The New Sudoku Players' Forum

by **SpAce** » Mon Jan 28, 2019 12:57 am

Related to this discussion, I have a question about the January 20 puzzle. I can usually figure out pretty quickly if a puzzle is a BUG+n or not and which ones the plus-candidates are, but this one is giving me a headache. Seems like it has two different BUG possibilities. Is that even possible? If so, how can it be exploited? Can they form together a big MUG or something, like two overlapping URs do? Or is it simply a case of no-BUG, as I originally deemed it? Or have I just overlooked something? I'm a bit confused and out of my depth here, so I would appreciate if someone could provide some clarity.

Here's as far as I get without ambiguity (or so I think):

Code: Select all: .--------------.------------------------.------------. | 2 1 9 | 4 8 5 | 36 7 36 | | 8 6 3 | 9 1 7 | 5 2 4 | | 7 45 45 | 26 3 26 | 8 9 1 | :--------------+------------------------+------------: | 15 9 6 | 7 25 13+2 | 4 8 23 | | 3 45 8 | 1(26)+5 9 4(26)+1 | 12 56 7 | | 14+5 2 7 | 35+16 46+5 8 | 13 56 9 | :--------------+------------------------+------------: | 9 8 1 | 3(26) 7 3(26) | 26 4 5 | | 6 7 2 | 15 45 14 | 9 3 8 | | 45 3 45 | 8 26 9 | 7 1 26 | '--------------'------------------------'------------'

Those four (26) combos are the problem. As far as I can see, I could arrange them in two ways and get two different BUG+11s:

BUG+11 (a):

Code: Select all: .--------------.------------------------.------------. | 2 1 9 | 4 8 5 | 36 7 36 | | 8 6 3 | 9 1 7 | 5 2 4 | | 7 45 45 | 26 3 26 | 8 9 1 | :--------------+------------------------+------------: | 15 9 6 | 7 25 13+2 | 4 8 23 | | 3 45 8 | 12+56 9 46+12 | 12 56 7 | | 14+5 2 7 | 35+16 46+5 8 | 13 56 9 | :--------------+------------------------+------------: | 9 8 1 | 36+2 7 23+6 | 26 4 5 | | 6 7 2 | 15 45 14 | 9 3 8 | | 45 3 45 | 8 26 9 | 7 1 26 | '--------------'------------------------'------------'

BUG+11 (b):

Code: Select all: .--------------.------------------------.------------. | 2 1 9 | 4 8 5 | 36 7 36 | | 8 6 3 | 9 1 7 | 5 2 4 | | 7 45 45 | 26 3 26 | 8 9 1 | :--------------+------------------------+------------: | 15 9 6 | 7 25 13+2 | 4 8 23 | | 3 45 8 | 16+25 9 24+16 | 12 56 7 | | 14+5 2 7 | 35+16 46+5 8 | 13 56 9 | :--------------+------------------------+------------: | 9 8 1 | 23+6 7 36+2 | 26 4 5 | | 6 7 2 | 15 45 14 | 9 3 8 | | 45 3 45 | 8 26 9 | 7 1 26 | '--------------'------------------------'------------'

That doesn't seem to make sense, but does it? Or have I made a rookie mistake somewhere? That's totally possible as I'm doing this manually, plus I don't often dig into other than obvious and useful BUG possibilities. This one is clearly not very useful, even if does have a BUG (or two), so take it as an academic exercise.

by **eleven** » Mon Jan 28, 2019 8:08 pm

SpAce wrote:Seems like it has two different BUG possibilities. Is that even possible?

Why not ? Actually i tried to find such one, when i read the other thread, but did not in the samples i looked at. So thanks for this one.
A BUG is just any collection of bivalue cells with strong links in all units. So in a grid with many candidates there should be a number of BUGs hidden.

by **SpAce** » Tue Jan 29, 2019 3:38 am

eleven wrote:
SpAce wrote:Seems like it has two different BUG possibilities. Is that even possible?

Why not ?

