This is kind of a twin puzzle to 8x8=9x9 which I posted few months ago. While the previous one was like the very serious, complicated and hard to understand twin, this one is more like the light-hearted, frisky one (yet still not easy to understand at times).

Huge thanks to polar, Piatato and Samish for their test solves and feedback.

Enjoy the puzzle, I will be happy to read your comments.

**Rules:**

**8x8 grid** - fill in the 8x8 grid with digits 0 to 9 with no repeats in any row or column.

**x-sums** - the outside numbers are x-sum clues, they indicate the sum of the first X cells in the row/column from that direction, where X is the first number in that row/column. Some of the clues (their position/value) must be found by the solver. **All clues are in the dotted area** (never in the corners). If X=9 then the sum includes 8 cells in the grid plus the outside clue on the opposite side.

**killer cages** - numbers in a cage sum to the clue in its top left corner. Numbers cannot repeat in a cage (but digits can, a cage is allowed to contain both 7 and 27 for instance). **No cell inside a killer cage can be empty**.

**regions** - draw ten orthogonally connected regions each containing digits 0 to 9 once each such that each cell in the 8x8 grid and each outside clue is a part of exactly one region and no empty cell is a part of any region. **The digits from the outside clues/killer totals are counted** into the region which contains their cell.

**Word of caution** (from both me and the testers): be careful not to forget the killer totals and also that the x-sums with 0 might allow some unexpected options.
Also note that the x-sums do not take into account the killer totals.

**Examples** (see the picture below):

x-sums

17=3+6+8 (X=3)

56=9+4+7+0+2+8+6+3+17 (X=9)

X=0 represents an empty sum with value 0

killer cages

98=56+32+3+7 (repeated 3 is ok since it is part of 32)

regions

the purple area represents a complete region, it contains

0, 1 and 7 as digits from the 8x8 grid

2, 3, 4, 5 and 6 from the outside clues

8 and 9 from the cage total

**Example puzzle** with five regions and digits 1 to 5 (penpa+ link)

Solve online witw this penpa+ link (answer check on digits/numbers and edges)

Huge thanks to polar, Piatato and Samish for their test solves and feedback.

Enjoy the puzzle, I will be happy to read your comments.

x-sums

17=3+6+8 (X=3)

56=9+4+7+0+2+8+6+3+17 (X=9)

X=0 represents an empty sum with value 0

killer cages

98=56+32+3+7 (repeated 3 is ok since it is part of 32)

regions

the purple area represents a complete region, it contains

0, 1 and 7 as digits from the 8x8 grid

2, 3, 4, 5 and 6 from the outside clues

8 and 9 from the cage total

Solve online witw this penpa+ link (answer check on digits/numbers and edges)

Lösungscode: rows 4 and 5 of the 8x8 grid (16 digits)

Zuletzt geändert am 13. November 2022, 09:22 Uhr

Gelöst von Samish, polar, Piatato, Jesper, KNT, jkuo7, ONeill, Vebby, amarins, wildbush7, lerroyy, Dandelo, MagnusJosefsson, ascension, Nordy, Prutsbeest

Zuletzt geändert am 7. Januar 2023, 10:37 Uhr

**am 7. Januar 2023, 00:08 Uhr von Prutsbeest**

Wow, what a puzzle! At first I had made an error and had to restart. After that the logic was clear and it started flowing well. The end however was hard to crack! Glad to have solved this! Thanks filuta for this gem!

...............

Thanks a lot!

Zuletzt geändert am 29. Dezember 2022, 11:59 Uhr

**am 29. Dezember 2022, 05:53 Uhr von Nordy**

A near-endless barrage of insane interactions… phenomenally good, but also super hard (yet never unfair). New record for number of restarts on a puzzle for me ;P Straight to my favorites!

...............

haha, thanks a lot, Nordy! and congrats on getting there in the end!

Zuletzt geändert am 19. November 2022, 07:43 Uhr

**am 18. November 2022, 14:00 Uhr von MagnusJosefsson**

Amazing! Consistently very challenging but the direction of the puzzle is always clear so you can keep working to push through. Very satisfying solve!

............

Thanks a lot for your comment! Happy you enjoyed it.

Zuletzt geändert am 18. November 2022, 11:47 Uhr

**am 17. November 2022, 18:27 Uhr von Dandelo**

Very nice, but very, very hard.

Would have been easier, if I hadn't forgotten that the number of regions is given ...

.........

Thank you!

Zuletzt geändert am 17. November 2022, 02:31 Uhr

**am 17. November 2022, 01:37 Uhr von lerroyy**

Incredible construction! The puzzle was very hard but I never felt stuck which was very nice. Thanks

.............

Thanks a lot, glad you liked it!

