A General Logic for Sudoku A basic guide to applying general logic |
Points.
The following summarizes everything about general logic from simple to complex.
Quick Description of general logic, how it applies. 1. Every elimination is based on
two groups of sets. A group of truths that contain all the candidates
in the logic and a group of links that contains the same candidates and
possibly others that are potential eliminations. 2. The number of links minus the
number of truths is called rank, which relates to the number of missing
constraints. Rank is a distributed property that applies everywhere within the
logic except as explained below. 3. Rank 0 eliminates any additional
candidates inside of links. Common examples of rank 0 logic include singles,
locked candidates, ALCs, X-wing, swordfish, etc. 4. Rank 1 eliminates any
additional candidates where two links overlap. Many Sudoku methods fall into
this category such as finned fish, chains, discontinuous nice loops, etc. 5. Ranks 2 (or 3) logic requires 3
(or 4) simultaneously overlapping links to cause eliminations. 6. Ranks higher than 3 can only cause
eliminations when combined with triplets, described next. 7. A triplet is a single candidate
that connects three sets. A strong-triplet connects two truths and one link, weak-triplet
connects two links and one truth. Triplets "point" in the direction
of the minority set, i.e., the link direction of a two truth triplet. The two
types of triplets are similar but have different properties. 8. Triplets can divide logic into
high rank and low rank regions and thus change the number of overlap links required
to eliminate a candidate, but this follows specific rules. When weak triplets
point in the direction of a candidate, it may reduce rank and the number of overlap
links required to eliminate a candidate.. 9. Strong triplets can increase
rank and the number of overlap links required to eliminate a candidate because
they can reduce the number of truths guaranteed to be in the logic. Weak triplets
cannot. 10. When a weak triplet can lower the rank of
an entire area, which usually extends from the triplet's truth to the first
bifurcation that has independent paths back to the triplet's two links. To understand
strong triplets, swap the terms truth and link in the preceding sentence. 11. The candidate in a strong triplet with an
unconnected link can be assigned in rank 0 logic. 12. When triplets are aligned correctly, their
effects can be additive. However, complex arrangements often require further
logical consideration. 13. When two blocks of logic are
(hypothetically) combined, each retains its original rank internally and the
rank of the combined logic is the sum of the individual ranks.
Eliminations can occur when links of different blocks overlap based on the
overall rank and triplets used to link blocks. 14. Logic containing odd length loops of truths can
contain dark logic, which works similar to broken wings. The dark logic part
contains only the candidates from the truths that logic illegal. (Illegal
logic means there is no valid arrangement of candidates that satisfies the
rules of Sudoku) 15. A dark loop has a rank of minus 1. For solving, a
dark loop can be ignored and the remaining candidates in truths can be used
as a single virtual truth. The virtual truth can then be used like a normal truth
and be combined with other logic. A virtual truth can have many candidates and
span rows, columns, and boxes. Referring to truths and links The
upper case letters RCNB are used to refer to truths rows, columns, cells, and boxes respectivly. Lower case letters rcnb usually refer to links or cover sets.Lower case
letters refer to links R(row, digit),
C(column, digit), N(row,
column), B(box, digit) truths r(row, digit),
c(column, digit), n(row,
column), b(box, digit) links These
are sometimes written with swapped digits. (digit)R(row), (digit)C(column, digit), (row)N(column),
(digit)B(box) truths (digit)r(row), (digit)c(column, digit), (row)n(column),
(digit)b(box) links Groups
like c51, c53, and c59 are often written as c5(139). Notation for eliminations The
base and cover sets for logic are sometimes written as: {R46 C5(139) C76 N23 N3(12)} r26*c56 => r2c5 <> 6. which
is of the form: {list of covering links}<overlap links> =>
<assignments> or <eliminations> In
some cases <overlap links> are replaced with a list of covering links: Rank1: {R46 C5(139) C76 N23 N3(12)}, {r26 r3(16) c66 n(146)5 b1(45)} => r2c5 <> 6. |