I guess you're right... why not indeed? Somehow I've just been fixated on a binary idea that either a puzzle has a single BUG or no BUG at all. I'm still betting that it works for all practical purposes (unless someone shows how to use multi-BUGs), but otherwise it doesn't seem to be correct.

Actually i tried to find such one, when i read the other thread, but did not in the samples i looked at. So thanks for this one.

I looked over all January puzzles and found one more: January 1. It's very similar to the Jan 20 example, having two possible configurations:

Code: Select all: .--------------.----------------.------------------. | 3 14+5 25 | 18 6 59 | 24 89+4 7 | | 6 8 9 | 2 47 47 | 5 1 3 | | 12 45+1 7 | 18 59 3 | 6 48+9 29+4 | :--------------+----------------+------------------: | 12 6.15 25 | 7 8 6.15+4 | 3 49 49+1 | | 4 6.15 3 | 9 25 6.15 | 8 7 12 | | 9 7 8 | 3 24 14 | 12+4 5 6 | :--------------+----------------+------------------: | 5 9 4 | 6 1 2 | 7 3 8 | | 7 2 1 | 4 3 8 | 9 6 5 | | 8 3 6 | 5 79 79 | 14 2 14 | '--------------'----------------'------------------'

Candidates preceding a dot are fixed BUG candidates; the indeterminate ones are the four (15)-pairs.

Then there was this: 17December18. If it has BUGs, it has many, but I won't even try to enumerate them manually. My detection process got stuck got quite early:

Code: Select all: .---------------------.-----------------.----------------. | 48+1 5 9.148 | 2 6 9.18 | 7 3 148 | | 37 9.18 6.148+9 | 17+9 5 3.18+9 | 2 49+1 6.148 | | 37 2 6.18+9 | 79+1 4 3.18+9 | 5 19 6.18 | :---------------------+-----------------+----------------: | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | :---------------------+-----------------+----------------: | 18 9.18 9.18 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | '---------------------'-----------------'----------------'

How many BUGs (if any) does that have?

A BUG is just any collection of bivalue cells with strong links in all units. So in a grid with many candidates there should be a number of BUGs hidden.

But what exactly do you mean by hidden BUGs? Do you count those BUGs that would become possible after some cells are solved (or candidates eliminated)? Otherwise it seems to me that a fixed PM state either has BUGs (i.e. BUG+Ns) or not, and in most cases it's easy to determine. However, a grid with no BUGs now (i.e. can't form a valid BUG+N with the current PM state) can turn into one with BUGs, so in that sense it has hidden BUGs. For example, I don't think any of the BUG puzzles in my tests were in a BUG+N state before basics were executed. Are some puzzles inherently BUG-free, though? With manual testing this kind of research gets difficult.

Anyway, here's my manual (i.e. error prone) results for the January(+one) puzzles, using the grid state after basics:

No BUGs: 19/29
One BUG: 7/29
Two BUGs: 2/29
Indeterminate (possibly >2 BUGs): 1/29

Individual Results: Show

Can anyone's software confirm those results?

by **eleven** » Tue Jan 29, 2019 6:06 pm

SpAce wrote:How many BUGs (if any) does that have?

This one was easy to find, but don't ask me, how many there are.

Code: Select all: +-------------+-------------+-------------+ | 48 5 49 | 2 6 19 | 7 3 18 | | 37 19 68 | 17 5 38 | 2 49 46 | | 37 2 16 | 79 4 38 | 5 19 68 | +-------------+-------------+-------------+ | 6 18 48 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | +-------------+-------------+-------------+ | 18 89 19 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | +-------------+-------------+-------------+

by **SpAce** » Tue Jan 29, 2019 11:35 pm

eleven wrote:This one was easy to find, but don't ask me, how many there are.

