7/8/07 Nightmare...a nice multiple-methods example
Posted: Sun Aug 05, 2007 1:50 pm
After basics:
A check of the single-digit grids leads to...
Grouped Turbot chain (labels a-h in Grid 1 below):
(3): r5c1 = r5c5 – r8c5 = r8c6 – r3c6 = r2c46 – r2c8
= r1c7 => r1c1 <> 3, or
= r4c8 => r4c1 <> 3.
Finned X-Wing (labels x and f in Grid 2 below):
X-Wing = r24c48 with r6c4 = fin => r4c6 <> 3,
or grouped Turbot fish:
(3): r46c4 = r2c4 – r2c8 = r4c8 => r4c6 <> 3.
.
After follow-up:
At this point, multi-digit methods abound for a single elimination (r4c4 <> 3):
1. WXYZ-Wing: W=6 and Z=3, using cells (36)r5c5 and (1369)r4c678 => r4c4 <> 3.
2. ALS-XZ rule:
ALS(A=[r5c5], B=[r4c678], X=6, Z=3) => r4c4 <> 3.
3. Grouped AIC:
(3=6)r5c5 – (6=193)r4c678 => r4c4 <> 3,
equivalent to both the WXYZ-Wing and the ALS-XZ rule.
4. Subset Counting: subset is (36)r5c5 and (1369)r4c678, with digit 3 having multiplicity of 2; all others have multiplicity of 1. Cell r4c4 = 3 would reduce total subset multiplicity from 5 to 3, which is one less than cell count => r4c4 <> 3.
5. APE (Aligned Pair Exclusion): cells (137)r4c4 and (16)r4c6 generate the two pair combinations, 3-1 and 3-6, and neither one is allowed => r4c4 <> 3.
6. Non-grouped AIC’s:
(3)r5c5 = (3-6)r5c1 = (6-7)r4c1 = (7)r4c4 => r4c4 <> 3.
(3=6)r5c5 - (6)r4c6 = (6-7)r4c1 = (7)r4c4 => r4c4 <> 3.
7. 3D Coloring: Coloring of digits in either of the above AICs quickly shows that (3)r5c5 and (7)r4c4 have opposite colors (parity) => r4c4 <> 3.
8. Discontinuous Nice loops:
[r4c4]-3-[r5c5]=3=[r5c1]=6=[r4c1]=7=[r4c4] => r4c4 <> 3.
[r4c4]-3-[r5c5]-6-[r4c6]=6=[r4c1]=7=[r4c4] => r4c4 <> 3.
These loops are equivalent to the above two non-grouped AIC’s.
Would anyone care to add to this “methods” list for the r4c4 <> 3 elimination? Contributions welcome!
Code: Select all
-------------------------------------------
135 14 345 | 6 7 2 | 349 8 49
9 26 7 | 13 8 134 | 5 34 26
38 26 348 | 5 9 34 | 267 27 1
--------------+-------------+--------------
367 8 2 | 137 4 136 | 139 39 5
36 5 1 | 9 36 8 | 24 24 7
4 79 39 | 137 2 5 | 13 6 8
--------------+-------------+--------------
1578 3 589 | 4 156 69 | 26789 279 26
178 1479 6 | 2 13 39 | 4789 5 49
2 49 459 | 8 56 7 | 469 1 3
-------------------------------------------
Grouped Turbot chain (labels a-h in Grid 1 below):
(3): r5c1 = r5c5 – r8c5 = r8c6 – r3c6 = r2c46 – r2c8
= r1c7 => r1c1 <> 3, or
= r4c8 => r4c1 <> 3.
Code: Select all
Grid 1
3* . 3 | . . . | 3h . .
. . . | 3f . 3f | . 3g .
3 . 3 | . . 3e | . . .
-------+----------+------
3* . . | 3 . 3 | 3 3h .
3a . . | . 3b . | . . .
. . 3 | 3 . . | 3 . .
-------+----------+------
. . . | . . . | . . .
. . . | . 3c 3d | . . .
. . . | . . . | . . .
X-Wing = r24c48 with r6c4 = fin => r4c6 <> 3,
or grouped Turbot fish:
(3): r46c4 = r2c4 – r2c8 = r4c8 => r4c6 <> 3.
Code: Select all
Grid 2
. . 3 | . . . | 3 . .
. . . | 3x . 3 | . 3x .
3 . 3 | . . 3 | . . .
------+---------+------
. . . | 3x . 3* | 3 3x .
3 . . | . 3 . | . . .
. . 3 | 3f . . | 3 . .
------+---------+------
. . . | . . . | . . .
. . . | . 3 3 | . . .
. . . | . . . | . .
After follow-up:
Code: Select all
--------------------------------------------
15 14 345 | 6 7 2 | 349 8 49
9 26 7 | 13 8 134 | 5 34 26
38 26 348 | 5 9 34 | 267 27 1
--------------+--------------+--------------
67 8 2 | 173* 4 16 | 139 39 5
36 5 1 | 9 36 8 | 24 24 7
4 79 39 | 137 2 5 | 13 6 8
--------------+--------------+--------------
1578 3 589 | 4 156 69 | 26789 279 26
178 1479 6 | 2 13 39 | 4789 5 49
2 49 459 | 8 56 7 | 469 1 3
--------------------------------------------
At this point, multi-digit methods abound for a single elimination (r4c4 <> 3):
1. WXYZ-Wing: W=6 and Z=3, using cells (36)r5c5 and (1369)r4c678 => r4c4 <> 3.
2. ALS-XZ rule:
ALS(A=[r5c5], B=[r4c678], X=6, Z=3) => r4c4 <> 3.
3. Grouped AIC:
(3=6)r5c5 – (6=193)r4c678 => r4c4 <> 3,
equivalent to both the WXYZ-Wing and the ALS-XZ rule.
4. Subset Counting: subset is (36)r5c5 and (1369)r4c678, with digit 3 having multiplicity of 2; all others have multiplicity of 1. Cell r4c4 = 3 would reduce total subset multiplicity from 5 to 3, which is one less than cell count => r4c4 <> 3.
5. APE (Aligned Pair Exclusion): cells (137)r4c4 and (16)r4c6 generate the two pair combinations, 3-1 and 3-6, and neither one is allowed => r4c4 <> 3.
6. Non-grouped AIC’s:
(3)r5c5 = (3-6)r5c1 = (6-7)r4c1 = (7)r4c4 => r4c4 <> 3.
(3=6)r5c5 - (6)r4c6 = (6-7)r4c1 = (7)r4c4 => r4c4 <> 3.
7. 3D Coloring: Coloring of digits in either of the above AICs quickly shows that (3)r5c5 and (7)r4c4 have opposite colors (parity) => r4c4 <> 3.
8. Discontinuous Nice loops:
[r4c4]-3-[r5c5]=3=[r5c1]=6=[r4c1]=7=[r4c4] => r4c4 <> 3.
[r4c4]-3-[r5c5]-6-[r4c6]=6=[r4c1]=7=[r4c4] => r4c4 <> 3.
These loops are equivalent to the above two non-grouped AIC’s.
Would anyone care to add to this “methods” list for the r4c4 <> 3 elimination? Contributions welcome!