# Mega-puzzle of many number puzzles: Nikoli Octathlon

This puzzle has been my Moby Dick of puzzles for so long. It's from MellowMelon's old puzzle site and is a Nikoli-based megapuzzle with eight sub-puzzles. I'm copying the puzzle image from there so as to follow the rules for this site, and I don't know the answer to the puzzle (I've only gotten about halfway through myself, after many many hours of attempts). I would especially love to see details about the logic you used to solve the puzzle.

Original source here.

And, I'll also copy the rules over from that site:

Akari (depends on Hitori, dependee of Shikaku) The walls are not given to you in this puzzle. The black cells in the solution to the Hitori puzzle are exactly the locations of the walls. Any wall with a number represents a typical numeric clue in Akari. Any number without a wall is a fake clue and should be ignored. Light bulbs are allowed to be placed on top of fake numbers.

Shikaku (depends on Akari, dependee of Heyawake) A lot of extra numbers are given in this puzzle. Overlay the light bulbs in the Akari solution on this puzzle. Only those numbers which have a lightbulb on top of them are true clues in this puzzle. All other numbers should be ignored; they may or may not denote the correct size of the rectangle that contains them.

Heyawake (depends on Shikaku, dependee of Nurikabe) The rooms are not given to you in this puzzle. The rectangles in the solution to the Shikaku are where the rooms are. Only those numbers appearing in the top left cell of a room should be used as clues. All other numbers should be ignored.

Nurikabe, featuring Kurodoko (depends on Heyawake, dependee of Fillomino) A Nurikabe and a Kurodoko puzzle have been superimposed on each other. The solution to the Heyawake puzzle and the Kurodoko puzzle are identical. Each number in this puzzle is the location of a clue in both the Nurikabe and Kurodoko, and the number is the sum of the clues in each of the puzzles (i.e. if there is a Nurikabe 1 and a Kurodoko 6 in the same space, a 7 will appear in this puzzle).

Fillomino (depends on Nurikabe, dependee of Country Road) A lot of extra numbers are given in this puzzle. Overlay the Nurikabe solution on this puzzle. All numbers covered by a black cell in the Nurikabe solution are liars and are not the correct number. All numbers covered by white cells are truth-tellers.

Country Road (depends on Fillomino, dependee of Corral) The rooms are not given to you in this puzzle. Overlay the Fillomino solution; it gives the shape of all of the rooms. Each room will contain no number or one number, which functions as a standard Country Road clue for that room. The number can appear in anywhere in the room.

Corral, featuring Masyu (depends on Country Road, dependee of Hitori) The grid is filled with Corral clues in black or white circles, but not all of the Corral clues are correct. Overlay the solution to the Country Road puzzle on this one. Cross off all of the circled numbers for which that colored circle would not be a valid Masyu clue in the Country Road solution. All of the numbers that are left are the clues to the Corral puzzle. (the numbers that are crossed off may or may not be correct)

Hitori, featuring Slitherlink (depends on Corral, dependee of Akari) A Hitori and a Slitherlink puzzle have been superimposed on each other. The solution to the Corral puzzle and the Slitherlink puzzle are identical. The Slitherlink puzzle gives a number in every square, just as the Hitori does. All of the Hitori clues are numbers from 1 to 10. Each number that appears in this puzzle is the units digit of the sum of the Hitori and Slitherlink clues. For instance, in a space where the Slitherlink has a 3 and the Hitori has an 8, a 1 would be written in this puzzle.

• Note that the source website has a second image with an alternate presentation; I found that version to be very useful when solving this a few years ago. (Unfortunately I don't have my solution/notes around anymore and I'm not going to do it again. Good luck!) Commented Aug 2, 2021 at 16:17

This has been a puzzle on my own 'hit list' for several years, but I had never made much progress. Thanks for posting it here, as I have now taken the plunge good and proper, and finally (after several false starts and dead ends) been able to push it all the way to completion! Using the alternative image provided on the host site (which makes it much easier to deal with the puzzles that have been overlaid on each other), the final solved state of the puzzle should look like this:

(Note here that I have merely used the three 'superimposed' puzzle grids as a reference for tracking progress in the constituent sub-puzzles - it is those other puzzle-specific grids where you will find the solutions.)

A few practical points on how I solved this:

• To get going, I found it easiest to print off the puzzle and attack it with pen-and-paper. In fact, I used a four-colour pen to help differentiate different aspects of the puzzles (shaded vs unshaded cells, cell annotations, Akari bulbs and non-bulbs, etc.).

• Whenever I found I had made a mistake and couldn't easily backtrack, I would print out another one and start over again. With this in mind as my plan, I kept a Word document alongside, in which I listed highly detailed instructions to help me recreate my progress without having to think from scratch. In all, I had to restart the puzzle 3 times, the final time when it was 98% complete (boy, am I glad I kept those instructions!). At that point I scrapped the paper approach and started to solve it in MS Paint instead, saving images at key interim timepoints to enable me to 'undo' any further errors.

• In all, this puzzle took me 2 weeks to solve, working on it most evenings (making sure to take some of them off altogether, in order to have a healthy, balanced family life also!). A lot of that time was probably absorbed by the simultaneous write-up itself.

Shortly, I will present in this answer the full instructions I kept. I have divided them into 20 steps with illustrations of progress so far at the end of each, for ease of following them. The logic in the instructions is very near complete, but occasionally I may add a marking to a cell in a grid without explicitly mentioning it - usually, these are straightforward knock-on deductions involving unshaded cells adjacent to shaded cells in the Akari, Hitori, Heyawake or Kurodoko, or cells that are confirmed inside/outside the Corral after updating the loop.

