## Quantums

Advanced methods and approaches for solving Sudoku puzzles

### Quantums

The term "quantum" has evolved. It's no longer just the fuzzy term we grew up with. Until the last year or so, there was little usage beyond Type 3 Unique Rectangles and related quantum sets. At this point there isn't even a searchable mention of the term "quantum" in Sudopedia.

The evolution began largely with Steve K's "Hybrid Sets" thread. The usage has expanded of late, and now seems to embrace a concept rather than simply another elimination technique. Some of it makes sense to me, and some is baffling. Rather than speak in general terms, I prefer to discuss some of the actual examples I've seen around the Net, and hope to get some clarity that way. In this first post there will be three examples that make some sense to me, with my layman description of what I think is going on. I'm way off base half the time, so it should be a learning experience! I'll follow up later with some of the "baffling" ones.

I think there's a lot of useful stuff under the "quantum" umbrella, stuff that is poorly understood, not clearly defined, and under-utilized. I'm just trying to get a handle on it from an Average Joe solver's perspective and get these ideas a bit more out in the open.

I'm not sure if this first one is the best quantum example, but itâ€™s a strong inference you donâ€™t see every day.

Ex 1: Quantum naked pair (QNP) from Steve K here (June 24, 2009).
Code: Select all
` *-----------------------------------------------------------------------------*  | 1       678     567     |*489     2      *789     | 5689    489     3       |  | 789     378     4       | 5       379     6       | 2       189     19      |  | 569     2       356     | 13489   349     1389    | 5689    7       4569    |  |-------------------------+-------------------------+-------------------------|  | 3       1467    12567   | 129     569     129     | 1679    1249    8       |  | 268     9       1268    | 7       36      4       | 136     5       126     |  | 24567   1467    12567   | 12389   3569    12389   | 13679   12349   124679  |  |-------------------------+-------------------------+-------------------------|  | 267     167     9       | 236     8       2357    | 4       23      1257    |  | 2478    5       12378   | 2349    3479    2379    | 1789    6       1279    |  | 24678   34678   23678   | 23469   1       23579   | 35789   2389    2579    |  *-----------------------------------------------------------------------------* (9)r1c46 = (QNPx8)r1c46 - (8)r3c46 = (8)r3c7 - (8)r2c8 = (np19)r2c89 => r2c5, r1c78<>9`

I think the "x" in (QNPx8)r1c46 stands for either 4 or 7 (or neither). It is refered to as a "quantum ALS(x89)r1c46." It doesn't seem to matter what value "x" is, since any possibility will allow a weak link to (8)r3c46, and the chain progresses.
Edit: I thought wrong. See preamble a few posts down.

Ex. 2: This one also seems to have one of the recurring themes of these quantums: the combination of an irregular set with an almost pattern. Quantum Naked Triple (QNT) from Steve K here.
Code: Select all
` *-----------------------------------------------------------*  | 239   39    5     | 1     69    8     | 7     246   46    |  | 4     1     29    | 359   5679  37    | 59    26    8     |  | 7     8     6     | 2     59    4     | 1     3     59    |  |-------------------+-------------------+-------------------|  | 36    3567  4     | 8     17    9     | 2     156   367   |  | 289   579   289   | 6     3     17    | 589   1459  47    |  |*3689  3679  1     | 4     2     5     |*89   *69    367   |  |-------------------+-------------------+-------------------|  | 189   2     7     | 59    4     6     | 3     589   19    |  | 5     4     89    | 39    189   13    | 6     7     2     |  | 1689  69    3     | 7     589   2     | 4     589   159   |  *-----------------------------------------------------------* (AUR36)r46c19 => sis[(89)r6c1, (6)r1c9] => (6)r1c9 = (QNT689)r6c178 => r46c9<>6 `

If I'm seeing this right, if both the (89) and the (6) are false, the (36) deadly pattern is forced. The pairs of (89) within the set will force (6) into r6c8, resulting in a naked triple. This move seems pretty accessible, spottable given awareness of underlying patterns; no "chain thinking" required other than for notation.

