This thread describes the process that makes use of the bilocation/bivalue plot described in this thread to identify triple chains, short for triple implication chains.

A triple implication chain is a chain with 3 implication streams and each of these stream results in the same exclusion or inclusion outcome(s).

Each triple implication chain involves exactly one trivalue cell or three trilocation cells.

A Trivalue cell is a cell containing only three candidates. For each trivalue cell, a trivalue tripod is drawn on the cell using 3 links of weak inference (broken lines) as follows:

Trilocation cells are cells with the same candidate appear exactly 3 times in a unit. For each set of trilocation cells, a trilocation triangle is drawn amongst the 3 cells using 3 links of stong inference (solid lines) as follows:

With the bilocation/bivalue plot, 2 of the 3 links in a trilocation triangle or trivalue tripod are used at a time to make implications following the same set of nice loop propagation rules and theorems for candidate exclusion and inclusion.

Triple implication chains can be continuous or discontinuous. Since a continuous chain is extremely rare, we will concentrate on discontinuous triple chains. As a matter of interest, one of the 322 swordfish patterns is a good example of a continuous triple implication chain with the trilocation triangle shown in red.

The diagram below shows a triple chain with a trivalue cell of discontinuous type. Since the chain is based upon a trivalue cell, the links in the trivalue node r6c2 form of a trivalue tripod (ie. 3 broken links shown in red diverge from node r6c2).

Nice loop notation: This chain has 3 nice loops with a trivalue tripod at r6c2.

[r3c3]-2-[r1c2]-1-[r6c2]-9-[r3c2]=9=[r3c3]

[r3c3]=9=[r3c2]-9-[r6c2]-2-[r7c2]=2=[r7c3]-2-[r3c3]

[r3c3]-2-[r1c2]-1-[r6c2]-2-[r7c2]=2=[r7c3]-2-[r3c3]

All imply r3c3<>2

Note: There are 3 implication streams in a triple chain. Since each line of nice loop notation contains 2 implication streams, any of the 2 lines above would be sufficient to express the triple chain.

This triple chain can expressed as a multiple nice loop with a multiple inference as described here.

[r3c3]-2-[r1c2]-1-[r6c2](-9-[r3c2]=9=[r3c3])-2-[r7c2]=2=[r7c3]-2-[(r3c3)] => r3c3<>2

Proof:

r6c2=1 => r1c2<>1 => r1c2=2 => r3c3<>2

r6c2=2 => r6c2=2 => r7c2<>2 => r7c3=2 => r3c3<>2

r6c2=9 => r3c2<>9 => r3c3=9 => r3c3<>2

The diagram below shows a triple chain with 3 trilocation cells of discontinuous type. Since the chain is based upon 3 trilocation cells, the links in the trilocation cells r4c5, r5c6 and r6c5 form of a trilocation triangle (ie. 3 solid links shown in red joining nodes r4c5, r5c6 and r6c5 together).

Nice loop notation: This chain has 3 nice loops with a trilocation triangle at r4c5, r5c6 and r6c5.

[r5c2]=9=[r5c6]-9-[r4c5]=9=[r8c5]-9-[r8c2]=9=[r5c2]

[r5c2]=9=[r5c6]-9-[r6c5]=9=[r8c5]-9-[r8c2]=9=[r5c2]

[r5c2]=9=[r5c6]-9-[r4c5]=9=[r6c5]-9-[r5c6]=9=[r5c2]

All imply r5c2=9

Note: There are 3 implication streams in a triple chain. Since each line of nice loop notation contains 2 implication streams, any of the 2 lines above would be sufficient to express the triple chain.

Multiple nice loop notation:

[r5c2]=9=[r5c6]-9-[r6c5](=9=[r4c5]-9-[r5c6]=9=[r5c2])=9=[r8c5]-9-[r8c2]=9=[(r5c2)] => r5c2=9

Proof:

r4c5=9 => r5c6<>9 => r5c2=9

r6c5=9 => r5c6<>9 => r5c2=9

r8c5=9 => r8c2<>9 => r5c2=9