Examples of braidsWARNING:

As almost any puzzle with SER ≤ 9.3 can be solved with nrczt-whips (i.e. chains or lassos) and as I'm very reluctant to rely on nets (even braids, which are a very special form of nets) when a chain is available, any interesting example of a braid will be obtained for a puzzle with SER > 9.3.

As a result, the solution to such a puzzle will require long chains and braids and it will seem very complex.

In the following, one should therefore not forget that any such puzzle is likely to be beyond normal human solving. I'm just giving an example of braids, I'm not stating that such complex puzzles should be proposed for human solving.

Contrary to the solutions I usually give with chains, I can't guarantee here that the braids used are the shortest ones available. (The contrary is very likely, didn't even try to optimise their length.) The only purpose is to show a few cases of how a braid can look.

The puzzle I'll use is #3263 in gsf's list of 8152 hardest:

20627, 094, 0640, 100800002003400050060005700000090040000006000009040000020000100700000006005080030, gsf-2007-05-24-0753, 0, 49.00s, C21.m/F10135.16143/N12760.27297/P3.33.6422.13.22.20618.18.4.2.230/M2.69.190/V2, C21.m/F15.57/N10.22/B8.18.18/H2.4.2/X2.3/Y1.30/K1.1.8.0.0.1/O1.1/G11.0.1/M1.27.1

- Code: Select all
`+-------+-------+-------+ `

| 1 . . | 8 . . | . . 2 |

| . . 3 | 4 . . | . 5 . |

| . 6 . | . . 5 | 7 . . |

+-------+-------+-------+

| . . . | . 9 . | . 4 . |

| . . . | . . 6 | . . . |

| . . 9 | . 4 . | . . . |

+-------+-------+-------+

| . 2 . | . . . | 1 . . |

| 7 . . | . . . | . . 6 |

| . . 5 | . 8 . | . 3 . |

+-------+-------+-------+

How did I proceed? As rules for braids are not (yet) defined in SudoRules, I used the following procedure (half manual):

- Code: Select all
`Input puzzle`

Loop until solution found:

Run SudoRules from the current state with the usual rules for chains.

When no rule is applicable, make ONE elimination with a braid.

End loop

***** SudoRules version 13.7wB *****

100800002003400050060005700000090040000006000009040000020000100700000006005080030

hidden-single-in-a-row ==> r1c2 = 5

interaction column c4 with block b8 for number 6 ==> r7c5 <> 6

hidden-pairs-in-a-block {n3 n4}{r1c7 r3c9} ==> r3c9 <> 9

hidden-pairs-in-a-block {n3 n4}{r1c7 r3c9} ==> r3c9 <> 8

hidden-pairs-in-a-block {n3 n4}{r1c7 r3c9} ==> r3c9 <> 1

hidden-pairs-in-a-block {n3 n4}{r1c7 r3c9} ==> r1c7 <> 9

hidden-pairs-in-a-block {n3 n4}{r1c7 r3c9} ==> r1c7 <> 6

At this point, the PM is:

- Code: Select all
`+-------------------------+-------------------------+-------------------------+ `

|1 5 47 |8 367 379 |34 69 2 |

|289 789 3 |4 1267 1279 |689 5 189 |

|2489 6 248 |1239 123 5 |7 189 34 |

+-------------------------+-------------------------+-------------------------+

|23568 1378 12678 |12357 9 12378 |23568 4 13578 |

|23458 13478 12478 |12357 12357 6 |23589 12789 135789 |

|23568 1378 9 |12357 4 12378 |23568 12678 13578 |

+-------------------------+-------------------------+-------------------------+

|34689 2 468 |35679 357 3479 |1 789 45789 |

|7 13489 148 |12359 1235 12349 |24589 289 6 |

|469 149 5 |12679 8 12479 |249 3 479 |

+-------------------------+-------------------------+-------------------------+

Now comes a special case of an nrczt-braid, in rc-space: an

yxzt-braid.

For easier reading, all the cells are numbered: C1 to C18. Each branch is written in a different line. The cell of the branching points (always an rlc or *) are recalled before the links (as C5 in "C5 ------- C8:{n3 n7 n6#1}r1c5"); "*" is the target.