My own notation is fairly consistent across the puzzles:

• Unshaded cells are coloured cream, shaded cells are coloured dark grey.

• Clues found to be true in Akari, Shikaku, Heyawake and Fillomino appear in green circles; false clues in Shikaku and Fillomino are crossed through with a grey line (I started doing this in Heyawake too, but dropped this pretty quickly as being overkill). False Fillomino clues later renumbered correctly appear with the correct number in front of a dark grey background. In Masyu, however, true/false clues are represented by dark grey/light green shading, to avoid adding yet more circles into a circle-dependent puzzle type.

• Walls between rooms are thick, black lines; loop segments are green and run straight through the middle of cells. I refer to Slitherlink and Corral walls as 'loop' or 'wall' segments interchangeably.

• One thing to bear in mind that is not necessarily immediately obvious to all from the instructions is that cells adjacent to Akari bulbs or adjacent to illuminated cells inline with a bulb cannot themselves contain a bulb. I mark all 'non-bulb' spaces with a small grey square in the top-left corner.

Also, note - not explicitly listed in the original instructions - that each Shikaku room must contain a single, correct clue. This is an important assumption for being able to solve the puzzle - no rooms exist without a clue.

At this point, you can choose whether to read on and immerse yourself in all the puzzle logic. Personally, I would strongly suggest you don't unless you are following the puzzle along yourself, solving as you go - it's heavy going otherwise, and you probably need to be invested in the puzzle yourself to feel any benefit from doing so. Ultimately, I think you would get the most benefit if you go away and try and solve this puzzle yourself first! Feel free to come back if you get stuck and use this answer as a 'hint sheet' to help you make progress when you can't see the way forward.

Of course, you're more than welcome to keep reading if you wish - but it's at the risk of your own sanity and time management!

Finally, huge kudos to Palmer Mebane, aka MellowMelon, for creating this masterpiece in the first place. When attempting to solve this puzzle in the past I have been unable to find any resource to help me when stuck (no solution write-up or video appears to exist already, as far as I could see), so I hope that this answer might help others who are searching the web for advice on how to solve this excellent puzzle, and come to an appreciation of its brilliance. (I know I have...)

I am of course happy to be contacted regarding possible errors in my write-up (or the solution itself - although that might actually make me cry...!), or for further clarifications of any points. In the case of the latter it may be better to ping me in The Sphinx's Lair, as it will probably take a bit of explaining.

# Full logical solution path:

Okay, a little further ado: Please note that this write-up has had to be presented using screenshots of my text, rather than as text within the body of this answer, as it comprises over 50,000 characters, whereas it turns out there is a character cap in this answer box of 30,000! My apologies to those with accessibility needs for whom this may be a problem - I will gladly provide a text transcript some other way upon request.

Note that it may be possible to make some inferences in the logic earlier than I have here, or to use alternative logic in some places - what I present below is just the path to the answer that I myself discovered. Very often the findings in one puzzle spin off into both of the ones either side of it, making it hard to focus the mind all ways at once - I have tried to present a solution path that provides focus and clarity, sacrificing speed, for the interested reader's benefit.

STEP 1: Some basic initial annotations:

STEP 2: A step that allows us to make some initial headway:

STEP 3: A significant deduction from Fillomino and Country Road:

STEP 4: Transfer the Country Road path segments into the Masyu…

STEP 5: Some more detailed Akari thinking leads to further Hitori and Slitherlink deductions…

STEP 6: Look now at the Shikaku and Heyawake…

STEP 7: Update the Corral loop:

STEP 8: Update the Masyu again using the updated Country Road loop:

STEP 9: Let’s make some progress in the Nurikabe…

STEP 11: Something else in the Fillomino…

STEP 12: NOW transfer the Corral loop to the Slitherlink…

STEP 13: Transfer Hitori shadings to the Akari:

STEP 14: There’s something else we can do in the Heyawake, with a chain of knock-ons elsewhere…

STEP 15: Update the Masyu, with consequences for Corral, Slitherlink, Hitori and Akari:

STEP 16: More work to be done in the Slitherlink…

STEP 17: Hitori and Slitherlink time again…

STEP 18: Now look at the Heyawake once more (it’s been a while!)…

STEP 19: Almost there now. Turn to the Fillomino:

STEP 20: The final deductions!

Phew, what a ride!

• Well done Stiv for resolving these unsolved questions! Having 25 Revival and 30 Necromancer badges is outstanding: data.stackexchange.com/puzzling/query/77471/… - (once this refreshes on Sunday).
– Tom
Commented Jun 30, 2022 at 18:04
• Thanks @Tom - wow, I knew I had a lot but didn't realise how disproportionate that was for the site as a whole! Have to confess I do like the challenge of wrapping up the old unsolved ones from time to time - clearly I was born for Necromancy :)
– Stiv
Commented Jun 30, 2022 at 18:16
• @Stiv! You're my hero! I've spent countless hours working on this puzzle (as yet unsolved for me) and our logic all matches up until step 4. How is Masyu R5C1 a real clue if the loop through its adjacent squares doesn't fit the rules for white squares? Commented Aug 21, 2022 at 20:06
• @kristinalustig Only one side is required, but it can be both. Think of it as 'at least one side' and that will open many doors to you :)
– Stiv
Commented Aug 21, 2022 at 21:29
• @kristinalustig No worries - and it happens. I can't tell you how many times I mucked up the Country Road rules along the way to solving this (I'd never done one of those before...)!
– Stiv
Commented Aug 21, 2022 at 21:38