Cross ALL the Streams

This grid deduction puzzle is a hybrid of three puzzle types: Cross the Streams, Tapa and Star Battle; rules for each are summarized below. In this grid, every cell is to be shaded either red, blue, or purple; a purple cell is to be considered shaded both red and blue in what follows.

The cells shaded red (including purple) form a solution to the Cross the Streams puzzle clued on the outside of the grid. The cells shaded blue (including purple) form a solution to the Tapa puzzle clued inside the grid; as usual with Tapa, cells with clues cannot form part of the Tapa shading, and thus cannot be shaded blue, forcing them to be shaded red. The cells shaded purple form a solution to the Star Battle puzzle clued by the outlined regions in the grid, with one "star" (purple shaded cell) per row, column, and region.

Note that each individual puzzle type may not be uniquely solvable on its own. However, there is only one shading of the grid that simultaneously solves all three; the puzzles are meant to be solved as a whole. I hope you enjoy!

Rules

These rules are extracted/adapted from the puzzles linked in the first paragraph above. Thanks to the original posters of these rulesets.

Cross the Streams

Shade cells red so that:

• Red shaded cells create a single orthogonally connected group. No 2x2 cell area within the grid consists entirely of cells shaded red.
• Numbers to the left/top of the grid represent the groups of consecutive red cells which are in that row/column in order, either from left to right or from top to bottom. (For example, a clue of "3" means the row or column has three consecutive red cells, and a clue of "3 1" means that the row or column has a group of three consecutive red cells followed by a single red cell, separated by at least one cell not shaded red.)
• A question mark (?) represents a group of consecutive red cells whose size is unknown; an asterisk (*) represents any number of unknown groups of red cells, including none at all.

Tapa

Shade cells blue so that:

• Blue shaded cells should form a single orthogonally connected group; no 2×2 cell area consists entirely of cells shaded blue.
• Some cells have clues in them. These cells cannot be shaded blue.
• Clues give the runs of shaded cells in the eight touching cells, in no particular order. (This is like a Nonogram/Picross clue, but instead of a row or column, it 'measures' a square around the clue.)

Star Battle

Shade cells purple so that:

• Each row and column contains exactly one purple cell.
• Each outlined region in the grid contains exactly one purple cell.
• No two purple cells can be adjacent, neither orthogonally nor diagonally.
• Damn it - I got a solution written up right to penultimate step, and then found it was wrong - trying to figure out how far I need to backtrack. The first couple of attempts at backtracking didn't go far enough, as they immediately led to either the same flawed deduction or another impossible case :( Commented Jul 20, 2021 at 12:28
• The solution referred to in the comment above got reduced to a single paragraph, as I completely restarted (albeit a bit quicker the second time, and having to pause overnight with the second attempt "almost done"). As I reached a conclusion that seemed to imply a non-unique solution, it turned out I'd completely ignored one of the rules due to being insufficiently familiar with Tapa. Would be fascinating to know if anyone could get similarly close ignoring one of the other rules! Commented Jul 21, 2021 at 8:12

Solution:

Step-by-step:

We can start with some obvious Tapa and Cross the Streams deductions:

Next, we can look at column 8. It already has one red cell and at least one of the last three cells also has to be red due to the 112 Tapa clue. Therefore, we can colour most of that column blue. This also leaves only one possible place for the 7 in the first row.

Some more basic deductions to follow. Also the top right corner can't be purple since it would force two purples in a row.

Now, let us look at column 2. The cells above and under the 5 must be blue to avoid a 2x2 red region. These blue cells must go around the 5 from the left, since otherwise all cells on the right side of the clue would be purple. Similar deductions also for the 6 clue in the same column.

Looking at row 6, the first cell must be purple in order to satisfy the Cross the Streams clue.

Now, R1C2 must also be purple to satisfy the Tapa clue.

After some more basic deductions, the only place left for the purple in the bottom left region is on row 9.

Again, some basic deductions lead us to a position where the only place for the purple in column 5 is on row 8. (Also connectivity for the blue cells would force it because purples in columns 4 and 6 must go on rows 3 and 11/12.)

In fact, the purple on the bottom rows, which allows the connection of the two red cells, must go on R11C4. This is because only one of the cells surrounding the red cells can be purple (the other one in those columns going on row 3), but since we must have a purple in the Star Battle region which contains the red cells, R11C4 is the only one out of those which can connect the two cells to the rest. This gives also the purple in R3C6. In addition, we can deduce some red cells on column 7 for the 6 clue. These together with the 15 Tapa clue on that column force another purple in that column.