As usual, #k after a candidate means it is justified by the rlc of cell Ck; * means it is justified by the target.

Remember that the ordering of the candidates is essential and that, in a braid, any t-candidate is still justified by the target or a PREVIOUS right-linking candidate (rlc) wrt to this ordering.

xyzt-braid[18]

* ------- C1:{n9 n6}r1c8 - C2:{n6 n8 n9*}r2c7 - C3:{n8 n7 n9*}r2c2 - C4:{n7 n4}r1c3 - C5:{n4 n3}r1c7 - C6:{n3 n4}r3c9 - C7:{n4 n7 n9*}r9c9 -

C5 ------- C8:{n3 n7 n6#1}r1c5 -

* ------- C9:{n9 n2 n8#2}r2c1 - C10:{n2 n8 n4#6}r3c3 - C11:{n8 n1 n4#4}r8c3 - C12:{n4 n6 n8#10}r7c3 -

C10 ------- C13:{n8 n9 n2#9 n4#6}r3c1 - C13:{n9 n4 n6#12}r9c1 - C15:{n4 n9 n1#11}r9c2 - C16:{n9 n2 n4#14}r9c7 - C17:{n2 n1 n4#14 n7#7 n9#15}r9c6 - C18{n1 . n2#9 n6#1 n7#8}r2c6

==> r2c9 <> 9

As in any chain, the llc of C1 is nrc-linked to the target.

But, contrary to a chain:

- the left-linking candidate of C9 is linked to the target instead of to the right-linking candidate of C8,

- the left-linking candidate of C8 is linked to the right-linking candidate of C5 instead of to the right-linking candidate of C7,

- the left-linking candidate of C13 is linked to the right-linking candidate of C10 instead of to the right-linking candidate of C12.

nrczt-whip-rn[11] n9{r1c8 r2c7} - n9{r2c2 r9c2} - n9{r7c1 r3c1} - n9{r3c4 r7c4} - n6{r7c4 r9c4} - n1{r9c4 r9c6} - n7{r9c6 r9c9} - {n7 n8}r7c8 - n8{r3c8 r2c9} - {n6 n1}r2c5 - {n6r2c5 .} ==> r8c8 <> 9

Alternative path:

the above braid and whip could have been replaced by the two whips below:

nrczt-rl-lasso[12] n9{r1c8 r1c6} - n9{r3c4 r3c1} - n9{r2c2 r9c2} - n9{r9c4 r7c4} - n6{r7c4 r9c4} - n1{r9c4 r9c6} - n7{r9c6 r9c9} - {n7 n8}r7c8 - n8{r3c8 r3c3} - n2{r3c3 r2c1} - {n2 n7}r2c6 - {n7 n9}r2c2 ==> r8c8 <> 9

nrczt-rl-lasso[14] n1{r2c9 r3c8} - n8{r3c8 r2c7} - n6{r2c7 r1c8} - n9{r1c8 r1c6} - n9{r3c4 r3c1} - n8{r3c1 r3c3} - n4{r3c3 r1c3} - {n4 n6}r7c3 - n6{r7c4 r9c4} - {n6 n4}r9c1 - {n4 n1}r8c3 - n1{r9c2 r9c6} - n1{r2c6 r2c5} - n6{r2c5 r2c7} ==> r2c9 <> 9

But I wanted to give an example of an xyzt-braid.

End alternative path.

At this point, the PM is:

- Code: Select all
`+-------------------------+-------------------------+-------------------------+ `

|1 5 47 |8 367 379 |34 69 2 |

|289 789 3 |4 1267 1279 |689 5 18 |

|2489 6 248 |1239 123 5 |7 189 34 |

+-------------------------+-------------------------+-------------------------+

|23568 1378 12678 |12357 9 12378 |23568 4 13578 |

|23458 13478 12478 |12357 12357 6 |23589 12789 135789 |

|23568 1378 9 |12357 4 12378 |23568 12678 13578 |

+-------------------------+-------------------------+-------------------------+

|34689 2 468 |35679 357 3479 |1 789 45789 |

|7 13489 148 |12359 1235 12349 |24589 28 6 |

|469 149 5 |12679 8 12479 |249 3 479 |

+-------------------------+-------------------------+-------------------------+

Now comes our second braid,

an nrczt-braid which uses the four types of 2D cells (rc, rn, cn and bn):

nrczt-braid-cn[14]

