In the 2nd post of this thread, I mentioned the rare case of a puzzle P with very different W and GW ratings: GW(P) = W(P) - 5. One might think that this can happen only for hard puzzles, but this example shows that it can happen also with relatively simple ones.

More precisely, we have SER(P) = 3.8, GW(P) = 2 and W(P) = 7.

The puzzle is the following:

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

| 1 . . | . . 6 | . . 9 |

| . . . | . . . | . . . |

| . 8 . | 1 . . | 5 6 . |

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

| . 3 . | 6 . . | 8 . 1 |

| . . . | . . . | . . 7 |

| . . . | . . 8 | 2 4 6 |

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

| . 4 . | 9 . . | . 2 . |

| 5 6 . | 8 4 1 | . . . |

| . . 7 | . . 3 | . . . |

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

If we accept g-whips, then there is a very short resolution path:

***** SudoRules version 15b.1.8-GW *****

1....6..9..........8.1..56..3.6..8.1........7.....8246.4.9...2.56.841.....7..3...

26 givens and 206 candidates

naked-single ==> r8c9 = 3

hidden-single-in-a-row ==> r8c3 = 2

interaction column c9 with block b9 ==> r9c8 <> 5

interaction row r8 with block b9 ==> r9c8 <> 9, r9c7 <> 9, r7c7 <> 7

interaction row r6 with block b5 ==> r5c4 <> 3, r5c5 <> 3

whip[2] b4n6{r5c1 r5c3} - b4n8{r5c3 .} ==> r5c1 <> 4

whip[2] b4n6{r5c1 r5c3} - b4n8{r5c3 .} ==> r5c1 <> 2

whip[2] b4n8{r5c1 r5c3} - b4n6{r5c3 .} ==> r5c1 <> 9

whip[2] b4n6{r5c3 r5c1} - b4n8{r5c1 .} ==> r5c3 <> 5

whip[2] b4n6{r5c3 r5c1} - b4n8{r5c1 .} ==> r5c3 <> 4

interaction row r5 with block b5 ==> r4c6 <> 4

whip[2] b4n6{r5c3 r5c1} - b4n8{r5c1 .} ==> r5c3 <> 1

whip[2] b4n8{r5c3 r5c1} - b4n6{r5c1 .} ==> r5c3 <> 9

;;; A

g-whip[2] r3n7{c1 c456} - c4n7{r2 .} ==> r6c1 <> 7singles

GRID 0 SOLVED. GW = 2, MOST COMPLEX RULE = G-Whip[2]

123456789

456789132

789132564

234697851

618524397

975318246

341975628

562841973

897263415

(remember that interactions = whip[1])

If we use only whips, the resolution path is much longer:

***** SudoRules version 15b.1.8-W *****

same path down to A

whip[3] c4n7{r1 r6} - c2n7{r6 r2} - r3n7{c1 .} ==> r1c5 <> 7

whip[3] c2n7{r1 r6} - c4n7{r6 r1} - r3n7{c6 .} ==> r2c1 <> 7

whip[3] c4n7{r2 r6} - c2n7{r6 r1} - r3n7{c1 .} ==> r2c5 <> 7

whip[3] c4n7{r2 r6} - c2n7{r6 r1} - r3n7{c1 .} ==> r2c6 <> 7

whip[3] r5n2{c6 c2} - r1n2{c2 c4} - b8n2{r9c4 .} ==> r4c5 <> 2

whip[3] b4n7{r6c2 r4c1} - r3n7{c1 c6} - b8n7{r7c6 .} ==> r6c5 <> 7

whip[3] r9c2{n1 n9} - r9c1{n9 n8} - r9c8{n8 .} ==> r9c7 <> 1

whip[3] r9c8{n8 n1} - r9c2{n1 n9} - r9c1{n9 .} ==> r9c9 <> 8

whip[5] r4n2{c1 c6} - r4n7{c6 c5} - b8n7{r7c5 r7c6} - r3n7{c6 c1} - r6c1{n7 .} ==> r4c1 <> 9