Some more basic deductions. The fact that row 6 must contain five separate red regions forces some blue cells around the 33 Tapa clue.

Again, there was only one spot left for the purple on column 8. That gives also the rest of the cells around the 112 Tapa clue.

Now, for connectivity reasons, R6C11 cannot be blue and therefore, due to the Tapa clue, neither can R4C11. Then, the only place for purple in column 11 is on row 7.

For connectivity reasons, the blue cells in the bottom right must continue upwards. Also, we must get one more red region on row 7.

Now, we see that to satisfy the 112 Tapa clue on row 11, either R12C9 or R12C10 must be purple. Also, either R2C9 or R2C10 must be purple, since those are the only possibilities in that region. The only place for purple in column 12 must then be R4C12. Also, due to connectivity, R8C10 and R9C11 cannot both be blue. Therefore, all the other cells surrounding the 7 Tapa clue must be blue.

R3C12 cannot be red because the Cross the Streams clue on that row would force all the remaining cells on that row to be red and it would create a 2x2 red region. However, because of connectivity, R3C11 must then be red forcing also R3C10.

Now, R2C10 must be blue forcing the remaining purples and after that the rest is easy.

• Argh, you've pipped me to the post (quite literally!). Well done for solving +1 I might finish my write-up anyway (I'm halfway through) as I visualised the solve differently and may have a few different deductive steps as a result...
– Stiv
Commented Jul 20, 2021 at 20:21
• That's it! Thanks for the detailed write-up! I did have a question in your logic for R11C4.: why could you not connect R11C5 through R11C6 instead of forcing R11C4? Ultimately you did get the right answer, of course, which proves all. Anyway, hope you enjoyed! Commented Jul 20, 2021 at 20:29
• @JeremyDover That's because only one of the cells surrounding the red cells can be purple (the other one on columns 4/6 must be on row 3) and if we chose R11C6 or R12C6, then there would be no purple cell in that star battle region. I will try to edit it to be more clear. Commented Jul 20, 2021 at 20:37
• Ah, I get it now. Thanks for the clarification! Commented Jul 20, 2021 at 21:11

My word, this was a tricky puzzle! I've been pipped to the post by @user39583 mid-writeup (fair play) but I have visualised the solve differently, using 3 separate grids for each of the 3 sub-puzzles, and it took me all evening anyway, so I shall follow through with my explanation nonetheless in case it is of use to others with a similar way of thinking...

The solved state of the puzzle is:

ARGH, my eyes!! (Apologies for the vividness of the colours - MS Paint defaults, I'm afraid...)

The solution is best explained in lots of little steps - brace yourself for a bit of a marathon. Also, please forgive the occasional slip of overlooking squares that could be shaded out as empty in any individual sub-puzzle - this happened fairly often and it is a monumental effort to go back and create perfection! Instead, what follows is a very honest explanation of my genuine thought process and solution path...

(Remember throughout that if a space is unshaded in the Tapa or Streams puzzle, it must be shaded in the other...)

Step 1:

First, we know that all Tapa clue squares must be red in the Streams grid:

We can then make some fairly straightforward Streams deductions (with knock-on effects in the other grids - I won't always point these out explicitly, as you soon get into a pattern of updating them automatically when you make changes in another of the grids):

And some initial Tapa deductions:

In fact, you can make a few more Tapa deductions at this stage by noting:

That the bottom 2 squares in Tapa C3 cannot both be shaded, or a 2x2 box will be formed - thus at least one of these must be blank in the Tapa grid.

Step 2:

Now turn your attention to Streams C8:

The red square already shaded must belong to the '2'. By contradiction, assume it belongs to the '1'. This would cause all squares below it to be unshaded in Streams and shaded in Tapa, producing a run of 3 shaded cells in the 1/1/2 group at the bottom - this is illegal:

Instead, we know it belongs to the '2' and can declare the 2 cells at the top of the column to be empty in Streams, allowing us to place the '7' in R1. This has a few knock-on deductions by following the no-2x2-blocks and unshaded-in-one-shaded-in-the-other rules:

Step 3:

Now look at the 4's in Stream C3.

We can fix 3 shaded cells for each and also resolve the Tapa 5 in the process...