Awesome, thanks! So with the plus-candidates it looks like:

BUG+24:

Code: Select all: .--------------------.----------------.----------------. | 48+1 5 49+18 | 2 6 19+8 | 7 3 18+4 | | 37 19+8 68+149 | 17+9 5 38+19 | 2 49+1 46+18 | | 37 2 16+89 | 79+1 4 38+19 | 5 19 68+1 | :--------------------+---------------+-----------------: | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | :--------------------+----------------+----------------: | 18 89+1 19+8 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | '--------------------'----------------'----------------'

I guess that's indeed solvable normally as a BUG+24, though I'm not going to try. Reduced to the minimum it's easy:

BUG+5:

Code: Select all: .----------------.-------------.-------------. | 48 5 49+1 | 2 6 19 | 7 3 18 | | 37 19 68 | 17 5 38 | 2 49+1 46 | | 37 2 16+8 | 79+1 4 38 | 5 19 68 | :----------------+-------------+-------------: | 6 18 48 | 5 7 14 | 3 2 9 | | 9 7 2 | 4-1 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | :----------------+-------------+-------------: | 18 89+1 19 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | '----------------'-------------'-------------'

(1)r1c3 - r1c6 = (1)r4c6
(1)r2c8 - r6c8 = (1)r5c9
(1)r3c4
(1)r7c2 - r4c2 = (1)r4c6
(8)r3c3 - (8=4)r4c3 - (4=1)r4c6

=> -1 r5c4; stte

So, can we conclude that in a multi-BUG situation it's enough to find/pick one of them and solve with that? (That is, if one really wants to use a complicated BUG solution -- again, this is more or less academic.) Originally I had a misconception of having to somehow combine all of the BUG variants, but I guess that's not necessary at all (and feels like a really dumb idea right now). A BUG is a BUG, and its derived SIS is valid, even if others are present, right? Seems perfectly logical now but somehow it didn't feel intuitive at first.

PS. I'm still wondering if there was some way to use the multi-BUG as a unit to reduce the list of plus-candidates, so that the combination would actually be simpler to solve than any of the individual BUGs. That'd be kind of cool, but I'm terrible with even regular MUGs.

Added. It should actually work, shouldn't it? At least in the simple case of two BUGs, if you remove all the known plus-candidates shared by both, don't you end up with a situation that can only be one or the other BUG, i.e. impossible? Doesn't it mean that we can simply use those shared plus-candidates as a SIS? If that's true, shouldn't the concept scale to bigger multi-BUGs as well (it's just harder to see if you have one or not)? Again, I'm just thinking aloud and it's totally possible that I'm missing something obvious here...

by **SpAce** » Wed Jan 30, 2019 1:27 am

In other words, I'm asking if this should be an acceptable solution to the January 1 puzzle:

Code: Select all: .--------------.----------------.------------------. | 3 14+5 25 | 18 6 59 | 24 89+4 7 | | 6 8 9 | 2 47 47 | 5 1 3 | | 12 45+1 7 | 18 59 3 | 6 48+9 29+4 | :--------------+----------------+------------------: | 12 6.15 25 | 7 8 6.15+4 | 3 49 49+1 | | 4 6.15 3 | 9 25 6.15 | 8 7 12 | | 9 7 8 | 3 24 14 | 12+4 5 6 | :--------------+----------------+------------------: | 5 9 4 | 6 1 2 | 7 3 8 | | 7 2 1 | 4 3 8 | 9 6 5 | | 8 3 6 | 5 79 79 | 14 2 14 | '--------------'----------------'------------------'

BIG-MUG+8

(1)r3c2 - (1=25)b1p73
(1-9)r4c9 = (95)r3c95
(4)r3c9,r6c7 - (4=25)r1c73
(4)r4c6 - r4c89 = r6c7 - (4=25)r1c73
(5)r1c2
(9-8)r1c8 = r3c8 - (8=12)r3c41 - (2=5)r1c3
(9)r3c8 - (9=5)r3c5

=> -5 r1c6; stte

(Would you like fries with that?