* ------- C1:n8r3{c1 c8 c3*} - C2:{n8 n2}r8c8 -

* ------- C3:n7{r2c2 r1c3} - C4:n4r1{c3 c7} - C5:{n4 n9 n2#2}r9c7 - C6:{n9 n7 n8#1}r7c8 - C7:{n7 n4 n9#5}r9c9 - C8:{n4 n1 n9#5}r9c2 -

C7 ------- C9:{n4 n6 n9#5}r9c1 -

C2 ------- C10:{n8 n6 n9#5}r2c7 - C11:{n6 n9}r1c8 - C12:{n9 n1 n2#2 n7#6 n8#1}r5c8 - C13:n1{r5c3 r4c3 r4c2#8 r5c2#8 r6c2#8} - C14:n6{r4 . r7#9}c3

==> r2c2 <> 8

Here again, we have a non-first left-linking candidate (in C3) which is linked to the target instead of the pevious right-linking candidate; and two left-linking candidates (in C9 and C10) which are linked to a right-linking candidate that is not the immediately previous one.

nrczt-whip-rc[11] {n4 n7}r1c3 - {n7 n9}r2c2 - {n9 n1}r9c2 - {n1 n8}r8c3 - {n8 n2}r3c3 - n4{r3c3 r3c1} - n8{r3c1 r3c8} - n9{r3c8 r1c8} - n6{r1c8 r6c8} - n1{r5c3 r5c8} - {n1r5c3 .} ==> r7c3 <> 4

nrczt-whip-rc[14] n8{r2c1 r3c3} - {n8 n6}r7c3 - n6{r7c4 r9c4} - n6{r9c1 r6c1} - n5{r6c1 r5c1} - n3{r5c1 r7c1} - n8{r7c1 r8c2} - {n8 n2}r8c8 - n2{r9c7 r9c6} - n1{r9c6 r9c2} - n9{r9c2 r9c1} - n4{r9c1 r8c3} - n4{r1c3 r1c7} - {n4r9c7 .} ==> r4c1 <> 8

nrczt-whip-rc[14] n8{r2c1 r3c3} - {n8 n6}r7c3 - n6{r7c4 r9c4} - n6{r9c1 r4c1} - n5{r4c1 r5c1} - n3{r5c1 r7c1} - n8{r7c1 r8c2} - {n8 n2}r8c8 - n2{r9c7 r9c6} - n1{r9c6 r9c2} - n9{r9c2 r9c1} - n4{r9c1 r8c3} - n4{r1c3 r1c7} - {n4r9c7 .} ==> r6c1 <> 8

At this point, the PM is:

- Code: Select all
`+-------------------------+-------------------------+-------------------------+ `

|1 5 47 |8 367 379 |34 69 2 |

|289 79 3 |4 1267 1279 |689 5 18 |

|2489 6 248 |1239 123 5 |7 189 34 |

+-------------------------+-------------------------+-------------------------+

|2356 1378 12678 |12357 9 12378 |23568 4 13578 |

|23458 13478 12478 |12357 12357 6 |23589 12789 135789 |

|2356 1378 9 |12357 4 12378 |23568 12678 13578 |

+-------------------------+-------------------------+-------------------------+

|34689 2 68 |35679 357 3479 |1 789 45789 |

|7 13489 148 |12359 1235 12349 |24589 28 6 |

|469 149 5 |12679 8 12479 |249 3 479 |

+-------------------------+-------------------------+-------------------------+

Now comes

a braid with two left-linking candidates (in C15 and C19) branching off the same right-linking candidate (in C12).

nrczt-braid-cn[23]

* ------- C1:{n9 n7}r2c2 - C2:{n7 n4}r1c3 - C3:{n4 n3}r1c7 - C4:{n3 n4}r3c9 -

* ------- C5:n3{r8c2 r7c1} - C6:n4r7{c1 c6 c9#4} - C7:n4r8{c6 c7 c2*} - C8:n5{r8c7 r7c9} - C9:{n5 n7 n3#5}r7c5 - C10:{n7 n6 n3#3}r1c5 - C11:{n6 n9}r1c8 - C12:{n9 n8 n7#9}r7c8 - C13:{n8 n2}r8c8 - C14:{n2 n9 n4#7}r9c7 -