This then enables us to resolve the Tapa 3 in the top-left corner (we can't use squares beneath it - they would form a 2x2 block - and we can't use both top spaces or we would have 2 purples). And, excitingly, this gives us our first purple star for the Star Battle puzzle too! We can then fully resolve the top-left 3x4 box in all 3 puzzles and the Streams 5 in R3.

Step 4:

We can now resolve the Tapa 6 in the bottom-left, which leaves one possible purple star in C3. We can then fully resolve the bottom-left shape in all grids:

Now R6C1 in Streams must be red, which also resolves another purple star and the whole left-central shape:

Step 5:

At this stage, there are a few 'easy pickings'...

- Tapa R10C6 for connectivity;
- Streams R6C7 to satisfy a row ? block;
- Tapa R8C8 and R10C8 to satisfy the 1/5 clue;
- R2 star battle must be in the rectangle shape, meaning that R2C4 must be blank in the Tapa;
- Resolve Tapa 1/5 in the middle;
- Tapa R8C5 must now be filled for connectivity - but it must be PURPLE!
- R2C6 Tapa for connectivity...

Now R3's purple star must be in the second shape along the top, and C6's star must be in that shape or the second shape along the bottom:

In fact, since for connectivity purposes Streams R11C4 must be shaded, this fixes the purple in position in the bottom shape, and then the one in the top one too. It also resolves the Tapa 6:

Step 6:

Streams R10C10-11 are red because of the '3' in that row. Then I spotted a couple more in Tapa R6 that I could have coloured appropriately already (oops). And we can also shade Streams R12C9-10 for connectivity.

Now look at Tapa 1/1/2 at the bottom... It is impossible to connect the bottom LH corner to the right side by a continuous line of 3 beneath this clue, so we must link via the already blue space in R10C8. Simultaneously, consider the nearby 1/5 clue - if the space beneath the 1/5 is unshaded, then the link must be via R11C7-8 (but then one of the 1's is not connected to the rest of the shaded squares). This is illegal, so the space beneath 1/5 must be shaded and the 1/5 can be resolved. As a knock-on effect, Streams R11C7 must now be shaded, which means R11C8 must be also (since there are only two ? blocks in this row). This resolves the outstanding 1 in Streams C8:

Step 7:

Now there's only one space for the purple star in C8. We can also shade Streams R12C7 due to only having 2 ? blocks in the row.

Now there's only one space for the purple star in C7. Also see that the Streams vertical '6' must be positioned towards the bottom of its column to avoid making a 2x2 block higher up. In a chain of knock-on effects the 1/1/2 at the top of the Tapa ends up getting resolved:

A few more miscellaneous steps:

- Streams R5 needs an empty cell in C10 for the right number of ? blocks;
- We can then shade Tapa R5C10&12 (and R4C10 for connectivity, and then R6C12 by symmetry of the 3/3 clue);
- This leaves only one way to resolve the remaining 2 ?'s in Streams R6;
- This in turn resolves the Tapa 3/3 fully;
- Note also that the 4 in the bottom-right of the Tapa must be connected to the rest up the RH edge (due to the 1/1/2 being impassable);
- This then resolves the final 3 in Streams R10.

Step 8:

Focus now on the bottom rightmost shape in the Tapa. Its purple cell must be in R12C9-10 (as the 1/1/2 needs another blue cell). This makes the bottom-right corner empty, so C10 must be purple and the rest of this box can be resolved across all 3 puzzles:

This leaves just one space for a purple star in R2. We can then place both the other remaining purples as there cannot be a purple star in R4C11 (see the Tapa - I just missed this while colouring the rest). Several knock-on deductions bring us to the following state:

Note now that in Streams either R8C10 or R9C11 must be shaded to link up the shaded squares; this means one of these must be the empty cell around the Tapa 7.

In fact, it must be R9C11, as otherwise both remaining squares in Tapa R9 would need to be shaded for connectivity, which then clashes with the remaining ? block to be resolved in Streams R9:

Step 9:

FINALLY, there is only one way to resolve the remaining corner and the puzzle is solved at last!