)

-----------------------------------------------

What about this for the January 20:

Code: Select all: .--------------.----------------------.------------. | 2 1 9 | 4 8 5 | 36 7 36 | | 8 6 3 | 9 1 7 | 5 2 4 | | 7 45 45 | 26 3 26 | 8 9 1 | :--------------+----------------------+------------: | 15 9 6 | 7 25 13+2 | 4 8 23 | | 3 45 8 | 1.26+5 9 4.26+1 | 12 56 7 | | 14+5 2 7 | 35+16 46+5 8 | 13 56 9 | :--------------+----------------------+------------: | 9 8 1 | 3.26 7 3.26 | 26 4 5 | | 6 7 2 | 15 45 14 | 9 3 8 | | 45 3 45 | 8 26 9 | 7 1 26 | '--------------'----------------------'------------'

BIG-MUG+7

(1)r5c6 - r5c7 = (1)r6c7
(1)r6c4
(2)r4c6 - (2=31)b6p47
(5)r5c4,r6c5 - (5=2)r4c5 - (2=31)b6p47
(5-4)r6c1 = (452)r684c5 - (2=31)b6p47
(6)r6c4 - (6=54)r6c85 - (4=52)r84c5 - (2=31)b6p47

=> -1 r6c1; stte

-----------------------------------------------

It would really simplify the 17December (+24 >> +9), but I have even less idea if this is valid logic or not:

Code: Select all: .---------------------.-----------------.----------------. | 48+1 5 9.148 | 2 6 9.18 | 7 3 148 | | 37 9.18 6.148+9 | 17+9 5 3.18+9 | 2 49+1 6.148 | | 37 2 6.18+9 | 79+1 4 3.18+9 | 5 19 6.18 | :---------------------+-----------------+----------------: | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | :---------------------+-----------------+----------------: | 18 9.18 9.18 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | '---------------------'-----------------'----------------'

BIG-MUG+9

(1)r1c1,r4c3 - (1=4*)r6c1 - b6p(8=6) - r1c9*1 = (4-9)r1c3 = (9)r1c6
(1)r2c8 - (1=4)r6c8 - r5c9 = (49)r58c4
(1)r3c4 - (1=49)r58c4
(9)r23c3 - r1c3 = (9)r1c6
(9)r2c4 - (9=4)r8c4 - r5c(4=9*) - r6c(8=1) - r1c1*9 = (4-9)r1c3 = (9)r1c6
(9)r23c6

=> -9 r8c6; stte

What do you think, eleven, are those valid SIS or not? In these examples they all did contain at least one true candidate, but is that guaranteed? You're the MUG expert. I'm flying blind here. (Anyone else is welcome to chime in as well, of course.)

Edit: added a missing chain (1r3c4) to the last example.

by **eleven** » Wed Jan 30, 2019 7:54 pm

Good question. We have to be careful.

If a mixed (undetermined) extra candidate always kills the alternative one in the other BUG, a non mixed candidate must be true.
But if you look at these BUG's (from the above sample), a 1r2c3 would kill both BUG's, or e.g. 1r1c3 would kill the first one, and 8r2c3 the second.

Code: Select all: .---------------------.-----------------.----------------. | 48+1 5 49+18 | 2 6 19+8 | 7 3 18+4 | | 37 19+8 68+149 | 17+9 5 38+19 | 2 49+1 46+18 | | 37 2 16+89 | 79+1 4 38+19 | 5 19 68+1 | :---------------------+-----------------+----------------: | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | :---------------------+-----------------+----------------: | 18 89+1 19+8 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | '---------------------'-----------------'----------------' .---------------------.-----------------.----------------. | 48+1 5 19+48 | 2 6 89+1 | 7 3 14+8 | | 37 19+8 46+189 | 17+9 5 38+19 | 2 49+1 68+14 | | 37 2 68+19 | 79+1 4 13+89 | 5 19 68+1 | :---------------------+-----------------+----------------: | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | :---------------------+-----------------+----------------: | 18 89+1 19+8 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | '---------------------'-----------------'----------------'

However this is no proof, that all of the fixed common extra candidates can be false.
We would need a counter-example - and probably a program to find it.

by **eleven** » Wed Jan 30, 2019 9:06 pm

Or we try this:
If we can show, that each mixed candidate is actually part of a BUG (and it seems so), then it should be proven (?), that one of the fixed extra candidates must be true.