C12 ------- C15:{n8 n1 n9#11}r3c8 - C16:{n1 n8}r2c9 -

C12 ------- C17:{n8 n6}r7c3 - C18:{n6 n4 n9#14}r9c1 - C19:{n4 n1 n9#14}r9c2 - C20:{n1 n8 n4#18}r8c3 - C21:{n8 n2 n4#4}r3c3 - C22:{n2 n9 n8#16}r2c1 - C23:n9{r2 . r1#11 r8* r9#14}c6

==> r8c2 <> 9

At this point, the PM is:

- Code: Select all
`+-------------------------+-------------------------+-------------------------+ `

|1 5 47 |8 367 379 |34 69 2 |

|289 79 3 |4 1267 1279 |689 5 18 |

|2489 6 248 |1239 123 5 |7 189 34 |

+-------------------------+-------------------------+-------------------------+

|2356 1378 12678 |12357 9 12378 |23568 4 13578 |

|23458 13478 12478 |12357 12357 6 |23589 12789 135789 |

|2356 1378 9 |12357 4 12378 |23568 12678 13578 |

+-------------------------+-------------------------+-------------------------+

|34689 2 68 |35679 357 3479 |1 789 45789 |

|7 1348 148 |12359 1235 12349 |24589 28 6 |

|469 149 5 |12679 8 12479 |249 3 479 |

+-------------------------+-------------------------+-------------------------+

nrczt-whip-rn[9] n9{r9c2 r2c2} - n9{r3c1 r3c8} - {n9 n6}r1c8 - n6{r2c7 r2c5} - n7{r2c5 r2c6} - n7{r9c6 r9c9} - {n7 n8}r7c8 - {n8 n6}r7c3 - {n6r9c1 .} ==> r9c4 <> 9

nrczt-whip-rc[11] n3{r3c9 r1c7} - n4{r1c7 r1c3} - n7{r1c3 r2c2} - n9{r2c2 r9c2} - n9{r9c9 r7c9} - n5{r7c9 r8c7} - n4{r8c7 r9c7} - {n4 n7}r9c9 - {n7 n8}r7c8 - n6{r9c1 r7c3} - {n6r9c1 .} ==> r5c9 <> 3

nrczt-whip-rc[12] n9{r3c8 r2c7} - n9{r2c1 r3c1} - n9{r9c1 r9c2} - {n9 n7}r2c2 - {n7 n4}r1c3 - n4{r1c7 r3c9} - {n4 n7}r9c9 - {n7 n8}r7c8 - {n8 n2}r8c8 - {n2 n4}r9c7 - {n4 n6}r9c1 - {n6r7c3 .} ==> r5c8 <> 9

nrczt-whip-rc[12] n9{r1c6 r1c8} - n9{r3c8 r3c1} - n9{r9c1 r9c2} - {n9 n7}r2c2 - {n7 n4}r1c3 - n4{r1c7 r3c9} - {n4 n7}r9c9 - {n7 n8}r7c8 - {n8 n2}r8c8 - {n2 n4}r9c7 - {n4 n6}r9c1 - {n6r7c3 .} ==> r2c6 <> 9

nrczt-whip-rc[13] n9{r1c6 r1c8} - n6{r1c8 r1c5} - n3{r1c5 r1c7} - {n3 n4}r3c9 - n4{r1c7 r1c3} - n7{r1c3 r2c2} - n9{r2c2 r9c2} - {n9 n7}r9c9 - {n7 n8}r7c8 - {n8 n2}r8c8 - {n2 n4}r9c7 - {n4 n6}r9c1 - {n6r7c3 .} ==> r1c6 <> 7

nrczt-whip-rn[11] {n3 n4}r3c9 - n4{r1c7 r1c3} - n7{r1c3 r1c5} - {n7 n5}r7c5 - n5{r7c9 r8c7} - n4{r8c7 r9c7} - n2{r9c7 r8c8} - {n2 n1}r8c5 - n1{r8c2 r9c2} - {n7 n9}r2c2 - {n7r2c2 .} ==> r3c5 <> 3

Nothing remarkable in the sequel.