The example grid position shown below was derived from the Sudocue Nightmare for 5/20/07.

The elimination of interest is r4c4 <> 7.

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28 2378 2348 | 1 9 457 | 235 2345 6

126 236 2346 | CD25 C234 8 | 9 7 A145

129 2379 5 | C247 C234 6 | 123 8 A14

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

5 26 126 | 47 16 3 | 8 9 A47

3 689 168 | 4789 5 479 | 167 146 2

89 4 7 | 289 16 29 | 156 156 3

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

27 1 23 | 6 28 59 | 4 235 789

267 5 9 | 3 248 24 | 1267 126 178

4 2368 2368 | BD59 7 1 | 236 236 A59

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Define the following ALS’s:

ALS A: (14579)r2349c9

ALS B: (59)r9c4

ALS C: (23457)r23c45

ALS D: (259)r29c4 (add cell r2c4 to ALS B)

Form a (conventional) grouped AIC using ALS’s A,B, and C:

(7=1459)r2349c9 – (9=5)r9c4 – (5=2347)r23c45 => r4c4 <> 7,

which is equivalent to an ALS-XY-Wing rule move.

Now, the confusion (for me) arises when one replaces ALS B with ALS D to form the new AIC,

(7=1459)r2349c9 – (9=52)r29c4 – (5=2347)r23c45.

At first look, this new chain would appear to imply the same elimination as before (r4c4 <> 7). However, ALS D and C share a common cell (r2c4), although two different (and single-occurrence) digits, 2 and 5, form the weak link between them.

Question 1: So, is this a properly linked AIC (and deduction)?

Going a step further, one can also simply switch the positions of digits 2 and 5 in ALS D to form another new AIC,

(7=1459)r2349c9 – (9=25)r29c4 – (5=2347)r23c45.

Again, ALS D and C share the common cell, r2c4, but now the weakly linking digit (5) appears more like a true “restricted common” between ALS D and C. However, it has also been stated in this forum that overlapping ALS cells cannot contain a restricted common of the two sets, in which case the above AIC is then incorrect.

Question 2: So, is this AIC improperly linked between ALS D and C because digit 5 is a restricted common?

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One other unrelated elimination of interest is r1c2 <> 2.

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G28 F2378 2348 | 1 9 457 | 235 2345 6

126 EF236 2346 | 25 234 8 | 9 7 145

129 2379 5 | 247 234 6 | 123 8 14

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

5 EF26 126 | 47 16 3 | 8 9 47

3 EF689 168 | 4789 5 479 | 167 146 2

G89 4 7 | 289 16 29 | 156 156 3

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

27 1 23 | 6 28 59 | 4 235 789

267 5 9 | 3 248 24 | 1267 126 178

4 EF2368 2368 | 59 7 1 | 236 236 59

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Define the following ALS’s:

ALS E: (23689)r2459c2

ALS F: (236789)r12459c2 (add cell r1c2 to ALS E)

ALS G: (289)r16c1

Form a grouped AIC using ALS’s E and G:

(2=3689)r2459c2 - (9=82)r16c1 => r1c2 <> 2 (also r3c2 <> 2).

which is equivalent to an ALS-XZ rule move.

Now, confusion again arises when one replaces ALS E with ALS F to form the new AIC,

(7=23689)r12459c2 - (9=82)r16c1.

This new chain clearly shows a strong-inference link between (single-occurrence) (7)r1c2 and (2)r1c1, which, in turn, seems to imply the desired elimination (r1c2 <> 2). However, the candidate-elimination cell, r1c2, is actually a part of (i.e., overlaps) ALS F.

Question 3: The chain is certainly valid, but does it properly imply the r1c2 <> 2 elimination?