Oriented 3D chains

ORIENTED 3D‑CHAINS: NRC‑, NRCT‑, NRCZ‑ AND NRCZT‑ CHAINS

1) NRC‑LINKS and GENERAL 3D‑CHAINS

The basic notion underlying all the 3D‑chains is that of an nrc‑link.

Definition

Two candidates n₁r₁c₁ and n₂r₂c₂ are nrc‑linked if they are different and

• either n₁ = n₂ and the two rc‑cells (r₁, c₁) and (r₂, c₂) are rc‑linked (i.e. share a unit ‑‑ same row, column or block) in rc‑space,
• or n₁ ≠ n₂ and the rc‑cells (r₁, c₁) and (r₂, c₂) are the same.

Remarks

• Being nrc‑linked is the most general, fully super‑symmetric support for the immediate detection of a contradiction between two nrc‑candidates.
• It is almost “physical” in the sense that it depends only on the grid structure, not on the truth values of the candidates (of course, if one is true the other is false, but that is the use of the link, not its definition).

In nrc‑notation, an nrc‑link is written as “—”.

Definition

A 3D‑chain is a sequence of candidates such that

• the first and last candidates are different (no global loop), and
• any two consecutive candidates are nrc‑linked.

Global loops are excluded by definition, but internal loops are allowed. (They are useless only for some chain types that do not contain the t extension, i.e. nrc and nrcz.)

Definition

A target of a 3D‑chain is a candidate that does **not** belong to the chain and is nrc‑linked to both endpoints of the chain.

Targets are external to the chain; this allows homogeneous patterns and several possible targets for the same chain. Interesting 3D‑chains (nrc‑, nrct‑, nrcz‑ and nrczt‑chains) add extra conditions that group adjacent candidates via stronger links than mere nrc‑links.

2) NRC‑CONJUGACY and NRC‑CHAINS

Definition

Two candidates are nrc‑conjugate if they are nrc‑linked and

• n₁ ≠ n₂, they share the same rc‑cell, and that cell is bivalue (only these two candidates); or
• n₁ = n₂, they are in different rc‑cells, and there is a row, column or block where they form the only two positions for that digit.

Remarks

• “nrc‑conjugate” unifies the bivalue property of cells and the conjugacy property of digits in rc‑space.
• Equivalently: “bivalue in any of the rc‑, rn‑, cn‑ or bn‑ 2‑D spaces”.

Definition

An nrc‑chain is a 3D‑chain of even length 2n where, for every odd k (1 ≤ k ≤ n), the pair (candidate 2k−1, candidate 2k) is nrc‑conjugate.

Two nrc‑conjugate candidates are written {n₁r₁c₁ n₂r₂c₂}. An nrc‑chain of length 6: {1 2} — {3 4} — {5 6}.

In an nrc‑chain, candidates group by twos, so the chain can be viewed as a chain of cells in varying 2‑D spaces.

• Odd‑indexed candidates ⇒ left‑linking
• Even‑indexed candidates ⇒ right‑linking

Theorem (nrc‑chain rule)

Given an nrc‑chain, any target candidate can be eliminated.

3) NRC‑CONJUGACY MODULO A SET OF CANDIDATES and NRCT‑CHAINS

Definition

Given a set S of candidates, two candidates are nrc‑conjugate modulo S if they are not in S, are nrc‑linked, and

• either they share the same cell (bivalue) with at most other values n such that (n,r,c) is nrc‑linked to an element of S; or
• they share a conjugate pair for the same digit along a unit, allowing extra cells (r,c) where that digit is nrc‑linked to an element of S.

Definition

An nrct‑chain is a 3D‑chain of even length 2n such that, for every odd k, the pair (2k−1, 2k) is nrc‑conjugate modulo the set of previous even candidates.

Pattern for length 6: {1 2} — {3 4 (2#2)} — {5 6 (2#2) (4#4)}

Theorem (nrct‑chain rule)

Given an nrct‑chain, any target candidate can be eliminated.

4) NRCZ‑CHAINS

Definition

Given a candidate C, an nrcz‑chain built on C is a 3D‑chain of even length 2n such that

• each conjugate pair is nrc‑conjugate modulo C; and
• C is nrc‑linked to both endpoints (C is the target).

Theorem (nrcz‑chain rule)

Given an nrcz‑chain, its target candidate can be eliminated.

5) NRCZT‑CHAINS

Definition

Given a candidate C, an nrczt‑chain built on C is a 3D‑chain of even length 2n such that

• each conjugate pair is nrc‑conjugate modulo the set {C + previous even candidates}; and
• C is nrc‑linked to both endpoints (target).

Pattern for length 6: {1 2 (*)} — {3 4 (2#2) (*)} — {5 6 (2#2) (4#4)} “(*)” marks an optional candidate conditioned on having an nrc‑link with the target.

Theorem (nrczt‑chain rule)

Given an nrczt‑chain, its target candidate can be eliminated.

6) PROOFS of the NRC‑, NRCT‑, NRCZ‑ and NRCZT‑CHAIN RULES

The proofs adapt those for xy‑, xyt‑, xyz‑ and xyzt‑chains: If the first candidate is false, all even candidates become true (induction on chain length). Thus, a target linked to both ends is contradicted and can be eliminated. For nrcz‑ and nrczt‑chains the target itself enters the proof, but the logic is identical.

7) NRCT‑ and NRCZT‑ LASSOS

With a partial nrct‑ or nrczt‑chain two extra contradiction patterns can arise (irrelevant for nrc‑ or nrcz‑chains due to no‑loop theorems):

• rl‑lasso – a right‑linking candidate equals a previous left‑linking one.
• lr‑lasso – a left‑linking candidate equals a previous right‑linking one.

In both cases the target can be eliminated even without finishing the chain.

8) SUBSUMPTION RELATIONSHIPS

• nrc‑chains subsume xy‑, hxy‑rn‑, hxy‑cn‑chains and basic Nice Loops/AICs (without subsets).
• nrct‑chains subsume nrc‑chains, xyt‑, xyt‑rn‑, cyt‑cn‑chains.
• nrcz‑chains subsume nrc‑chains, xyz‑, xyz‑rn‑, xyz‑cn‑chains.
• nrczt‑chains subsume nrc‑chains, xyzt‑, xyzt‑rn‑, xyzt‑cn‑chains and most fish patterns.
• nrcz‑chains subsume Broken Wings and basic row/column interactions with blocks.

(Original reference: The Hidden Logic of Sudoku. Further web references forthcoming.)