Ex.3: This one's pretty cool, IMO. Quantum Naked Triple (QNT) from Steve K here (Nice graphic at this link.)
Code: Select all
` *--------------------------------------------------------------------*  | 3      9      1      | 2458   2458   7      | 58     25     6      |  | 2      7      48     | 6      58     1      | 358    9      34     |  | 6      5      48     | 289    289    3      | 1      7      24     |  |----------------------+----------------------+----------------------|  | 15     36     7      |*235    2356   8      | 9      4      123    |  | 59     4      36     |*235    1     *569    | 7      256    8      |  | 159    8      2      | 7      34569 *4569   | 35     56     13     |  |----------------------+----------------------+----------------------|  | 7      23     9      | 1      346    46     | 246    8      5      |  | 8      1      56     | 45     7      2      | 46     3      9      |  | 4      236    356    | 3589   35689  569    | 26     1      7      |  *--------------------------------------------------------------------* (AUR 89)r39c45, => sis[(2)r3c45, (9)r9c6] =>(2)r3c45=(9-5)r9c6=(QNT235)r56c6, r45c4 =>r1c4<>2`

Once again, the combination of an irregular set with an almost pattern. In looking at the set, it seems that there are four cells that can be treated as three, so I'm guessing that is the "quantum" nature of it. Noting (5)r9c6=(5)r56c6, it won't matter if a 5 ends up in r5c6 or r6c6; either way, a naked triple will result because of the (235) in r45c6. Seems a pretty simple and useful idea, but for some reason not widely practiced.

Is anyone else interested in exploring this kind of thing? It goes a lot deeper than the above, I'm sure of that much .

Luke
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### Re: Quantums

Luke451 wrote:Ex. 2:
Steve K wrote:(AUR36)r46c19 => sis[(89)r6c1, (6)r1c9] => (6)r1c9 = (QNT689)r6c178 => r46c9<>6

If I'm seeing this right, if both the (89) and the (6) are false, the (36) deadly pattern is forced. The pairs of (89) within the set will force (6) into r6c8, resulting in a naked triple.

Correct, but I see nothing that justifies the quantum term. The extra AUR candidates being considered are in one cell, not two. We simply have derived sis (6)r1c9 = (NT689)r6c178.

Luke451 wrote:[b]Ex. 3:
Steve K wrote:(AUR 89)r39c45, => sis[(2)r3c45, (9)r9c6] =>(2)r3c45=(9-5)r9c6=(QNT235)r56c6, r45c4 =>r1c4<>2

Once again, the combination of an irregular set with an almost pattern. In looking at the set, it seems that there are four cells that can be treated as three, so I'm guessing that is the "quantum" nature of it. Noting (5)r9c6=(5)r56c6, it won't matter if a 5 ends up in r5c6 or r6c6; either way, a naked triple will result because of the (235) in r45c6. Seems a pretty simple and useful idea, but for some reason not widely practiced.

I don't think this is proper usage of the quantum term either. Any naked subset should be comprised of cells -- 100 percent. When a "naked subset" has both cell sets and a portion of a hidden set (as in this example), [b]Steve Kurzhal's original hybrid qualifier term seems much more appropriate.

[edit: After addition of a missing premise, comments as to Ex. 1 deleted]
ronk
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### Re: Quantums

Luke451,

when you start a thread about a new technique/concept or whatever, your first example should not be flawed.
I neither understand the notation nor know something about the concept behind, but obviously you cant eliminate 9 in [edit typo] r1c8 from the cells in the chain, not even from the first band at all.
eleven