• Great solve with a very nice write-up, Stiv! Hope you enjoyed! Commented Jul 20, 2021 at 21:18
• @JeremyDover Oh, I absolutely did :) Took me my whole evening, but it was an evening well spent!
– Stiv
Commented Jul 20, 2021 at 21:31

[I'd paused my solve overnight, and the browser didn't tell me other solutions had been posted when I woke the PC back up - it was later determined that I'd had the somewhat unique approach of completely ignoring one of the rules making for a few detours from the intended solving path until I finally needed to invoke the missing rule at the end]

A solution that seems to fulfil all of the clues given:

I initially tried to use @bobble's Penpa link, but didn't really get on with the interface, so returning to MS Paint, I got most of the way to an incorrect solution...

At which point I realised I'd messed up... both of the '1 1 2' clues aren't met, and the remaining greens are unresolveable...

Using the experience from failing to solve it once, I started again from an empty grid...

I'm using the following colour scheme:

[dark] red/blue : this colour exclusively
[pale] red/blue : either the corresponding colour or possibly purple.
Purple : definitively known to be purple.
[pale] green : NOT purple (may be red or blue)
white/empty : could be anything
coloured blobs : identifying one of a group - especially in cases where one must be purple.
coloured lines : at least one of the cells either side of the line must be the specified colour.

First we can mark the really obvious stuff from the Tapa clues themselves, and the adjacent squares where two of them are close or one is near the edge and sufficiently restricted:

Next, look at the clues for the "Cross the streams" puzzle:

The '4' in column 5 is placeable immediately, several other columns/rows cause some cells to be known red, and others to have a known red adjacent to a given grid line. This also allows a few more blues to be completed to avoid a 2x2 region of red. The "* 6 *" clue is particularly annoying - as either * could represent nothing at all, the 6 could be precisely either half of the grid so it gives us nothing at this stage.

At this point row 11 is particularly significant:

one of the "outer" reds must be connected to the "middle" red, because there are only 2 red regions. This must overlap the blue, and so gives us a choice of two locations for a purple, both of which "see" the two squares above the right hand possibility:

Next, observe in column 4

There must be 3 adjacent red cells, at least one of which must be adjacent to the '6' Tapa clue in column 5. It already has a red cell adjacent to it, and they cannot both avoid overlapping with blue. Indeed, there's only one place to put that '3' at the top of the column to avoid having more than 1 overlap with blue, so this gives us another purple that must be in columns 4 or 6.
Having a kind-of 'x-wing' formation of purples, plenty of other cells can be marked as "not purple" (or definitely red/blue if already known which), row 12 has two red regions identified, and needs the right-hand "red" to be able to connect, and a few other simple deductions to:

In the above image, observe that the cells marked with green ovals

are the balance of the bottom 4 rows - the starbattle must have 4 purple cells in the bottom 4 rows, and 4 purple cells in the bottom 4 regions, so one of the upper "green oval" cells is purple if and only if the lower one is.
However, the lower "green oval" cell sees both remaining possible locations for a purple in column 5 (empty "pink" cells above), so in fact all the green-oval cells must be non-purple.
Also observe in row 6 that one of the "pink" cells must also be purple to allow for 5 blues connected around the '5' Tapa clue above it.

Next steps:

Still in row 6, the "3 3" Tapa clue allows for only two directly opposing non-blue cells, so in particular one of these must be the red for the final '?' of the row 6 clue. The Tapa clue in row 9 has a definite red, to the left, so the other non-blue must be either above or below it, with all remaining cells being at least blue.
Row 12, the purple must be in the right-hand region (this has been sitting here for a while...!), which means the purple on row 11 must be in column 4, also disambiguating some other purples... the immediate effect of this deduction (after marking remaining cells near a known purple as "not-purple") is:

Next clues:

"Cross the streams" row 11 gives us a definite red next to the Tapa 6, so we can complete the positions of blues next to that, which also fixes the Tapa 3 on row 9.
"Cross the streams" row 10 is then somewhat restricted, more importantly, after marking as definite blue the cells which prevent a 2x2 red region near the Tapa 3 and 6, there are only two cells in the lower-left region that could possibly be purple, giving only two possibilities for the starbattle in rows 1 and 3:

We can then disambiguate

which side of the Tapa 5 in row 5 the blue cells are (they are to the left), because of the red in column 3 that became a definite red due to not being one of the two possible positions for a purple...
After ensuring connectivity of reds in the left-hand side without making a 2x2 square, and also respecting the '* 4 4 *' clue in column 3, we reach the following:

The top-left section is easily resolved

as the other alternative for connecting to the red would break the '3' tapa clue. We can also resolve several cells around the '6' tapa clue in row 4, and note that row 5's purple cell is now restricted to one of 3 positions in the same region.

