The New Sudoku Players' Forum

[Wecoc]

I don't know how to rate this exactly but this is probably the hardest Killer Sudoku I have ever made... and still, has the shape of the petals of a beautiful and innocent rose.
This is the real Flower Power!

Code: Select all

.-----.--------.-----------.
|24   |25      |26         |
|  .--'  .--.--'-----.--.  |
|  |     |20|19      |26|  |
|  :-----:  '-----.  |  '--:
|  |20   |        |  |     |
|  |  .--+--.-----+--'--.  |
|  |  |20|13|9    |20   |  |
:--:  |  |  :--.--:  .--:  |
|24|  |  |  |1 |5 |  |20|  |
|  :--'  :--'--:  |  |  :--:
|  |     |17   |  |  |  |26|
|  '--.--+-----'--+--'  |  |
|     |21|20      |     |  |
:--.  |  '-----.  :-----:  |
|24|  |        |  |25   |  |
|  '--'-----.--'--'  .--'  |
|           |        |     |
'-----------'--------'-----'

Try it on Sudoku Wiki here

[Wecoc]

Has someone tried this? I wasn't joking when I said it was hard, am I right?

First steps: Show

Up to this point it's easy but from there it becomes full madness.

SumoCue can solve it by guessing, but it takes more than 10 minutes to get the right solution (by the way, that's how I checked there was only one valid solution)
SudokuSolver directly crashes when trying to solve this, so there's no way to get the SSscore of the puzzle but I'll say it has to be over 4.0

I've tried looking for an alternative strategy to solve this but I can't find anything ingenious.
For example, by hand you can get many houses (some more obvious than others), but none gives much valid information.

[Mathimagics]

You are indeed right, it seems … this one is very hard indeed

SS (v3.6.6) appears to have an issue with single-cell cages, but will happily accept the puzzle if you remove the central cell cage (see definition string below).

But the solver makes very little progress - it identifies the "Sybil Fawlty" cases (the "1" in the central cell and the 4 naked pairs in the central box), then gives up.

Thus no SS rating is possible, sad to say.

SS puzzle def:

Hidden Text: Show

3x3::k:6150:6150:6401:6401:6401:6658:6658:6658:6658:6150:6401:6401:5123:4868:4868:4868:6661:6658:6150:5120:5120:5123:5123:5123:4868:6661:6661:6150:5120:5127:3336:2313:2313:5130:5130:6661:6155:5120:5127:3336:0000:1299:5130:5134:6661:6155:5127:5127:4367:4367:1299:5130:5134:6672:6155:6155:5393:5138:5138:5138:5134:5134:6672:6156:6155:5393:5393:5393:5138:6413:6413:6672:6156:6156:6156:6156:6413:6413:6413:6672:6672:

I can confirm that, by my solver, it does have a unique solution:

Solution: Show

Code: Select all

821376945795481362634529718183654297956712834247893156519267483478135629362948571

[tarek]

Confirmed single solution too!

[Mathimagics]

SumoSue can solve it by guessing, but it takes more than 10 minutes to get the right solution (by the way, that's how I checked there was only one valid solution)

I assume that should read SumoCue?

Finally found the trick to installing this (needs Windows Feature ".NET 3.5" enabled), and on my system SumoCue takes about 5 minutes to solve this puzzle …

I might try to beat that, if at all possible ...

[Wecoc]

I assume that should read SumoCue?

Yes, sorry I misspell. Thank you for testing this! And thank you too, tarek!

Those are some houses I found playing a bit with that shape but as I said none gave me enough information to remove candidates without guessing anyways.

Houses: Show

Those are fairly easy, using the two regions forming a rectangle with height (or width) 2 and knowing the 2 implicated rows (columns) add up to 90 altogether.


Same with other two regions that combined form a shape with height (or width) 2.


This one is more tricky. Let's focus on the red one; we know first row adds up to 45 and we know the sum of the regions that touch that row so we can substract that to get the sum on the red region.


What if we do the same but using boxes (the ones in the corners) instead of rows or columns? Well, then we get this first one.
The second one is using the L shape box that is completely inside of a box so we also know the sum of the rest of the box.


You can see the previous ones affect the same cells, so adding for example the green and the yellow houses from the first and removing the yellow one from the second you get a similar case but with width (or height) 2 which is more convenient.
You can also get those same houses using the same thing I used on the subtraction of the row (column) but with 3 rows (columns) instead of only one.


Finally using same substracting trick as before plus knowing the sum of some involved cages in the middle box, we get this. These final houses start to be a bit more useful to remove candidates, but sadly still not enough as far as I tried.

[Mathimagics]

Nice work!

The identification of hidden (implicit) houses is, as you have found, not necessarily a major benefit for P&P solvers - but it can be extremely useful for software solvers!


I do have a fast Sudoku solver that can use arbitrary/hidden houses, but it will take some time to create a Killer-capable version. I'm currently tied up on another project (as usual ) but will report any progress when it happens!

[creint]

You have nice examples of hidden regions, a solver could identify them all if there are more. Can you also provide the sum values of those extra regions?

[Wecoc]

Houses: Show

Note last ones don't seem much of an improvement but one of their cells always has only 2 candidates (23, 45, 67, 89)

[Mathimagics]

Ok, I've only had half an eye on this, so I missed something important!

I interpret both "house" and "region" as meaning a set of cells that must all be different values. So a region plus a sum gives you a cage.

These implicit-sum examples are not regions by this definition, and thus not as useful (for software solvers) as I first imagined.

Perhaps pseudo-cage might be a better term?

[Wecoc]

Yes, you are right. I wasn't sure what terminology should I use since the Killer Cage Convention applies on the cages but not on these extra regions.
Maybe KCC vs. non-KCC cages or something like that would be more explicit

Even with that, sometimes these pseudo-cages can be useful to eliminate candidates.
For this particular 'monstuous' puzzle nothing else I tried seems to work besides guessing.

[tarek]

I had a play with creating “the most difficult killer” before. I didn’t know how to rate them beyond a point and relied on processing time (the length of time requires to solve)

Tarek

[Mathimagics]

It's very much like one of my diabolical Kakuro puzzle examples. Kakuro's can be made that are effectively impossible to solve by P&P, and this is probably a Killer Sudoku in the same class.

It is noteworthy for that very reason!

I had thought previously that this was unique to Kakuro's, but it makes sense that it could also apply to Killer Sudoku. They are, after all, closely related.

A challenge therefore suggests itself - making puzzles that are even worse! How far can we go, making valid puzzles that take even more time for software solvers?

[Mathimagics]

I had a play with creating “the most difficult killer” before

Now why does not surprise me?

I didn’t know how to rate them beyond a point and relied on processing time (the length of time required to solve)

It's probably the only basis on which we can compare them ...

[Mathimagics]

JSudoku takes 5m 30s to solve that puzzle, so that's very similar to the time I had for SumoCue.

Curiously, JSudoku reports "Unique solution found with 35 guesses" ??