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### Re: Quantums

eleven, I couldn't follow Steve's preamble describing how we have a QNP in r1c46, but using a derived inference I can validate the equivalent chain:
Code: Select all
` *-----------------------------------------------------------------------------*   | 1       678     567     |*489     2      *789     | 5689    489     3       |  | 789     378     4       | 5       379     6       | 2       189     19      |  | 569     2       356     | 13489   349     1389    | 5689    7       4569    |  |-------------------------+-------------------------+-------------------------|   | 3       1467    12567   | 129     569     129     | 1679    1249    8       |  | 268     9       1268    | 7       36      4       | 136     5       126     |  | 24567   1467    12567   | 12389   3569    12389   | 13679   12349   124679  |  |-------------------------+-------------------------+-------------------------|   | 267     167     9       | 236     8       2357    | 4       23      1257    |  | 2478    5       12378   | 2349    3479    2379    | 1789    6       1279    |  | 24678   34678   23678   | 23469   1       23579   | 35789   2389    2579    |  *-----------------------------------------------------------------------------*  (9)r1c46 = (QNPx8)r1c46 - (8)r3c46 = (8)r3c7 - (8)r2c8 = (np19)r2c89 => r2c5, r1c78<>9`

1: (8|9=4)r1c4 - (4)r89c4 = (4-7)r8c5 = (7)r789c6 - (7=8|9)r1c6 => *[A] (8=9)r1c46
at least one of r1c4 and r1c6 must contain 8 or 9, so taken together these cells must contain at least one of these digits = derived inference [A].

2: (9=[A]=8)r1c46 - (8)r3c46 = (8)r3c7 - (8=19)AHS:r2c89 => r2c5, r1c78<>9
starts with the derived inference to build up the rest of the deduction.

For my own purposes I try to notate chains so that a) every component that contributes to a deduction is shown & b) any nested sub-chains are replaced by labelled derived inferences as above. This at least gives me a fighting chance of being able to follow some of the complex combinational logic. With some of Steve's 'super patterns' I don't know the logic well enough to know how to set about this though. I'll therefore be following this thread with interest.

In the second example Steve's chain:
(AUR36)r46c19 => sis[(89)r6c1, (6)r1c9] => (6)r1c9 = (QNT689)r6c178 => r46c9<>6

is equivalent to:
(6=4)r1c9 - (4=7)r5c9 - (7=36)AHS:r46c9 -[UR]- (36)r46c1 = (89)AHS:r6c17 - (9=6)r6c8 => r46c9 <> 6

And in the third example:
(AUR 89)r39c45, => sis[(2)r3c45, (9)r9c6] =>(2)r3c45=(9-5)r9c6=(QNT235)r56c6, r45c4 =>r1c4<>2

is functionally equivalent to:
(2=89)AHS:r3c45 -[UR]- (89)AHS:r9c45 = (9-5)r9c6 = (5)r56c6 - (5=23)AHS:r45c4 => r1c4 <> 2

So in neither case is there any sort of derived inference involved.

Steve, you have an amazing way of visualising inferences in terms of strong inference sets that often loses me. This gives you an insight into finding eliminations by eye that I can only discover far more ponderously by tracking chains. Having found them however, it would be helpful if they were translated (if possible) into valid AICs - that is with strictly alternating inferences and with clear linking digits between consecutive nodes for the benefit of us mere mortals.

You were the first one to use the term "quantum" which seems to cover a range of similar conditions. No one yet has ever liked anything that I've tried to define (apart from the space in "proving loops") so I bite my tongue and await your clarification.
David P Bird
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### Re: Quantums

David P Bird wrote:For my own purposes I try to notate chains so that a) every component that contributes to a deduction is shown & .
Yes, otherwise the "solutions" are more a riddle than a help. This was the reason that i stopped to follow Carcul's chains. Sometimes it was easier to solve the puzzle than to understand, what he left hidden.

Your first step can be found easily, when you notice the strong links for 4 and 7 in column 5 (one of r2c5=7 and r3c5=4 must be true). This is what i would expect to be mentioned.
eleven

