Apart from grouped x-node and almost locked set, there are also other grouped nodes that can be used to make inferences within grouped nice loops. Together, these grouped nodes form a list including but are not limited to the following:
- grouped x-node, Note: a grouped x-node nice loop is equivalent to an empty rectangle deduction.
almost locked set (Lockedset+1)
almost unique rectangle (AUR)
almost x-wing, Note: an almost x-wing nice loop is equivalent to a fillet-O-fish.
almost NxN fish (finned fish)
almost BUG-Lite (BUG-Lite+2), Note: all almost unique patterns are almost BUG-Lites.
- Grouped x-cycle - an x-cycle with grouped x-node(s) as grouped node(s).
Grouped xy-chain - an xy-chain with almost locked set(s) as grouped node(s).
Strong Grouped Nice Loop - a simple nice loop with grouped x-node(s) as grouped node(s).
Weak Grouped Nice Loop - a simple nice loop with almost locked set(s) as grouped node(s).
AUR Nice Loop - a simple nice loop with almost unique rectangle(s) as grouped node(s).
Almost x-wing Nice Loop - a simple nice loop with almost x-wing(s) as grouped node(s).
I will demonstrate how to spot this type of grouped nice loop via a few examples, including an xyz-wing, a grid posted by Bennys for the almost locked set xy-wing rule and an almost x-wing.
- Code: Select all
Example 1 - Grouped xy-chain (xyz wing)
+----------------+----------------+----------------+
| 7 1 4 | 2 3 9 | 8 6 5 |
| 2 5 9 | 8 6 1 | 4 7 3 |
| 3 6 8 | 57 4 57 | 2 1 9 |
+----------------+----------------+----------------+
| 4 38 7 | 56 1 2568 | 35 9 28 |
| 9 38 6 | 47 58 247 | 35 24 1 |
| 5 2 1 | 3 9 *48 | 7 48 6 |
+----------------+----------------+----------------+
| 8 9 3 | 1 7 45 | 6 25 24 |
| 6 4 2 | 9 58 3 | 1 58 7 |
| 1 7 5 |*46 2 *468 | 9 3 48 |
+----------------+----------------+----------------+
An xyz-wing is regarded as a chain with triple implication streams due to the inference made by the naked pair. With the definition of grouped nice loop, this triple chain can now be expressed as a double chain with a grouped node [r9c6|r9c4]. This grouped nice loop, also specifically called an grouped xy-chain, has links of weak inference associated with the grouped node containing an almost locked set (lockedset+1).
Nice loop notation:
[r7c6]-4-[r6c6]-8-[r9c6|r9c4]-4-[r7c6] => r7c6<>4
where
[r9c6|r9c4] is a lockedset+1 grouped node
[r6c6]-8-[r9c6|r9c4]-4-[r7c6] are grouped inferences meaning: r6c6=8 => r9c6=r9c4=46 => r7c6<>4
- Code: Select all
Example 2 - Grouped xy-chain (almost locked set xy-wing rule)
+-------------------+-------------------+-------------------+
| 459 1 %479 |%78 3 %89 | 2 %58 6 |
| 59 569 679 | 278 289 4 |^35 1358 13 |
| 8 3 2 | 5 6 1 | 7 9 4 |
+-------------------+-------------------+-------------------+
|*139 7 *139 | 6 4 5 | 8 123 1239 |
| 6 28 *134 | 9 28 7 |^34 13 5 |
| 24 289 5 | 238 1 238 |^49 6 7 |
+-------------------+-------------------+-------------------+
| 7 4 39 | 23 259 6 | 1 235 8 |
| 12359 2569 1369 | 4 2589 2389 | 3569 7 239 |
| 2359 2569 8 | 1 7 239 | 3569 4 239 |
+-------------------+-------------------+-------------------+
In this example, Bennys proved that r6c2<>9 by a technique called almost locked set xy-wing rule. The same deduction can be viewed as a grouped nice loop with both multiple inference and grouped inference where the grouped node is an almost locked set [r5c3|r4c1|r4c3].
Nice loop notation:
[r6c2]-9-[r6c7]-4-[r5c7]-3-[r2c7]-5-[r1c8](-8-[r1c4]-7-[r1c3])-8-[r1c6]-9-[(r1c3)]-4-[r5c3|r4c1|r4c3]-9-[r6c2] => r6c2<>9
where
(-8-[r1c4]-7-[r1c3]) and (r1c3) is a multiple inference meaning: r1c8=8 => r1c6=9 & r1c4=7 => r1c3=4
[r5c3|r4c1|r4c3] is a lockedset+1 grouped node
[(r1c3)]-4-[r5c3|r4c1|r4c3]-9-[r6c2] are grouped inferences meaning: r1c3=4 => r5c3, r4c1 & r4c3 is a locked set of 139 => r6c2<>9
As a matter of interest, if this deduction is to be represented by a chain, it would be a forcing chain with 5 implication streams.
- Code: Select all
Example 3 - Strong Grouped Nice loop
*---------------------------------------------------------------------*
| 7 156 456 | 2 458 146 | 3 458 9 |
| 19 3 459 | 14589 7 149 | 2 458 6 |
| 269 269 8 | 3 45 69 | 14 7 15 |
|----------------------+-----------------------+----------------------|
| 3 2678 267 | #1478 9 *1247 | 5 1246 128 |
| 2689 4 25679 | 1578 258 127 | 1689 3 128 |
| 289 2589 1 | #458 6 3 | 49 249 7 |
|----------------------+-----------------------+----------------------|
| 12689 12689 269 | ^469 ^24 5 | 7 1269 3 |
| 4 2679 3 | 679 1 ^279 | 689 2569 258 |
| 5 12679 2679 | 679 3 8 | 169 1269 4 |
*---------------------------------------------------------------------*
This example is really a strong grouped nice loop with a grouped x-node. It is included here to clarify any wrong impression from the 2 examples above that a grouped nice loop should have all links with weak inferences. In fact, the inferences within a grouped nice loop could be a mixture of strong and weak inferences as shown here.
Nice loop notation:
[r4c6]-2-[r8c6]=2=[r7c5]=4=[r7c4]-4-[r6c4|r4c4]=4=[r4c6] =>r4c6<>2
where
[r6c4|r4c4] is a grouped x-node.
[r7c4]-4-[r6c4|r4c4]=4=[r4c6] are grouped inferences.
- Code: Select all
Example 4 - Almost x-wing Nice Loop (Filet-O-Fish)
This example is a grouped nice loop with an almost x-wing and a grouped x-node as grouped nodes. Other almost patterns work in the same principle.
Nice loop notation:
[r5c4]-1-[almost x-wing:r24c24]=1=[r4c56]-1-[r5c4] => r5c4<>1
where both [almost x-wing:r24c24] and [r4c56] are grouped nodes.
