While I'm still on a roll, here is an M grade variant of the previous puzzle. Look for a symmetric hidden triple of the type I mentioned below.


is.gd/Leren_490_Four_Kites_X_M

Wow, very nicely constructed. I was stuck for a long time at one solved cell. Used CE, the hint that gives a bit more CE and X logic, then setti is necessary (which is not usual in X-str8ts). Then I still got stuck a bit before finding the unlocking moves. It's very nice how each move is necessary for the following one. And you need the X constraints until the very end.

I think the decisive step for me was to figure out where the 1 column 8 and what implications this will have for column 9. and from there, it just worked out. :)

33'; not too difficult. An early Setti, a conditional pair in row G, and an early single in J1. That clears the relevant hi/low decisions.

After the early setti, Leren's symmetric hidden triple generates a single on the other diagonal. After resolving its implications, there is a big round of singles on both diagonals and finally the impasse is resolved by one last X implication.

I just wonder how Leren managed to create a puzzle with that symmetric hidden triple as the important move on cue! 8-)

@ Jan

The answer to your question about the symmetric hidden triple is that it just happened to be there for this puzzle :) Not so fluky though, it comes up from time to time, so it's not as rare as you might think

Thank you Leren for this very nice one. I like the way it solves very much. I wonder if it can be done without the symmetric hidden triple. With the early Setti, CE, a naked triple in J, a single in G and a naked double in 9 you get quite far. But elimination of 4 from F4 is the unlocking move for me.

@ Olfinho

Check solves as M without the SHT but requires a few more wings.

@Jan and Olfinho
I need some help. What is the "symmetric hidden tripel"?

@ Frank

I can help you out. Early on, with just E1 solved you should find three 4's on the \ diagonal at D4, E5 and F6. Since all three cells can see D6 and F4 you can remove 4 from both cells.

You then get a hidden pair of 4's in E5 and G3 on the / diagonal, which removes 4's from E3 and G5 and solves E3.

Hope this helps.

@Leren
Thank you. Until now a tripel, naked or hidden, consisted of three different numbers for me. So I thought I might have been missing something.

Annual goat donations (and advent anticipation):
I am happy to continue the tradition to donate goats to poor people in the name of our puzzle creators. Thank you all for your pleasant service during the year! I cordially invite Klaus and all the other creators to retrieve their donation certificates from

is.gd/AnnualGoat_Klaus
and from
is.gd/AnnualGoat_creators

to put it under your Christmas trees.
At the same time, I may express my hope and expectation that Santa Klaus may again provide us with a daily calendar during the forthcoming advent period. I, as many others too, enjoyed that provision very much during the last years, and I would be glad if ii could continue in 2019 as well.

Do we already have a name for the following X-puzzle move:
If a certain digit can only appear in two cells Xn or Ym on a diagonal, then this digit can be excluded from cells Xm and Yn.
In words: If a certain digit can only occur in two cells of a diagonal, then it can not occur in the other two corner cells of a square over the section between the two cells on the respective diagonal.
This is logically not very complex, but it is often very helpful. I wonder whether somebody already named this logical figure.

I personally think of them as Apex Candidates - since you are effectively defining a triangle and removing the candidates from one of the apex's.

There are of course other scenarios - such as a binary pair on a row - Xm & Xn, where Xm is on a diagonal, you can remove the candidate along the same diagonal at Yn.

I just call this move a hidden pair on a diagonal. It's important to note that you have to know that the digit, call it N, is required on the diagonal.

If one of the cells is E5 there are 4 elimination cells. eg If N is required in E5 or G7 it can be eliminated from C7, E7, G3 and G5.

Some related moves are :

Symmetric hidden triple on a diagonal. eg If there are 3 N's in B2, E5 and H8 then N can be eliminated from B8 and H2.

Hidden pair in a row or column touching a diagonal. eg If N is required in A7 or G7 it can be eliminated from A1.

Hidden pair in a row or column touching both diagonals. Eg if N is required in H2 and H8 it can be eliminated from E5, B2 and B8.

That's enough for the moment. If I think of anything else I'll post again.

I know what I missed. Symmetric hidden triple in a row or column touching both diagonals. eg if N is required in one of H2, H5 or H8 it can be removed from E5.

I appear to be on a roll this week, so here goes:

is.gd/Leren_490_Four_Kites_X

Very nice one, once again. Thank you. Lots of X-arguments, singles, SI, no "real chains" needed. Yet, in the very end, it seems that the solution is not unique. There are some 3/4 and 4/5 "square" compartments interchangeable. Nevertheless, great fun!