Too short of time now ...

by **SpAce** » Wed Jan 30, 2019 10:30 pm

eleven wrote:If we can show, that each mixed candidate is actually part of a BUG (and it seems so), then it should be proven (?), that one of the fixed extra candidates must be true.

I guess that's sort of my hypothesis, but I have no idea of its validity. The more I think about it the more my head wants to explode. I think you're right that a program is probably needed. Finding a positive proof seems difficult anyway, but a single counter-example would quickly disprove it, as you said. That shouldn't be too hard if someone has the infrastructure ready and easily modifiable for such testing.

by **SpAce** » Thu Jan 31, 2019 4:25 am

The December puzzles seem like a more fruitful set to find both normal and potential multi-BUGs:

Code: Select all: No BUG : 16 Single BUG : 10 Multi-BUG? : 6 --------------- Total : 32

Individual results: Show

They're still manually checked, so errors are quite possible, but it seems that at least these might include some more test cases for multi-BUGs:

by **blue** » Thu Jan 31, 2019 6:32 pm

SpAce wrote:you're right that a program is probably needed. Finding a positive proof seems difficult anyway, but a single counter-example would quickly disprove it, as you said.

Here are two puzzles showing counterexamples:

Code: Select all: .86.75...3.......47...........2891.3...5..2...6...1.....21...97..3...4..67..4.... 186475932395628714724913685457289163931564278268731549842156397513897426679342851 .......4.6.38...1...1...5........8.3..7..5......2.8.5.7.23....4.34....258....4.6. 978513642653842719241967538526491873487635291319278456762359184134786925895124367

For the first one: After basics, there are two BUG+31's with 25 shared +N candidates.
The *'s in the PMs, designate +N candidates that are true in the solution.
None of the shared (+N) candidates are true in the solution.

Code: Select all: +----------------+------------------+-----------------------+ | 19 8 6 | 4 7 5 | 39 23+1 12+9 | | 3 29+1 5 | 68+9 19+2* 26+8*| 7 18 4 | | 7 12+9 4 | 39+8 12+9 38+2 | 69+58 58+1 56+189 | +----------------+------------------+-----------------------+ | 4 5 7 | 2 8 9 | 1 6 3 | | 19 3 18+9 | 5 6 4 | 2 7 89 | | 2 6 89 | 7 3 1 | 58+9 4 59+8 | +----------------+------------------+-----------------------+ | 8 4 2 | 1 5 36 | 36 9 7 | | 5 19 3 | 68+9 29 7 | 4 12+8 68+12 | | 6 7 19 | 39+8 4 28+3 | 58+3 35+128 12+58 | +----------------+------------------+-----------------------+ +----------------+------------------+-----------------------+ | 19 8 6 | 4 7 5 | 39 23+1 12+9 | | 3 29+1 5 | 69+8 12+9 68+2 | 7 18 4 | | 7 12+9 4 | 38+9* 19+2 23+8 | 69+58 58+1 56+189 | +----------------+------------------+-----------------------+ | 4 5 7 | 2 8 9 | 1 6 3 | | 19 3 18+9 | 5 6 4 | 2 7 89 | | 2 6 89 | 7 3 1 | 58+9 4 59+8 | +----------------+------------------+-----------------------+ | 8 4 2 | 1 5 36 | 36 9 7 | | 5 19 3 | 68+9 29 7 | 4 12+8 68+12 | | 6 7 19 | 39+8 4 28+3 | 58+3 35+128 12+58 | +----------------+------------------+-----------------------+

For the second one: After basics, there are two BUG+4's that don't share any +N candidates.

