Moderate level examples of nrc(z)(t) chainsElementary and extreme puzzles are interesting but, if you want to get familair with these new chains, it is also worth having intermediate level examples.

Here are three pedagogic examples, in increasing order of complexity, taken from the Ocean collection, a vey nice collection, with all its puzzles starting with one or more swordfish.

http://forum.enjoysudoku.com/viewtopic.php?t=4144Puzzle #1uses only nrc chains (of length no more than 4).

Our first example, which is also #1 in the list uses only nrc-chains.

Remember that nrc-chains are merely another view of "basic" AICs, i.e. AICs with no subsets of any kind (groups, hinges, …).

The chains used in this example should therefore be familiar.

000010002

001000030

040005600

000006700

300000005

008400000

007800040

050000900

200030000

swordfish-in-columns n3{r7 r1 r6}{c2 c6 c7} ==> r7c9 <> 3, n3{r7 r1 r6}{c2 c6 c7} ==> r6c9 <> 3, r1c4 <> 3, ==> r1c3 <> 3

swordfish-in-columns n4{r4 r8 r2}{c1 c5 c9} ==> r8c6 <> 4, r8c3 <> 4, r4c3 <> 4, r2c7 <> 4, ==> r2c6 <> 4

swordfish-in-rows n5{r2 r6 r7}{c7 c1 c5} ==> r9c7 <> 5, r4c5 <> 5, r4c1 <> 5, r1c7 <> 5, r1c1 <> 5

nrc3-chain n5{r9c8 r1c8} - {n5 n8}r2c7 - {n8 n1}r9c7 ==> r9c8 <> 1

nrc3-chain {n1 n8}r9c7 - {n8 n4}r1c7 - n4{r1c6 r9c6} ==> r9c6 <> 1

nrc3-chain n2{r8c8 r7c7} - n3{r7c7 r7c2} - {n3 n6}r8c3 ==> r8c8 <> 6

nrc4-chain n3{r1c2 r1c6} - n4{r1c6 r2c5} - n4{r8c5 r8c1} - n8{r8c1 r9c2} ==> r1c2 <> 8

nrc3-chain n8{r9c2 r2c2} - {n8 n5}r2c7 - n5{r7c7 r9c8} ==> r9c8 <> 8

nrc4-chain n5{r1c3 r2c1} - n5{r2c7 r7c7} - n3{r7c7 r7c2} - {n3 n6}r8c3 ==> r1c3 <> 6

nrc4-chain {n9 n5}r1c3 - n5{r4c3 r4c4} - n3{r4c4 r3c4} - n3{r3c3 r1c2} ==> r1c2 <> 9

nrc4-chain n3{r1c6 r3c4} - n3{r4c4 r4c9} - n4{r4c9 r2c9} - n4{r2c5 r1c6} ==> r1c6 <> 9, r1c6 <> 8, r1c6 <> 7

nrc4-chain {n6 n3}r8c3 - n3{r3c3 r3c4} - {n3 n4}r1c6 - n4{r9c6 r8c5} ==> r8c5 <> 6

nrc3-chain n6{r2c5 r7c5} - n5{r7c5 r7c7} - n5{r2c7 r2c1} ==> r2c1 <> 6

nrc3-chain n6{r2c5 r7c5} - n5{r7c5 r7c7} - {n5 n8}r2c7 ==> r2c5 <> 8

nrc4-chain n2{r8c8 r7c7} - n5{r7c7 r7c5} - n6{r7c5 r2c5} - n4{r2c5 r8c5} ==> r8c5 <> 2

nrc4-chain {n8 n5}r2c7 - n5{r7c7 r7c5} - n6{r7c5 r2c5} - n4{r2c5 r2c9} ==> r2c9 <> 8

nrc4-chain n6{r2c5 r7c5} - {n6 n1}r7c9 - {n1 n8}r9c7 - n8{r9c2 r2c2} ==> r2c2 <> 6