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### Re: Quantums

Hi, all, thanks for the observations. Yes, I see I have misinterpreted a preamble that was provided by Steve K in the first example. There is, as David pointed out, a derived inference that must be included in order for the starting link to work. Here is the one provided by Steve:
Code: Select all
`*-----------------------------------------------------------------------------*   | 1       678     567     |*489     2      *789     | 5689    489     3       |  | 789     378     4       | 5       379     6       | 2       189     19      |  | 569     2       356     | 13489   349     1389    | 5689    7       4569    |  |-------------------------+-------------------------+-------------------------|   | 3       1467    12567   | 129     569     129     | 1679    1249    8       |  | 268     9       1268    | 7       36      4       | 136     5       126     |  | 24567   1467    12567   | 12389   3569    12389   | 13679   12349   124679  |  |-------------------------+-------------------------+-------------------------|   | 267     167     9       | 236     8       2357    | 4       23      1257    |  | 2478    5       12378   | 2349    3479    2379    | 1789    6       1279    |  | 24678   34678   23678   | 23469   1       23579   | 35789   2389    2579    |  *-----------------------------------------------------------------------------*  (9)r1c46 = (QNPx8)r1c46 - (8)r3c46 = (8)r3c7 - (8)r2c8 = (np19)r2c89 => r2c5, r1c78<>9`

Steve K wrote:6) Note the AALS (4789)r1c46. Note the Almost Hub (47)r8c5, with two spokes (7)r82c5, (4)r83c5 => almost Hub spoke rim, with AALS r1c46 as almost rim. We can, equivalently (for the purpose of the elimination it is equivalent) note that we have the quantum ALS(x89)r1c46, whereas x is no more than one of, but possibly none of,(47). With the ALS(189)r2c89, and the newly stronger (8)r3, we have a Wwing with ALSâ€™s:

(9)r1c46 = (QNPx8)r1c46 â€“ (8)r3c46 = (8)r3c7 â€“ (8)r2c8 = (np19)r2c89 => r2c5, r1c78<>9, depth 7

=> (lc9)r1c46 => r3c456<>9

I'm sorry for the omission, but like I said at the start, I'm just trying to understand some new things and I'm likely to make mistakes along the way.

David P Bird wrote:Steve, you have an amazing way of visualising inferences in terms of strong inference sets that often loses me. This gives you an insight into finding eliminations by eye that I can only discover far more ponderously by tracking chains.

This is exactly why I started this thing up! Steve has said that "Quantums are a way to shorthand the relationships between sis. They place sis restrictions into recognizable patterns." The term "quantums" seems to me to be about visualization and better recognizing what any given puzzle is offering. Learning about new and better ways to attack a puzzle is what I'm really after.

Luke
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### Re: Quantums

Luke451 wrote: This is exactly why I started this thing up! Steve has said that "Quantums are a way to shorthand the relationships between sis. They place sis restrictions into recognizable patterns." The term "quantums" seems to me to be about visualization and better recognizing what any given puzzle is offering. Learning about new and better ways to attack a puzzle is what I'm really after.

Nice subject Luke. Will be interested to see where it leads us. In my experience with Steve's many solving innovations, sometimes it's difficult to follow the notation he uses when he's still experimenting, but I can't recall a situation where his overall logic ended up not being accurate.
DonM
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### Re: Quantums

ronk wrote:Correct, but I see nothing that justifies the quantum term. The extra AUR candidates being considered are in one cell, not two. We simply have derived sis (6)r1c9 = (NT689)r6c178.

...I don't think this is proper usage of the quantum term either. Any naked subset should be comprised of cells -- 100 percent. When a "naked subset" has both cell sets and a portion of a hidden set (as in this example), Steve Kurzhal's original hybrid qualifier term seems much more appropriate.

So what is the proper use of the quantum term since afaik, it hasn't been expressed in actual solving by anyone but Steve. (Otherwise, link would be appreciated and IMO, MadOverlord's use of the term in Oct 2005 doesn't count.) Also, he has defined his use of it in his AU forum blog and has been posting quantum-based solutions there & over at the UK forum for some time -some of which are similar to the above examples, but I've never seen this question raised.

Maybe Steve still sees the use of 'quantum' the same way he did suggested alternate names in the Hybrid sets thread (Apr 1, 2008): Again, this is a technique for finding chains. Perhaps the name, hybrid is not the best. Maybe pseudo is better. In my head, the labels are not that darn important.
DonM
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### Re: Quantums

DonM wrote:MadOverlord's use of the term in Oct 2005 doesn't count.