Code: Select all: +-----------+-----------+---------------+ | 29 79 8 | 5 1 3 | 26+7 4 67+2*| | 6 5 3 | 8 4 27 | 79+2 1 29+7 | | 24 47 1 | 9 6 27 | 5 3 8 | +-----------+-----------+---------------+ | 5 2 6 | 4 9 1 | 8 7 3 | | 14 8 7 | 6 3 5 | 24 9 12 | | 3 14 9 | 2 7 8 | 46 5 16 | +-----------+-----------+---------------+ | 7 6 2 | 3 5 9 | 1 8 4 | | 19 3 4 | 17 8 6 | 79 2 5 | | 8 19 5 | 17 2 4 | 3 6 79 | +-----------+-----------+---------------+ +-----------+-----------+---------------+ | 29 79 8 | 5 1 3 | 67+2 4 26+7 | | 6 5 3 | 8 4 27 | 29+7* 1 79+2 | | 24 47 1 | 9 6 27 | 5 3 8 | +-----------+-----------+---------------+ | 5 2 6 | 4 9 1 | 8 7 3 | | 14 8 7 | 6 3 5 | 24 9 12 | | 3 14 9 | 2 7 8 | 46 5 16 | +-----------+-----------+---------------+ | 7 6 2 | 3 5 9 | 1 8 4 | | 19 3 4 | 17 8 6 | 79 2 5 | | 8 19 5 | 17 2 4 | 3 6 79 | +-----------+-----------+---------------+

by **eleven** » Thu Jan 31, 2019 7:56 pm

Great, thanks ! Easy to see now, what i have missed.
Could you also tell us, how many BUG's the 17December puzzle has, please ?

by **blue** » Thu Jan 31, 2019 8:22 pm

eleven wrote:Could you also tell us, how many BUG's the 17December puzzle has, please ?

Seven BUG+24's (according to the code).
I can list them if you're interested.

by **SpAce** » Thu Jan 31, 2019 8:43 pm

blue wrote:Here are two puzzles showing counterexamples:

Thanks, blue! Awesome work!! Fortunately there's not much to be disappointed about, because it probably wouldn't have had much practical value as a solving technique anyway. It's good to be sure, though, so we can close the ticket.

blue wrote:
eleven wrote:Could you also tell us, how many BUG's the 17December puzzle has, please ?

Seven BUG+24's (according to the code).
I can list them if you're interested.

Yes, please! That'd be nice. What about December 9 and December 14? With those I also got stuck with quite a few indeterminate candidates left so I suspected they might have > 2 BUGs as well.

by **blue** » Thu Jan 31, 2019 9:20 pm

SpAce wrote:What about December 9 and December 14?

For December 9: 8 BUG+35's
For December 14: 8 42 BUG+58's

I checked your full result lists for the January and December puzzles -- good job !

Hidden Text: Show

The BUG+24's for the "17December18" puzzle:

Hidden Text: Show

Code: Select all: +--------------------+----------------+----------------+ | 48+1 5 19+48 | 2 6 19+8 | 7 3 48+1 | | 37 19+8 46+189 | 17+9 5 38+19 | 2 49+1 68+14 | | 37 2 68+19 | 79+1 4 38+19 | 5 19 16+8 | +--------------------+----------------+----------------+ | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | +--------------------+----------------+----------------+ | 18 89+1 19+8 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | +--------------------+----------------+----------------+ +--------------------+----------------+----------------+ | 48+1 5 89+14 | 2 6 19+8 | 7 3 14+8 | | 37 19+8 46+189 | 17+9 5 38+19 | 2 49+1 68+14 | | 37 2 16+89 | 79+1 4 38+19 | 5 19 68+1 | +--------------------+----------------+----------------+ | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | +--------------------+----------------+----------------+ | 18 89+1 19+8 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | +--------------------+----------------+----------------+ +--------------------+----------------+----------------+ | 48+1 5 49+18 | 2 6 19+8 | 7 3 18+4 | | 37 19+8 68+149 | 17+9 5 38+19 | 2 49+1 46+18 | | 37 2 16+89 | 79+1 4 38+19 | 5 19 68+1 | +--------------------+----------------+----------------+ | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | +--------------------+----------------+----------------+ | 18 89+1 19+8 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | +--------------------+----------------+----------------+ +--------------------+----------------+----------------+ | 48+1 5 19+48 | 2 6 19+8 | 7 3 48+1 | | 37 89+1 46+189 | 17+9 5 38+19 | 2 49+1 16+48 | | 37 2 16+89 | 79+1 4 38+19 | 5 19 68+1 | +--------------------+----------------+----------------+ | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | +--------------------+----------------+----------------+ | 18 19+8 89+1 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | +--------------------+----------------+----------------+ +--------------------+----------------+----------------+ | 48+1 5 49+18 | 2 6 19+8 | 7 3 18+4 | | 37 89+1 16+489 | 17+9 5 38+19 | 2 49+1 46+18 | | 37 2 16+89 | 79+1 4 38+19 | 5 19 68+1 | +--------------------+----------------+----------------+ | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | +--------------------+----------------+----------------+ | 18 19+8 89+1 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | +--------------------+----------------+----------------+ +--------------------+----------------+----------------+ | 48+1 5 19+48 | 2 6 89+1 | 7 3 14+8 | | 37 19+8 46+189 | 17+9 5 38+19 | 2 49+1 68+14 | | 37 2 68+19 | 79+1 4 13+89 | 5 19 68+1 | +--------------------+----------------+----------------+ | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | +--------------------+----------------+----------------+ | 18 89+1 19+8 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | +--------------------+----------------+----------------+ +--------------------+----------------+----------------+ | 48+1 5 19+48 | 2 6 89+1 | 7 3 14+8 | | 37 89+1 46+189 | 17+9 5 13+89 | 2 49+1 68+14 | | 37 2 16+89 | 79+1 4 38+19 | 5 19 68+1 | +--------------------+----------------+----------------+ | 6 18 48+1 | 5 7 14 | 3 2 9 | | 9 7 2 | 14 3 6 | 8 5 14 | | 14 3 5 | 8 9 2 | 6 14 7 | +--------------------+----------------+----------------+ | 18 19+8 89+1 | 3 2 7 | 4 6 5 | | 5 6 3 | 49 8 49 | 1 7 2 | | 2 4 7 | 6 1 5 | 9 8 3 | +--------------------+----------------+----------------+

---

For kicks ...

For anyone to verify/refute: this puzzle has 5643 BUG+66's (after basics).
It was the largest count in a ~30 min search, using randomly generated puzzles.

Code: Select all: .85.......4.9...17.9.4...5.4.7.....3....5.....1....64.9..2.837....6...8.......... +-------------------+-----------------+-----------------+ | 1267 8 5 | 17 1267 1267 | 49 3 49 | | 236 4 236 | 9 2368 5 | 28 1 7 | | 1237 9 123 | 4 12378 1237 | 28 5 6 | +-------------------+-----------------+-----------------+ | 4 25 7 | 8 1269 1269 | 15 29 3 | | 2368 236 23689 | 13 5 4 | 7 29 18 | | 2358 1 2389 | 37 279 279 | 6 4 58 | +-------------------+-----------------+-----------------+ | 9 56 146 | 2 14 8 | 3 7 145 | | 1235 2357 1234 | 6 13479 1379 | 1459 8 12459 | | 1238 237 12348 | 5 13479 1379 | 149 6 1249 | +-------------------+-----------------+-----------------+

Added: Supressing the console ouptut, I got some larger counts in a 10 min run ...

Code: Select all: ..62.....2.......3.35.8..9..5.....3.6.7.....4...4.2....8.6..9.....1.......3..964. : 7785x BUG+87 ......4...6.4.85.........964....68.....7.3.2.......65..526.......3..9...74..3...8 : 9492x BUG+86 .8.1..5.9....5..78..6..7.4.1....2....65..8..1..25..4...3...5........4....9.....3. : 15482x BUG+106 .1.....37...4.1.854.........47.3...2.5.1................2..7....8..6.5..9..84..6. : 60968x BUG+99

The New Sudoku Players' Forum

Double-BUG???

Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???

Re: Double-BUG???