row r2 interaction-with-block b2 ==> r1c4 <> 6

nrc3-chain n6{r1c1 r1c2} - n3{r1c2 r7c2} - {n3 n6}r8c3 ==> r8c1 <> 6, r7c1 <> 6

nrc3-chain n6{r1c1 r6c1} - n5{r6c1 r2c1} - {n5 n9}r1c3 ==> r1c1 <> 9

nrc4-chain n6{r1c1 r6c1} - n5{r6c1 r4c3} - {n5 n9}r1c3 - {n9 n7}r1c4 ==> r1c1 <> 7

nrc4-chain {n7 n9}r1c4 - {n9 n5}r1c3 - n5{r4c3 r4c4} - n3{r4c4 r3c4} ==> r3c4 <> 7

nrc4-chain {n6 n3}r8c3 - n3{r3c3 r3c4} - {n3 n4}r1c6 - n4{r9c6 r9c3} ==> r9c3 <> 6

nrc3-chain n4{r4c1 r5c3} - {n4 n9}r9c3 - {n9 n1}r7c1 ==> r4c1 <> 1

nrc4-chain {n1 n8}r9c7 - {n8 n4}r1c7 - {n4 n3}r1c6 - n3{r6c6 r6c7} ==> r6c7 <> 1

nrc3-chain {n2 n3}r6c7 - n3{r4c9 r4c4} - n5{r4c4 r6c5} ==> r6c5 <> 2

nrc4-chain {n2 n3}r6c7 - n3{r7c7 r8c9} - n3{r8c3 r3c3} - n2{r3c3 r2c2} ==> r6c2 <> 2

nrc4-chain {n2 n3}r6c7 - n3{r6c6 r4c4} - n5{r4c4 r9c4} - n5{r9c8 r7c7} ==> r7c7 <> 2

…(NS+HS)…

635914872

981267534

742385619

419526783

326178495

578493261

197852346

853641927

264739158

Puzzle #3 uses nrct chains (still of length no more than 4)

The second example, #3 in the list, requires various kinds of chains, including nrct-chains of length upto 4.

Interestingly, it also uses 2D chains, i.e. chains in the rc-, rn- and cn- spaces. Remember that these chains are particular case of nrc(z)(t) chains.

I keep have in SudoRules as independent chains because they are easier to spot (in my view of things, more general does not mean better).

Remember that 2D chains are enough to solve 97% of the randomly generated puzzles - although nrc(z)(t) chains sometimes allow using shorter chains (but 3D instead of 2D).

This is a very good example for understanding what is allowed by the t extension.

000010002

001000030

040005600

000007400

300000006

008900000

006300080

050000100

200040000

swordfish-in-columns n3{r9 r1 r6}{c2 c6 c7} ==> r9c9 <> 3, r9c3 <> 3, r6c9 <> 3, r6c5 <> 3, r1c3 <> 3

nrc3-chain n4{r6c6 r6c1} - n4{r5c3 r8c3} - n4{r8c8 r1c8} ==> r1c6 <> 4

nrc4-chain n3{r1c6 r3c5} - n3{r3c3 r8c3} - n4{r8c3 r5c3} - n4{r6c1 r6c6} ==> r6c6 <> 3

hidden singles ==> r4c5 = 3, r1c6 = 3, r3c3 = 3, r9c2 = 3, r8c9 = 3, r6c7 = 3, r8c1 = 8, r2c2 = 2, r1c2 = 8

column c2 interaction-with-block b4 ==> r6c1 <> 6, r4c1 <> 6

row r9 interaction-with-block b8 ==> r7c6 <> 1

hidden-pairs-in-a-block {n1 n8}{r9c4 r9c6} ==> r9c6 <> 9, r9c6 <> 6, r9c4 <> 7, r9c4 <> 6