Prior usage by MadOverlord, myself, and others is exactly the basis for my opinion. If the above is your opinion, there's not much for me to say except ...

I'm beginning to regret ever suggesting the quantum term to Steve K. If my suggestion implied all its current usages, mea culpa, mea culpa.

BTW I haven't visited Steve's AU blog in a long long time ... and have virtually stopped visiting the Eureka forum since its search utility broke. [edit: Besides, the activity on this one forum is enough to keep me busy, my wife says too busy. ]
ronk
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### Re: Quantums

ronk wrote:
DonM wrote:MadOverlord's use of the term in Oct 2005 doesn't count.

Prior usage by MadOverlord, myself, and others is exactly the basis for my opinion. If the above is your opinion, there's not much for me to say except ...

You raised the question of the term's usage. I'm quite happy with the term as Steve uses it so the ball's in your court. The basis of raising the question pending some indication of '[yourself] and others' use of it (preferably practical vs theoretical solving examples since the former is what is being discussed) and how that relates to the examples above, leaves it (IMO) unfounded. One thing I do know is that any relationship to the term as used in Oct 2005, considering the [lack of] state of the art of solving at the time, compared to what Steve is doing with these structures now is, well, irrelevant.
DonM
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### Re: Quantums

DonM wrote:
ronk wrote:
DonM wrote:MadOverlord's use of the term in Oct 2005 doesn't count.

Prior usage by MadOverlord, myself, and others is exactly the basis for my opinion. If the above is your opinion, there's not much for me to say except ...

You raised the question of the term's usage. I'm quite happy with the term as Steve uses it so the ball's in your court ... yada ... yada ... yada

You don't seem to understand historical precedence.

Besides, I already gave a reason for objecting to each usage, on a case by case basis. However, my last objection is really to use of the "quantum naked triple" term, where "quantum hybrid triple" would be better, because it has properties of both naked and hidden sets. Even here though, I'm not comfortable with the "quantum" term.
ronk
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### Re: Quantums

ronk wrote:
DonM wrote:
ronk wrote:Prior usage by MadOverlord, myself, and others is exactly the basis for my opinion. If the above is your opinion, there's not much for me to say except ...

You raised the question of the term's usage. I'm quite happy with the term as Steve uses it so the ball's in your court ... yada ... yada ... yada

You don't seem to understand historical precedence.

Actually, what I don't understand is the repeated practice of, in threads where the main subject is an interesting manual solving technique, raising questions of terminology that are, in the grand scheme of things, of little importance, if not irrelevant or which should have been raised elsewhere (and inexplicably weren't). Unless the terms in question can actually to be shown to, in practice, be misleading or confusing then all that raising the question does is unnecessarily detract from the main subject. Which is unfortunate because these types of solving based threads take work and I'd like to see more people be encouraged to post them.

edit: (in response to ronk edit above: I already gave a reason for objecting to each usage, on a case by case basis....): Such as 'but I see nothing that justifies the quantum term.? Based on what? Where is the link to the standardized use of 'quantum' as used in sudoku that you must be referring to. Otherwise, raising the question of the use of the term continues to be irrelevant and an affectation. It's the technique that's important here.
DonM
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### Re: Quantums

DonM, it appears praise of manual solvers and praise of Steve K summarizes your counterpoints. I agree both deserve praise, but neither is a substantive counterpoint IMO.
ronk
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### Re: Quantums

Luke451 wrote: Ex. 2: This one also seems to have one of the recurring themes of these quantums: the combination of an irregular set with an almost pattern. Quantum Naked Triple (QNT) from Steve K here.
Code: Select all
` *-----------------------------------------------------------*  | 239   39    5     | 1     69    8     | 7     246   46    |  | 4     1     29    | 359   5679  37    | 59    26    8     |  | 7     8     6     | 2     59    4     | 1     3     59    |  |-------------------+-------------------+-------------------|  | 36    3567  4     | 8     17    9     | 2     156   367   |  | 289   579   289   | 6     3     17    | 589   1459  47    |  |*3689  3679  1     | 4     2     5     |*89   *69    367   |  |-------------------+-------------------+-------------------|  | 189   2     7     | 59    4     6     | 3     589   19    |  | 5     4     89    | 39    189   13    | 6     7     2     |  | 1689  69    3     | 7     589   2     | 4     589   159   |  *-----------------------------------------------------------* (AUR36)r46c19 => sis[(89)r6c1, (6)r1c9] => (6)r1c9 = (QNT689)r6c178 => r46c9<>6 `

