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Puzzle start
*-----------*
|8..|.4.|...|
|.95|..3|...|
|...|2..|..6|
|---+---+---|
|6..|...|2..|
|.2.|7.1|.4.|
|..3|...|..9|
|---+---+---|
|7..|..9|...|
|...|5..|68.|
|...|.2.|..1|
*-----------*
Puzzle point before pattern exists. A non-trivial enterprise just to get here!
*-----------*
|8..|.4.|...|
|.95|6.3|...|
|...|2.8|..6|
|---+---+---|
|6..|...|2..|
|.2.|761|.4.|
|..3|..2|.69|
|---+---+---|
|7..|..9|...|
|...|5..|68.|
|...|.26|..1|
*-----------*
This is a fairly difficult puzzle. After the
simple sudoku solving set is appled to the last puzzle position above, one must make the following eliminations to get to the point where I found the pattern that I wish to discuss:
1) r7c4=1 == r1c4=1 -- r1c4=9 == r3c5=9 -- r3c4=5 == r6c5=5 -- r6c5=8 == r7c5=8 forbids r7c5=1 and forbids r7c4=8
2) Y wing style elimination:
r9c3=4 == r9c3=8 -- r9c4=8 == r6c4=8 -- r6c4=4 == r4c6=4 forbids r4c3=4 (called Y wing style as it has a vertex and two endpoints, but in this case one endpoint and one vertex are strong in location, not cells)
The puzzle is now advanced to:
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Puzzle possibility matrix where pattern occurs.
*-----------------------------------------------------------------------------*
| 8 1367 1267 | 19 4 57 | 13579 123579 2357 |
| 124 9 5 | 6 17 3 | 1478 127 2478 |
| 134 1347 147 | 2 1579 8 | 134579 13579 6 |
|-------------------------+-------------------------+-------------------------|
| 6 14578* 178 | 39 39 45& | 2 157 578 |
| 59 2 89 | 7 6 1 | 358 4 358 |
| 145 1457# 3 | 48 58& 2 | 157 6 9 |
|-------------------------+-------------------------+-------------------------|
| 7 13468# 12468 | 134 38& 9 | 345 235 2345 |
| 12349 134 1249 | 5 137 47 | 6 8 234 |
| 345 3458* 48 | 348 2 6 | 79 79 1 |
*-----------------------------------------------------------------------------*
* = multi hub.
# = Spokes
& = AIC chain pieces that act like an ALS.
Perhaps this is almost off topic. I have found patterns similar to the hub and spoke idea in puzzles, but often they involve mulitple hubs. In this example, I see an almost hidden pair 58 at r49c2. But for the 5 at r6c2 and the 8 at r7c2, it would be a hidden pair. If there is no hidden pair at r49c2, then r6c5=8, thus r4c6=5, thus r4c2=4. This allows one to eliminate 17 from r4c2 and 4 from r9c2. A AIC chain representation of this step:
{Hidden pair 58 at r49c2} == {r6c2=5 == r7c2=8 -- r7c5=8 == r6c5=8} -- r6c5=5 == r4c6=5 -- r4c6=4 == r4c2=4.
Forbids r4c2 from being anything but 458.
forbids r9c2 from being 4.
These eliminations consider the following strong sets:
1) 5's in column 2: {r9c2, r6c2, r4c2} = 5
2) 8's in column 2: {r9c2, r7c2, r4c2} = 8
3) 8's in column 5: {r7c5, r6c5} = 8
4) 5's in box 5: {r6c5, r4c6} = 5
5) 4's in row 4: {r4c6, r4c2} = 4
Logic:
a) Strong sets 1) and 2) form an Almost Almost Hidden pair 58.
b) If hidden pair 58 at r49c2, then r49c2 is nothing but candidates 58.
c) If not hidden pair 58 at r49c2, then at least one of the following is true:
{r7c2=8, r6c2=5}
d) Clearly, at this point, we can combine b,c to write:
{hidden pair 58 at r49c2} =={r7c2=8 == r6c2=5}
e) But, because of strong set 3): no matter what, if the second part of d be true, we can have no more 5 at r6c5. In fact, r6c5 would be 8, but this is really not material to the chain, rather the fact tht r6c5 is not 5 is material.
f) Because of strong set 4), e) => r4c6=5 =>r4c6<>4.
g)Because of strong set 5), f) => r4c2=4.
i)Conclusion: b) == g): {Hidden pair 58 at r49c2} == {r4c2=4} forbids:
r4c2=17, r9c2=4.
j)Intermediate conclusion: {r4c6=5} == {r4c6=5}, but this forbids nothing as we already have that. This intermediate conclusion is perhaps the most difficult to understand. A whole bit of theory about partial loop closure would have to be introduced as to why this weak link in the chain must be strong, but not some of the other ones. Careful study of the bracketing that I use in the forbidding chain representation of the step shows that although the weak link:
{r6c2=5 == .... == r6c5=8} -- r6c5=5
is indeed strong, it does not directly forbid anything, (ever, IMO).
How does this fit the pattern?
1) The Almost Hidden Pair is the hub(or double hub).
2) The neccessary and sufficient conditions for not having the hidden pair are the spokes(58 in c2 outside of the hub)
3) A short conditional AIC chain shows if not hidden pair 58, then r4c2=4 - this acts like an ALS.
Alternate view:
1) cell r4c6 is the hub, with the weak link between 4 & 5.
2) strong 4's and 5's from that hub are the spokes.
3) Almost Almost Locked set Hidden pair 58 is the rim.
4) A short AIC chain connects the dots, reducing the Almost Almost Locked Hidden pair in fact to an Almost Hidden Pair, conditionally.
The alternate view is best seen reading the chain backwards. Naturally, a properly developed chain has
no direction, so reading this chain backwards should make the same sense as forwards.
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Alternate viewpoint of the hub
*-----------------------------------------------------------------------------*
| 8 1367 1267 | 19 4 57 | 13579 123579 2357 |
| 124 9 5 | 6 17 3 | 1478 127 2478 |
| 134 1347 147 | 2 1579 8 | 134579 13579 6 |
|-------------------------+-------------------------+-------------------------|
| 6 14578S* 178 | 39 39 45 HUB | 2 157 578 |
| 59 2 89 | 7 6 1 | 358 4 358 |
| 145 1457* 3 | 48 58S^ 2 | 157 6 9 |
|-------------------------+-------------------------+-------------------------|
| 7 13468* 12468 | 134 38^ 9 | 345 235 2345 |
| 12349 134 1249 | 5 137 47 | 6 8 234 |
| 345 3458* 48 | 348 2 6 | 79 79 1 |
*-----------------------------------------------------------------------------*
HUB = hub.
S = Spokes endpoints
* = Almost Almost Locked Set (HIDDEN)
^ = Spoke extension
To understand this alternate viewpoint, the tempation would be to use the existing strong link at the HUB. Suppose, for a moment that I do not do that:
a) If r4c2=4 then clearly (r4c2<>17 and r9c2<>4).
b) If r4c2<>4, then r4c6=4 => r6c5=5 => r7c5 = 8.
c} (r6c5=5 and r7c5=8) => Hidden pair 58 at r49c2, which also forbids:
r4c2=17 and r9c2=4.
Just thought I might introduce some further complications into this idea.