hidden-single-in-a-row ==> r9c8 = 6

hidden-pairs-in-a-block {n1 n8}{r9c4 r9c6} ==> r9c4 <> 5

hidden-single-in-a-block ==> r7c5 = 5

block b8 interaction-with-row r8 ==> r8c8 <> 7, r8c3 <> 7

nrc2-chain n8{r2c7 r5c7} - n8{r4c9 r4c4} ==> r2c4 <> 8

nrczt2-chain n8{r2c7 r5c7} - n8{r5c5 r3c5} ==> r2c6 <> 8

nrc3-chain {n9 n2}r7c6 - n2{r7c7 r8c8} - n4{r8c8 r7c9} ==> r7c9 <> 9

nrct3-chain n2{r3c4 r3c5} - {n2 n8}r5c5 - n8{r2c5 r3c4} ==> r3c4 <> 7

nrczt3-chain n4{r6c1 r7c1} - {n4 n7}r7c9 - n7{r7c2 r5c2} ==> r6c1 <> 7

nrczt3-chain n5{r4c4 r5c4} - n5{r5c3 r1c3} - n5{r1c8 r6c8} ==> r4c9 <> 5

nrczt3-chain n5{r5c4 r4c4} - n5{r4c3 r1c3} - n5{r1c8 r6c8} ==> r5c7 <> 5

hxyt-cn4-chain {r1 r8}c8n4 - {r8 r5}c3n4 - {r5 r4}c3n2 - {r4 r1}c3n5 ==> r1c8 <> 5

column c8 interaction-with-block b6 ==> r6c9 <> 5

nrc3-chain {n1 n7}r6c9 - {n7 n4}r7c9 - n4{r7c1 r6c1} ==> r6c1 <> 1

nrc4-chain n5{r6c8 r6c1} - n4{r6c1 r7c1} - {n4 n7}r7c9 - {n7 n1}r6c9 ==> r6c8 <> 1

nrct4-chain {n2 n8}r5c5 - n8{r5c7 r4c9} - n8{r3c9 r3c4} - n2{r3c4 r3c5} ==> r8c5 <> 2, r6c5 <> 2

naked-single ==> r6c5 = 6

hidden-single-in-a-row ==> r4c2 = 6

naked-pairs-in-a-row {n1 n7}r6{c2 c9} ==> r6c8 <> 7, r6c6 <> 1

hidden-pairs-in-a-column {n1 n8}{r5 r9}c6 ==> r5c6 <> 4, r5c6 <> 2

swordfish-in-columns n4{r7 r6 r2}{c1 c6 c9} ==> r2c4 <> 4

nrc2-chain n2{r6c8 r6c6} - n2{r7c6 r7c7} ==> r8c8 <> 2

naked and hidden singles ==> r7c7 = 2, r7c6 = 9, r8c5 = 7, r5c2 = 9

column c1 interaction-with-block b1 ==> r1c3 <> 9

xy3-chain {n2 n4}r6c6 - {n4 n5}r6c1 - {n5 n2}r4c3 ==> r4c4 <> 2

xy3-chain {n1 n8}r5c6 - {n8 n7}r5c7 - {n7 n1}r6c9 ==> r5c8 <> 1

row r5 interaction-with-block b5 ==> r4c4 <> 1

xy3-chain {n2 n5}r4c3 - {n5 n8}r4c4 - {n8 n2}r5c5 ==> r5c3 <> 2

hidden-single-in-a-block ==> r4c3 = 2

nrc3-chain {n5 n4}r6c1 - n4{r5c3 r5c4} - n5{r5c4 r4c4} ==> r4c1 <> 5

naked and hidden singles ==> r4c1 = 1, r6c2 = 7, r7c2 = 1, r6c9 = 1, r3c8 = 1

x-wing-in-rows n7{r3 r7}{c1 c9} ==> r9c9 <> 7, r2c9 <> 7, r2c1 <> 7 r1c1 <> 7

nrc3-chain {n5 n9}r9c9 - {n9 n4}r8c8 - n4{r7c9 r2c9} ==> r2c9 <> 5

hidden-single-in-a-column ==> r9c9 = 5

nrc3-chain {n7 n5}r1c3 - n5{r1c7 r2c7} - n7{r2c7 r2c4} ==> r1c4 <> 7

hidden-single-in-a-block ==> r2c4 = 7

nrc3-chain n6{r2c1 r2c6} - n4{r2c6 r6c6} - {n4 n5}r6c1 ==> r2c1 <> 5

…(NS+HS)…

685413792

921786534

743295618

162837459

394521876

578964321

416359287

859672143

237148965

Puzzle #6 uses nrczt chains (still of length no more than 4)

Our third example, #6 in the list, uses nrczt4 chains. After you have understood with the previous example what is allowed by the t extension, it will allow you to understand what is allowed by the z extension.