Zuletzt geändert am 15. November 2022, 08:03 Uhr

**am 14. November 2022, 19:28 Uhr von wildbush7**

Amazing!! Absolutely loved this! Not sure if it was the word of caution, but this puzzle seemed to flow rather nicely for me.

.......

Happy you enjoyed it so much, thanks for your comment!

**am 13. November 2022, 09:19 Uhr von filuta**

expanded the word of caution part

Zuletzt geändert am 12. November 2022, 16:15 Uhr

**am 12. November 2022, 16:06 Uhr von ONeill**

Wonderful puzzle once you wrap your head around the complex rules. Thanks!

.........

Thanks a lot!

**am 11. November 2022, 14:51 Uhr von filuta**

added example puzzle

Zuletzt geändert am 11. November 2022, 11:58 Uhr

**am 11. November 2022, 05:15 Uhr von madhupt**

A couple of clarification please.

1. Are the digits of killer cage totals inside(repeat, inside) the grid also counted into the region?

2. Are the digits inside the regions supposed to be appearing ONLY once? E.g. can the x-sum of R1 be 1 and be a part of region with x-sum of R2 (16)?

3. Is every cell of x-sum (whether given or determined and not 0) required to be necessarily part of a region or some of these can be left out also?

Thanks.

.............

1. yes, they are counted

2. no this (your example) is not allowed, all the digits within a region (8x8 grid, killer totals and x-sum clues) must make exactly ten digits - 0 to 9 once each

3. yes, each x-sum clue (even 0) must be part of some region, the cells outside the grid which are not inside any region are considered empty

Thanks a lot. One more clarification. Does this mean that any non-zero cell outside the grid , e,g., 1 in cell outside R1 can be not part of any region and can be considered empty? The only constraint being that such an empty cell cannot be inside a cage? The confusion is arising as an empty cell is not clearly defined in the rule set. The only place it is mentioned is in the example where it is zero. Its non-zero constraint if any is not defined.

........

The empty cells are the cells outside the grid that contain no value in the finished solution of the puzzle. Think about it this way - the finished solution will contain 64 digits in the 8x8 grid, then the digits from the killer totals and the digits from the given x-sum clues, the possible remaining digits which you need in order to draw the 10 regions must be then obtained from valid x-sum clues which are to be located in the dotted area. The rest of the outside cells will remain empty. So for instance as you say the possible 1 clue on the left of 1 in r1c1 will certainly be an empty cell in the finished solution since it can't be connected to any region.

**am 11. November 2022, 00:06 Uhr von KNT**

while i have mild reason to assume the people below saying this was smooth were bribed by filuta, it really was a very fun and memorable solve despite being so difficult. extremely impressive construction, thanks

Zuletzt geändert am 11. November 2022, 12:20 Uhr

**am 10. November 2022, 22:49 Uhr von Jesper**

Awesome puzzle, thanks! Much easier to wrap your head around than 8x8=9x9, but still really challenging :)

.........

thanks, happy you liked it.

Zuletzt geändert am 10. November 2022, 22:26 Uhr

**am 10. November 2022, 22:12 Uhr von Elliptical**

I have a question. At the beginning of the rules it states "fill in the 8x8 grid with digits 0 to 9" which makes me think there are no empty cells, but further down it states "...no empty cell is a part of any region..." which makes me think there are some empty cells. Can you help clear that up? Are there empty cells in this puzzle?

..........

the 8x8 grid must be completely filled in, but some of the cells outside the grid (in the dotted area) may end up being empty

Zuletzt geändert am 10. November 2022, 23:58 Uhr

**am 10. November 2022, 19:28 Uhr von Piatato**

Awesome puzzle! The unconventional nature of the ruleset means that one really needs to be alert — I tried my best, yet at every single possible moment I managed to forget about the cage totals and their many roles in this puzzle. Nonetheless I had a great time solving it, there's so many amazing moments throughout the solve! :-D

Zuletzt geändert am 10. November 2022, 21:05 Uhr

**am 10. November 2022, 19:26 Uhr von polar**

An (almost) approachable 5* Filuta puzzle is not something I thought I'd ever type, but this one comes close despite the bonkers ruleset! Stunning setting - super smooth solve with some marvelous deductions along the way. Thank you :)

......

And I thank you once again for your kind words (and the testing too).

Zuletzt geändert am 10. November 2022, 21:07 Uhr

**am 10. November 2022, 17:45 Uhr von Samish**

Despite the unusual ruleset, for once it was on the easier side of 5/5 in my opinion (as long as the words of caution are read carefully), but also one of the funniest puzzles I've solved and elegant as always.

..........

Thank you, Samish!

Schwierigkeit: | |

Bewertung: | 100 % |

Gelöst: | 16 mal |

Beobachtet: | 4 mal |

ID: | 000BXJ |