Check if any of the "interchangeable" sets has a cell on a diagonal, or has a clue. If so, it's not interchangeable.

the 4/5 compartment includes the diagonal (top right/lower left), yet it is still interchangeable - as far as I can see it.

@ Toni

Best to detail the relevant cells and I'll have a look at it

All cells can be solved uniquely except for B679, D6789, E78, F24, H2479. Potential values are only between 1-5. I assume you then use a UR argument that gives you F4=1. Then you get to one of the solutions in a unique way. Yet, if you say F4=4/5, i.e. D6=1, you get for example the 4/5 square in the 4 fields FH24 which cannot be resolved uniquely.

sorry Leren and all that went through the above conversation, I am stupid - the diagonal isn't correct in the version with F4=4/5. Hope you don't mind.

@ Toni

No worries - I set f6 = 1 as a clue and got a no solution, so I think that all is well

Leren, you definitely are on a roll!
This was the toughest X I´ve ever managed. Took me a while to sort out the 6es. That defined hi/lo in the center, but even then I needed countless skewed fishes and VCE to find the solution - thank you!

Thanks. Took me about one hour. Very elegant X-puzzle! First, the 6s decide on many hi/lows, then the 8s, then the 9s, and the rest falls down. 3x3 skewed wings become less difficult to see with practice.

Like jgrab. Excellent challenge. Thanks, Leren!

I don't quite manage to see the skewed 3-wings...I haave done the 6s (I think), can you please geive me an example for the 8s?

OK, after deciding on the two 8s on the /-diagonal with a short chain, I can see the skewed 3-wing on 9 and arrive at the solution (after another skewed 2-wing and some X-eliminations). Still can't see this for the 8s.

@Jan: When the 6 in F is clear: Look for a skewed x-wing on 8 between / and row F.

...but it isn't, and it is only resolved by tackling the 8s!

With apologies to Hone :)


is.gd/Leren_490_White_Widower

Another very nice one! Solved in 12 min, once again with CE and settis. Thank you, Leren.

Thanks, Leren! Easier than hone´s BlackWidow. However, I still needed to SIs on my way. At the start to exclude 7 from J9 and at the some thinking about the 4s in CFH.

Thanks. Like pets, SI on J9. The 4s in CFH, however, are wings. My second HC was on the 1 in row E.

Thanks, Leren, very enjoyable for me. Early exclusion of 7 from J9 then wings and careful setti analysis.

@pets, @jgrab: I'm never sure about the the difference between HC/SI and CE: In this case at some point it's clear that 6 in G89 and that J9=7 would immediately imply B9=6 and J8=6, a contradiczion. I would call that careful elimination, and therefore would rate the puzzle an M.

sfter 12 solved cells it was the 5 in col 9 that helped; one of the two "5" lead immediately to a conflict in col 8, the correct 5 solves the puzzle (says my solver ;-)
(Glad to be no widower yet, no matter what color ...)

@BP,

From my perspective, the difference between CE and SI is clear.

Careful Eliminations (CE) - logic restricted to a single row or column. Generally requires consideration of how multiple compartments on the line interact.

Street Interaction (SI) - logic determined by how two (occasionally three) "streets" (ie. rows/columns) interact.

Head Chains (HC) is a more general term - ultimately ALL strategies can be described in terms of "chains", so both SI's and CE's would be subset's of HC. but people often use the term if the chain is very short but does not neatly fall into another named strategy.

The logic you identify is therefore an SI - as it requires consideration of two columns.

Whether a puzzle is moderate of not is quite subjective. Many people would have more easily spotted the SI you describe, but would not have seen the big-fish in hone's puzzle - even though fish are a more common pattern used to solve puzzles.

@Ben: Thank you very much for your detailed explanation!!!

@Susan: Did you also use CE in the way Ben defines it? In that case I'd be really interested in your solution!

@Ben:
Great explanation! I agree completely to that...

@ Ben:
very good Explanation, iagree entirely...

After deciding J9 - you can also start from the assumption that the large compartment in the column is low, which leads to the contradiction that J9=3 - two wings clear up the 6s, with the additional information (compared to the setti just solving J5) that F6=4 is impossible in the column, which solves. So only one SI/HC is required.

Nice, I used CE, some wings, settis and several SIs

is.gd/hone490_BlackWidow

Wow, that was a nice one! Done in 27 min with settis and CE. No chains. Thank you!

Too hard a spider for me. Borrowed A2 and B1 after desparately seeking Settis.