If I'm seeing this right, if both the (89) and the (6) are false, the (36) deadly pattern is forced. The pairs of (89) within the set will force (6) into r6c8, resulting in a naked triple. This move seems pretty accessible, spottable given awareness of underlying patterns; no "chain thinking" required other than for notation.

In practice, this simple move made a 2 step out of what would have been a 3 step solution of the puzzle. One preceding w-wing (producing 3 placements) had brought it to this point:

Adding to the thought behind Luke's comment above: IMO, the particularly attractive feature of these 'quantum' moves is that, not only are they powerful, but contrary to techniques such as AAICs where one has to almost justify their use by restricting them to more difficult puzzles, these make no assumptions (and I don't consider uniqueness an assumption if they're used with AURs) and can be used in ER=7.1 or ER=9.0 puzzles.
DonM
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### Re: Quantums

An aside: My glasses are broken, thus writing/reading is extremely laborious. Please accept that some typos/errors may be invisible to me.
Perhaps the use of the term quantum is overstretched. For this I apologize. Here is an example of a derived group of inferences used. Commonly when one uses a dervied inference, especially those which spring from uniqueness, the shape of the resultant sis is atypical. Hence, the shape of AIC components is also atypical. Below is an example of an atypically shaped Almost X Wing from the .au tough of August 10:

Code: Select all
` *-----------*  |..1|6..|3..|  |.4.|..3|2..|  |...|1..|...|  |---+---+---|  |.5.|...|.43|  |..9|...|1..|  |87.|...|.9.|  |---+---+---|  |...|..8|...|  |..5|9..|.7.|  |..2|..5|4..|  *-----------*    After ssts:     *-----------------------------------------------------------------------------*  | 2579    29      1       | 6       245789  2479    | 3       58      45789   |  | 5679    4       78      | 58      5789    3       | 2       1       56789   |  | 235679  2369    378     | 1       245789  2479    | 56789   568     456789  |  |-------------------------+-------------------------+-------------------------|  | 1       5       6       | 2       789     79      | 78      4       3       |  | 234     23      9       | 3458    345678  467     | 1       568     5678    |  | 8       7       34      | 345     13456   146     | 56      9       2       |  |-------------------------+-------------------------+-------------------------|  | 34679   1369    347     | 34      1346    8       | 569     2       1569    |  | 346     1368    5       | 9       12346   1246    | 68      7       168     |  | 69      1689    2       | 7       16      5       | 4       3       1689    |  *-----------------------------------------------------------------------------* `

The hidden pervious MUG (16) r789c259 => sis [(6)r3c2, (6)r235c9, (16)r8c6]
Add to this MUG sis the following typical sis: (6)r35c8, (6)r568c6, (1)r68c6
One can write a short AIC, with the derived sis in color:
(16)r8c6 = (XWing6)[r5c9=r23c9r3c2 - r3c8 = r5c8 loop] - (6)r5c6 = (HP16)r68c6 => r8c6<>24, ssts to end
It would be nice to just write the AIC as follows:
(16)r8c6 = (QXW6)r5c9r89c9.r3c2 , r35c8 - (6)r5c6 = (HP16)r68c6 => r8c6<>24, sstste
The dot in this case denotes the partitioning of the MUG sis. the comma used to seperate the Xwing sis. Q meaning only that a derived inference is being used. However, other suggested descriptive manners are certainly possible.

Edited, in green unforgivable errors in the presentation above. I vow to not post until my glasses are repaired. Please, I am sorry for the confusion caused by my errors.
Steve K

Posts: 98
Joined: 18 January 2007

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