000010002001000030040005600000007800100000005003400000006300090050000400200080000

swordfish-in-columns n3{r9 r1 r5}{c2 c6 c7} ==> r9c9 <> 3, r5c5 <> 3, r1c1 <> 3

swordfish-in-columns n4{r4 r7 r2}{c1 c5 c9} ==> r7c6 <> 4, r4c8 <> 4, r4c3 <> 4, r2c6 <> 4

swordfish-in-rows n5{r2 r6 r7}{c7 c1 c5} ==> r9c7 <> 5, r4c5 <> 5, r4c1 <> 5, r1c7 <> 5, r1c1 <> 5

nrczt3-chain n1{r4c4 r6c6} - n1{r6c7 r7c7} - n1{r7c2 r9c2} ==> r9c4 <> 1

nrc4-chain n5{r7c7 r7c5} - n5{r6c5 r4c4} - n1{r4c4 r8c4} - {n1 n2}r7c6 ==> r7c7 <> 2

hidden-single-in-a-block ==> r8c8 = 2

column c8 interaction-with-block b3 ==> r3c9 <> 8, r2c9 <> 8

nrc4-chain n3{r1c6 r3c5} - n3{r4c5 r4c9} - n4{r4c9 r2c9} - n4{r2c5 r1c6} ==> r1c6 <> 9, r1c6 <> 8, r1c6 <> 6

nrc4-chain n4{r1c8 r2c9} - n4{r2c5 r7c5} - n5{r7c5 r7c7} - n5{r9c8 r1c8} ==> r1c8 <> 8

hidden singles ==> r3c8 = 8, r3c9 = 1

row r8 interaction-with-block b8 ==> r9c6 <> 1, r7c6 <> 1

naked-single ==> r7c6 = 2

nrct3-chain n3{r5c7 r4c9} - n4{r4c9 r2c9} - n9{r2c9 r6c9} ==> r5c7 <> 9

nrc4-chain n4{r1c8 r2c9} - n4{r2c5 r7c5} - n5{r7c5 r7c7} - n5{r9c8 r1c8} ==> r1c8 <> 7

nrc4-chain n3{r4c9 r5c7} - n3{r5c6 r1c6} - n4{r1c6 r1c8} - n4{r5c8 r4c9} ==> r4c9 <> 9

block b6 interaction-with-row r6 ==> r6c6 <> 9, r6c5 <> 9, r6c2 <> 9, r6c1 <> 9

nrc4-chain n9{r6c9 r2c9} - n4{r2c9 r4c9} - n3{r4c9 r5c7} - n2{r5c7 r6c7} ==> r6c7 <> 9

hidden-single-in-a-block ==> r6c9 = 9

nrc3-chain {n7 n4}r2c9 - {n4 n5}r1c8 - n5{r2c7 r2c1} ==> r2c1 <> 7

nrc4-chain n5{r9c4 r9c8} - {n5 n4}r1c8 - {n4 n7}r2c9 - {n7 n6}r9c9 ==> r9c4 <> 6

nrc4-chain n3{r1c2 r1c6} - n4{r1c6 r1c8} - {n4 n7}r2c9 - {n7 n9}r1c7 ==> r1c2 <> 9

nrc4-chain n3{r4c9 r5c7} - n3{r5c6 r1c6} - n4{r1c6 r1c8} - n4{r5c8 r4c9} ==> r4c9 <> 6

column c9 interaction-with-block b9 ==> r9c8 <> 6

nrc4-chain n4{r7c5 r9c6} - n4{r1c6 r1c8} - n5{r1c8 r9c8} - n5{r9c4 r7c5} ==> r7c5 <> 7