A very nice one, correct. But I also didn't find really setties from the start, opened the puzzle with a lengthy head chain.

After 24 solved cells just with CE setti 8 solves nicely.

@Susan: Just as pax I didn´t find a proper start with settis. So I started my chain with J1=8. How did you start with settis & CE?
@hone: Rather tough, but enjoyable - thanks!

Though solvable with standards tools, it seems to be a bit challenging. So one can use the following hint: one setti and one sixfish at zero solved cells will lead to the solution.

Saw hone's fish hint too late. I agree with Susan that the puzzle can be solved with VVCE (or tiny HCs?) and careful setti analysis, however it took me much longer than 27min to even solve the first cell. Chapeau!, Susan!!! And thanks, hone, for this hard challenge!

A real masterpiece, thanks a lot, hone!

Thank you hone for the hint. As others I did not find a beginning (I am no fan of trial and error). Even after the sixfish it tookk me a while to find a threewing that was crucial for me.

Also big thanks to hone for this beast!

OK, I believe I have found the initial setti - it eliminates two digits in row J. But I can't find the 6-wing...perhaps a little hint, I could be missing some little elimination?!

@Jan: after CE and the setti and the elimination of 8 from J45 mentioned by yourself, there is a col-six-fish on 8 ...

@Jan: The setti eliminates two 8's in J, leading to a 6-wing for 8 in columns 234578.

Thanks - in retrospect, it's obvious that after eliminating two 8s, I should have looked at that digit! 8-(

But progress from this wing is minimal, still stuck...

There is a reasonable head chain which shows that, no matter what you select for J1, B1-4 must be high. But that doesn't help that much either...

@Jan: Then look for the swordfish which Olfinho has mentioned.

Uh-oh...yes, that is unexpected, and the 6-wing is indeed required to enable it!

is.gd/Leren_490_Kettle_of_Kites_X

Thanks, nice, relaxing, the proper X for a gray day in november

Thank you, Leren.
Not too difficult, nice!

Thanks, Leren! A really nice one, liked the way it develops step by step by repeatedly using diagonal arguments in the middle square.

Thanks. Quite difficult (on paper during a train ride), but doable with X-power.

Lovely - thank you, Leren!
With a naked quadruple in /, a xwing on 5 and six solved cells (B1, B3, C4, G6, G7, G1) I could´t see a simple way: H3 = 7, —> H2 = 69, J1 = 8 —> no 8 in H1-3. Is this a head chain or a SI? Any hints for a more elegant solution?

Early on 6 is easily removed from H2 because of 6 in HJ1.

At 6 solved cells J1 = 8 => H2 = 9 => H1 = 8, so H1 <> 8 and B8 = 8 is a single on the / diagonal. Hope this helps.

yep - I did not take care of 6 in HJ1. Your way stays in the / diagonal and is easier. Thanks...

@pets
You can proceed with x-implications all the way:
- 1 on diagonal must be in H8 or J9, this excludes both 1 and 5 from H9 and J8. For me this elimintes a lof of candidates in the lower right and on the diagonal.
- Most important: D4=78. This restricts the range of both D3-D7 and C4-G4. This creates a naked pair on the diagonal (F4 and D6).
- The pair also excludes 8 from D4 and F6 and both cells along with C3 are solved.
From here on it collapses.

Leren, very nice puzzle indeed! Thank you. Took me ages...

Thanks for the tip, Olfinho - took me quite a while to understand what you meant!

The biggest surprise for me is that eliminating the 1 from C1 is crucial, as it creates the 1 pair in H8J9, as described above, and sets all the other steps in motion.

Nice one. CE, quadruple, wings, X-logic, and some SI using the diagonal.

is.gd/Leren_490_Wake_of_Kites_M

Very nice. Needed no chains but used several UR arguments. M grading justified.

A bit more than M. Mainly a wing cascade, after CE.

Excellent! As wings and a few obvious URs are sufficient, it is clearly a M from my perspective.

I'd call it an M grade. As you say, a wing cascade, as well as a double large gap and a naked quintuple, which is quite rare.

Phew, not easy for me. CE, one UR, wings and the quintuple, but that was not enough. I used some SIs and unicity of J8 to unlock.

For discussion of Andrew's puzzle.

I can't remember having seen a previous puzzle that is so well balanced, only for a hidden pair to get you started. After that, three singles, an x-wing and a setti are all you need (but love).

There were no major difficulties in solving the puzzle. The process was pleasant. A good helper, of course, was Setti.