Next steps:

Considering the tapa clues with "1 1 2", these must have at least one red in each adjacent column and row. In particular column 8 with it's "2 1" cross the streams clue must have a red adjacent to the lower "1 1 2" tapa clue, so the red starting on the upper "1 1 2" tapa clue must be of length 2, and there must be no other reds in the column... this also disambiguates the '7' clue for the top row:
Row 7 is also much more restricted, and in particular gives a red in column 7, which reduces the possible positions of the 6 connected reds from the cross-the-streams clue from "* 6 *"
After marking the two possible positions for the purple in the right-of-centre region at the top, it's noticeable that each remaining purple is restricted to a single row within a single region, but none seem definite yet...

It was also around here I went wrong on my first attempt, failing to notice the possibility of a purple just below the upper "1 1 2" Tapa clue.

Further steps:

the "1 5" tapa clue needs the purple to complete the group of 5 blues.
We cannot have purple in row 12 column 8, as the "2 1" clue in column 8 would make 3 "blues" to the left of the "1 1 2" clue in row 11, so the purple in column 8 must be in row 5.
For the "3 3" Tapa clue, if the non-blue cells are diagonal, this would require a purple cell in row 4 column 11. This quickly leads to a contradiction, as the top right becomes fully enclosed in blue so the red cannot connect... so the non-blue cells must be vertical.

At this point, the only possibility for a purple

in column 11 is in row 7, just above the "7" Tapa clue, which must then be surrounded mostly by blue, but can connect left or down with a red if needed (or otherwise place a red anywhere "out of the way"!)
Considering the positioning of the red near the bottom of column 8, there seem to be two possibilities - exploring the red in the lowest position, this forces this layout...

Attempting to connect the disconnected red regions

the only ways legally possible whilst respecting the clues in the vicinity, I reached this near-solution, which fails due to not respecting the "? ? ?" clue in column 9 (variations to make it respect that clue will break others) nor indeed the "? ?" clue in column 12 (this false solution was "forced" by row clues and tapa clues).

place the red in column 8 at row 11, leading to the following, which allows more flexibility because there are fewer disconnected red regions:

Looking at row 3

the red region represented by the second '?' needs to connect to column 11 or to row 4 column 10 in order to come back down to the remainder. This gives us a 2x2 block which must contain a pure blue and a purple above the 2 reds.
Considering the Tapa clue 7 in column 11, this needs to connect either to the reds to its left or the reds below. Either way we get several blues from this...
The '3' in row 10 cannot be at the right, as this would disconnect the red regions, so it must be to the left, so we reach the following:

One way to proceed from here could be:

To loop the red into the right column and also have a red in the bottom-right to fulfil the "? ?" clue in column 12, however due to the two '4' in tapa clues in column 11, this forces 2 purples into column 10 which isn't allowed!

If we choose the bottom-right square to be

blue, the remaining purples are forced, and in order to fulfil the "? ?" for column 12, we need to do this:
at which point the not-purple in row 3 cannot be blue as it would break the tapa clue and cannot be red as it would create a 2x2 block.

Instead, the following layout seems to fulfil all of the clues:

reached by making the bottom-right square red, which forces the positions of all remaining purples, then choosing a layout that gets a "1 4" tapa clue fulfilled and connecting everything together.

Unfortunately,

The puzzle does not seem to have a unique solution - the pair of "not purple" in row 2 must contain at least one blue, but can otherwise be freely chosen without breaking any clues, and the other "not purple" can be arbitrarily chosen as red or blue. From these possibilities, I arbitrarily chose the most aesthetically pleasing to present as the final chosen solution at the top of this answer.

As pointed out by @deusovi, this was because

I'd completely missed the rule "Blue shaded cells should form a single orthogonally connected group; no 2×2 cell area consists entirely of cells shaded blue.", and done the entire solve without relying on that rule at all...
therefore the presented solution is indeed unique as others had found before I saw their solutions.

• Your remark on nonuniqueness isn't correct - you would break blue's connectivity if either was red.
– Deusovi
Commented Jul 21, 2021 at 7:32
• @Deusovi no wonder it was so hard - I'd completely missed one of the rules out, and managed to almost get a solution anyway! (updated answer to acknowledge) Commented Jul 21, 2021 at 7:35