nrct4-chain {n8 n7}r7c9 - {n7 n4}r2c9 - n4{r4c9 r4c1} - {n4 n8}r7c1 ==> r7c2 <> 8

nrczt4-chain {n7 n9}r1c7 - {n9 n5}r2c7 - n5{r1c8 r9c8} - n7{r9c8 r5c8} ==> r6c7 <> 7

nrc3-chain {n2 n1}r6c7 - n1{r4c8 r4c4} - n5{r4c4 r6c5} ==> r6c5 <> 2

nrc4-chain {n6 n5}r6c5 - {n5 n4}r7c5 - n4{r2c5 r1c6} - n3{r1c6 r5c6} ==> r5c6 <> 6

nrczt4-chain {n7 n9}r1c7 - {n9 n5}r2c7 - n5{r1c8 r9c8} - n7{r9c8 r6c8} ==> r5c7 <> 7

block b6 interaction-with-column c8 ==> r9c8 <> 7

nrc3-chain n1{r4c4 r4c8} - {n1 n5}r9c8 - n5{r9c4 r4c4} ==> r4c4 <> 9, r4c4 <> 6

nrc3-chain n1{r4c4 r4c8} - {n1 n5}r9c8 - n5{r9c4 r4c4} ==> r4c4 <> 2

xy3-chain {n6 n5}r6c5 - {n5 n1}r4c4 - {n1 n6}r4c8 ==> r6c8 <> 6, r4c5 <> 6

nrc3-chain {n1 n5}r9c8 - n5{r9c4 r4c4} - n1{r4c4 r4c8} ==> r6c8 <> 1

naked-single ==> r6c8 = 7

nrc3-chain {n2 n3}r5c7 - {n3 n4}r4c9 - n4{r5c8 r5c3} ==> r5c3 <> 2

nrczt3-chain n2{r6c2 r6c7} - n1{r6c7 r4c8} - n6{r4c8 r4c1} ==> r6c2 <> 6

nrczt3-chain {n2 n3}r5c7 - n3{r5c6 r4c5} - n2{r4c5 r4c3} ==> r5c2 <> 2

hxy-rn4-chain {c2 c6}r1n3 - {c6 c8}r1n4 - {c8 c3}r5n4 - {c3 c2}r5n7 ==> r1c2 <> 7

nrc4-chain {n8 n2}r6c2 - n2{r6c7 r5c7} - n3{r5c7 r5c6} - n3{r1c6 r1c2} ==> r1c2 <> 8

nrc4-chain {n6 n3}r1c2 - {n3 n4}r1c6 - n4{r1c8 r5c8} - n6{r5c8 r4c8} ==> r4c2 <> 6

nrc3-chain n6{r4c1 r4c8} - {n6 n4}r5c8 - n4{r5c3 r4c1} ==> r4c1 <> 9

nrc4-chain n6{r4c1 r4c8} - {n6 n4}r5c8 - {n4 n5}r1c8 - n5{r2c7 r2c1} ==> r2c1 <> 6

nrc4-chain {n9 n2}r4c2 - n2{r6c2 r6c7} - {n2 n3}r5c7 - n3{r9c7 r9c2} ==> r9c2 <> 9

nrc4-chain {n9 n2}r4c2 - n2{r6c2 r6c7} - {n2 n3}r5c7 - n3{r5c6 r4c5} ==> r4c5 <> 9

row r4 interaction-with-block b4 ==> r5c3 <> 9, r5c2 <> 9

nrc3-chain {n2 n3}r4c5 - n3{r3c5 r1c6} - n4{r1c6 r2c5} ==> r2c5 <> 2, r2c5 <> 2

nrc4-chain n1{r7c2 r7c7} - {n1 n2}r6c7 - {n2 n3}r5c7 - n3{r9c7 r9c2} ==> r9c2 <> 1