whip[5] r4c8{n9 n5} - r4c5{n5 n7} - b8n7{r7c5 r7c6} - r3n7{c6 c1} - r6c1{n7 .} ==> r4c3 <> 9

whip[3] r9c2{n9 n1} - c3n1{r7 r6} - c3n9{r6 .} ==> r2c2 <> 9

whip[6] b3n1{r2c7 r2c8} - r9c8{n1 n8} - r9c1{n8 n9} - r6c1{n9 n7} - b1n7{r3c1 r1c2} - c4n7{r1 .} ==> r2c7 <> 7

whip[7] c3n6{r2 r5} - c1n6{r5 r2} - c1n4{r2 r4} - c1n2{r4 r3} - b3n2{r3c9 r2c9} - c9n8{r2 r7} - c3n8{r7 .} ==> r2c3 <> 4

whip[7] r3n3{c1 c5} - c3n3{r3 r7} - b7n1{r7c3 r9c2} - b7n9{r9c2 r9c1} - r6c1{n9 n7} - r3n7{c1 c6} - c4n7{r1 .} ==> r2c1 <> 3

whip[7] r1n2{c5 c2} - c1n2{r2 r4} - c6n2{r4 r5} - r2n2{c6 c9} - r3c9{n2 n4} - c6n4{r3 r2} - c1n4{r2 .} ==> r3c5 <> 2

whip[7] r4n2{c1 c6} - r4n7{c6 c5} - b8n7{r7c5 r7c6} - r3n7{c6 c1} - c1n2{r3 r2} - r1c2{n2 n5} - r2c2{n5 .} ==> r4c1 <> 4

hidden-single-in-a-block ==> r4c3 = 4

whip[4] b5n4{r5c6 r5c4} - r1n4{c4 c7} - c7n7{r1 r8} - c7n9{r8 .} ==> r5c6 <> 9

whip[4] r3c3{n9 n3} - r3c5{n3 n7} - c4n7{r1 r6} - r6c1{n7 .} ==> r3c1 <> 9

whip[5] b3n7{r1c7 r2c8} - c8n1{r2 r9} - r7c7{n1 n6} - r9c7{n6 n4} - r1n4{c7 .} ==> r1c4 <> 7

;;; now we get the crucial elimination with a whip[2]:

whip[2] c4n7{r6 r2} - r3n7{c6 .} ==> r6c1 <> 7singles to the end

GRID 0 SOLVED. W = 7, MOST COMPLEX RULE = Whip[7]

Interestingly, this puzzle can also be solved with subset rules, but it has to go one level deeper than with g-whips

***** SudoRules version 15b.1.8-WS *****

1....6..9..........8.1..56..3.6..8.1........7.....8246.4.9...2.56.841.....7..3...

26 givens and 206 candidates

naked-single ==> r8c9 = 3

hidden-single-in-a-row ==> r8c3 = 2

interaction column c9 with block b9 ==> r9c8 <> 5

interaction row r8 with block b9 ==> r9c8 <> 9, r9c7 <> 9

interaction row r8 with block b9 ==> r7c7 <> 7

interaction row r6 with block b5 ==> r5c4 <> 3, r5c5 <> 3

hidden-pairs-in-a-row r5{n6 n8}{c1 c3} ==> r5c3 <> 9, r5c3 <> 5, r5c3 <> 4, r5c3 <> 1, r5c1 <> 9, r5c1 <> 4

interaction row r5 with block b5 ==> r4c6 <> 4

hidden-pairs-in-a-row r5{n6 n8}{c1 c3} ==> r5c1 <> 2

;;; same situation as A (all the whips[2] in the W or GW resolution paths correspond to hidden pairs)

naked-triplets-in-a-row r9{c1 c2 c8}{n8 n9 n1} ==> r9c9 <> 8, r9c7 <> 1

swordfish-in-rows n7{r3 r4 r7}{c6 c1 c5} ==> r6c5 <> 7

The crucial elimination is now obtained with a swordfish:

swordfish-in-rows n7{r3 r4 r7}{c6 c1 c5} ==> r6c1 <> 7singles to the end

GRID 0 SOLVED. WS = 3, MOST COMPLEX RULE = SHT