play the roles of left-linking candidates,
- for the most classical complementarity reasons: NS(5) =
HS(4), HS(5) = NS(4), NS(6) = HS(3) ... (here, NS= Naked
Subset, HS=Hidden Subset)
- for supersymmetry reasons: for n = 1, 2, 3, 4, NS(n),
HS(n) and SHS(n) (i.e. Fish(n)) are all counted as a NS(n).
This rating is supersymmetric and consistent with the
NRCZT-rating: whenever a chain can be viewed according to
the two points of view (which, according to the general
subsumption results, is the case for almost all the chains
of ALS, AHS and A-Fish), its length will be the same
according to the two points of view.
Although there is currently no program computing ratings of
ALS-chains in a way consistent with this definiton, it is
almost certain (due to the subsumption theorems) that this
LS-rating is very strongly correlated with the NRCZT-rating.
3.3.2) Now, consider the more general case of
nrczt-whips(Subsets), as defined in section 2 of the zt-ing page. In such
chains, a naked, hidden and super-hidden (fish) Subset
(modulo the target and all the previous right-linking
candidates) can be taken as a right-linking object, in lieu
of a mere candidate.
Define the length of such a whip(Subsets) as the sum of all
the sizes of the right-linking objects (Subsets) it contains
(a Single candidate being still considered as a subset of
Define the NRCZT(Subsets)-rating as the length of the
maximal whip(Subsets) necessary to solve it.
Again, this rating is supersymmetric and consistent: any
chain that can be viewed according to several points of view
(whether some parts of it are considered as subsets modulo
the target and the previous rlc's or as mere nrczt-chains)
will have the same ratings for all these points of view.
We know that, most of the extremely rare puzzles that cannot
be solved with simple whips (fewer than one in 10 millions)
can be solved with zt-whips(Subsets).
It is thus very likely that most of the puzzles that can be
solved by mere nrczt-whips will have the same ratings when
they can be considered according to the two points of view.
The nrczt(Subsets)-rating is thus almost an extension of the
3.3.3) This definition obviously applies to Paul Isaacson's
whips with ALS inserts, which are a secial case of
3.3.4) Finally, this definition should apply to Allan
Barker's set cover patterns (networks of 2D-cells, i.e. rc-,
rn-, cn- and bn- cells - "base sets" in set cover
terminology, renamed "truths" in his vocabulary), for which
one could take the number of 2D-cells as the "length" of the
pattern, disregarding the number of "cover sets" (in set
cover terminology - renamed "linksets" in Allan's
This definition of the NRCZT-rating, thus extended to other
patterns in a consistent way, allows comparisons of the
complexities of the solutions obtained with the