hidden-single-in-a-block ==> r7c2 = 1

naked-triplets-in-a-column {n7 n9 n5}{r1 r2 r7}c7 ==> r9c7 <> 7

nrc3-chain n3{r8c9 r9c7} - {n3 n7}r9c2 - {n7 n6}r9c9 ==> r8c9 <> 6

hidden-single-in-a-block ==> r9c9 = 6

nrczt3-chain n3{r8c9 r9c7} - {n3 n7}r9c2 - n7{r7c1 r7c7} ==> r8c9 <> 7

block b9 interaction-with-row r7 ==> r7c1 <> 7

nrc3-chain {n8 n4}r7c1 - {n4 n5}r7c5 - n5{r6c5 r6c1} ==> r6c1 <> 8

naked-pairs-in-a-row {n5 n6}r6{c1 c5} ==> r6c6 <> 6

hxyzt4-rn-chain-type-1 {c3 c2}r5n7 - {c2 c4}r9n7 - {c4 c8}r9n5 - {c8 c3}r1n5 ==> r1c3 <> 7

nrc4-chain n7{r2c9 r7c9} - n8{r7c9 r8c9} - n3{r8c9 r8c1} - {n3 n7}r9c2 ==> r2c2 <> 7

nrc4-chain {n6 n5}r6c1 - n5{r6c5 r4c4} - n1{r4c4 r4c8} - n6{r4c8 r4c1} ==> r5c2 <> 6

column c2 interaction-with-block b1 ==> r1c1 <> 6

nrc3-chain n6{r1c4 r1c2} - n3{r1c2 r1c6} - n4{r1c6 r2c5} ==> r2c5 <> 6

xyzt4-chain {n7 n4}r2c9 - {n4 n5}r1c8 - {n5 n9}r2c7 - {n9 n7}r2c5 ==> r2c4 <> 7

nrc4-chain n6{r1c4 r1c2} - n3{r1c2 r1c6} - n4{r1c6 r1c8} - {n4 n6}r5c8 ==> r5c4 <> 6

block b5 interaction-with-column c5 ==> r8c5 <> 6

hidden-pairs-in-a-block {n1 n6}{r8c4 r8c6} ==> r8c6 <> 9, r8c4 <> 9, r8c4 <> 7

nrc3-chain n7{r8c5 r9c4} - n5{r9c4 r7c5} - n4{r7c5 r2c5} ==> r2c5 <> 7

row r2 interaction-with-block b3 ==> r1c7 <> 7

naked-single ==> r1c7 = 9

nrc2-chain n7{r8c5 r3c5} - n7{r1c4 r1c1} ==> r8c1 <> 7

column c1 interaction-with-block b1 ==> r3c3 <> 7

nrc3-chain {n8 n5}r1c3 - {n5 n4}r1c8 - n4{r5c8 r5c3} ==> r5c3 <> 8

block b4 interaction-with-column c2 ==> r2c2 <> 8

nrc3-chain n7{r1c1 r1c4} - n6{r1c4 r1c2} - n3{r1c2 r3c1} ==> r3c1 <> 7

hidden-single-in-a-block ==> r1c1 = 7

hidden-pairs-in-a-block {n5 n8}{r1c3 r2c1} ==> r2c1 <> 9

hidden-pairs-in-a-column {n3 n9}{r3 r8}c1 ==> r8c1 <> 8

xy3-chain {n7 n9}r8c5 - {n9 n3}r8c1 - {n3 n7}r9c2 ==> r9c4 <> 7

hidden singles ==> r8c5 = 7, r3c4 = 7

row r8 interaction-with-block b7 ==> r9c3 <> 9

xy3-chain {n9 n4}r2c5 - {n4 n5}r7c5 - {n5 n9}r9c4 ==> r2c4 <> 9

nrc3-chain {n8 n6}r1c4 - n6{r1c2 r2c2} - n2{r2c2 r2c4} ==> r2c4 <> 8

xy4-chain {n8 n7}r5c2 - {n7 n3}r9c2 - {n3 n6}r1c2 - {n6 n8}r1c4 ==> r5c4 <> 8

…(NS+HS)…

765813942

821649537

349725681

692537814

174298365

583461279

416352798

958176423

237984156