This is a puzzle about solving reverse star battle puzzles. In star battle, solvers place 2 stars in each region, row and column with no stars sharing a corner. In this puzzle, we create regions which each contain 2 stars (note that there are 2 stars per column and row and no touching stars) in addition to following the rules of the given puzzle. The constraints of the additional puzzles are slightly relaxed: in Corral and Slitherlink, we don't create a single loop, and in Hidato we make 10 snakes instead of of 1.

- For some larger numbers, we can quickly establish a general shape when they run into smaller clues or quickly pick up 2 stars. This lets us place boundaries on the 11 in the top left and 9 in the bottom right very quickly.
- Since every region needs exactly 2 stars, we can rule out scenarios and shapes that isolate an odd number of stars in an area. This is key to solving the left hand side of the grid, as well as the bottom portion.
- Most of the late deductions revolve around separating which clues must be part of separate regions. For instance, the realizations that the 8 and 7 diagonally adjacent to the lower-right 9 cannot be in the same region leads to the shapes of those regions.

6 | ⭐ | ⭐ | 4 | ||||||

11 | ⭐ | 3 | ⭐ | ||||||

⭐ | 3 | ⭐ | 5 | 6 | 6 | ||||

⭐ | ⭐ | ||||||||

⭐ | 6 | ⭐ | |||||||

4 | ⭐ | ⭐ | |||||||

⭐ | ⭐ | 3 | 7 | 3 | |||||

8 | 9 | ⭐ | 8 | ⭐ | |||||

8 | ⭐ | 5 | ⭐ | 9 | |||||

⭐ | 7 | ⭐ |

6 | ⭐ | ⭐ | 4 | ||||||

11 | ⭐ | 3 | ⭐ | ||||||

⭐ | 3 | ⭐ | 5 | 6 | 6 | ||||

⭐ | ⭐ | ||||||||

⭐ | 6 | ⭐ | |||||||

4 | ⭐ | ⭐ | |||||||

⭐ | ⭐ | 3 | 7 | 3 | |||||

8 | 9 | ⭐ | 8 | ⭐ | |||||

8 | ⭐ | 5 | ⭐ | 9 | |||||

⭐ | 7 | ⭐ |

- Since only orthogonal adjacency is used, if two numbers are an odd number of squares away and have an even difference (or vice versa), they cannot be a part of the same snake.
- Since there can be at most 10 snakes, finding 10 numbers which must all be parts of different snakes (due to their parity or simply being too far apart) tells us that every other square, including other given numbers, must be in the same snake as one of these 10 numbers.
- This key step lets us connect the central 5 to the top 8, which then forces the 3 next to it to connect to the 8 in the top right. After this, we can force the postion of the top left quickly.

1 | 2 | 3 | 10 | ⭐ | 8 | 6 | 7 | 8 | ⭐ |

6 | ⭐ | 4 | 11 | 12 | 7 | ⭐ | 10 | 9 | 8 |

7 | 8 | 11 | 12 | ⭐ | 6 | 4 | 1 | ⭐ | 7 |

⭐ | 9 | ⭐ | 3 | 4 | 5 | 3 | 2 | 3 | 6 |

2 | 1 | 2 | 2 | 1 | ⭐ | 10 | ⭐ | 4 | 5 |

3 | ⭐ | 3 | ⭐ | 5 | 6 | 9 | 2 | 3 | 4 |

6 | 5 | 9 | 9 | 8 | 7 | 8 | ⭐ | 1 | ⭐ |

7 | 7 | ⭐ | 10 | 7 | ⭐ | 10 | 11 | 2 | 6 |

⭐ | 6 | 5 | 5 | 6 | 8 | 7 | 6 | ⭐ | 7 |

2 | 3 | 4 | ⭐ | 3 | 2 | ⭐ | 5 | 4 | 8 |

1 | 2 | 3 | 10 | ⭐ | 8 | 6 | 7 | 8 | ⭐ |

6 | ⭐ | 4 | 11 | 12 | 7 | ⭐ | 10 | 9 | 8 |

7 | 8 | 11 | 12 | ⭐ | 6 | 4 | 1 | ⭐ | 7 |

⭐ | 9 | ⭐ | 3 | 4 | 5 | 3 | 2 | 3 | 6 |

2 | 1 | 2 | 2 | 1 | ⭐ | 10 | ⭐ | 4 | 5 |

3 | ⭐ | 3 | ⭐ | 5 | 6 | 9 | 2 | 3 | 4 |

6 | 5 | 9 | 9 | 8 | 7 | 8 | ⭐ | 1 | ⭐ |

7 | 7 | ⭐ | 10 | 7 | ⭐ | 10 | 11 | 2 | 6 |

⭐ | 6 | 5 | 5 | 6 | 8 | 7 | 6 | ⭐ | 7 |

2 | 3 | 4 | ⭐ | 3 | 2 | ⭐ | 5 | 4 | 8 |

- An important realization is that since there are 9 given islands and we're looking for 10 regions, we should be treating the sea as a region, which should also contain 2 stars.
- The first key step to this puzzle is identifying what possible stars each island can connect to. You want to do this counter-clockwise, starting from the 4 on the left of the board, moving to the lower 5, and so on.
- Many Nurikabe tricks are still useful here. For instance, using the knowledge that the top-left and bottom-left corners cannot be part of an island is key to deducing the shapes of the nearby islands.
- The key breakthrough after the beginning placements is the center six. We can deduce which stars it contains (C4 and E3), and once we do this, we can deduce that it must contain one of C3 or C4 (to avoid a collection of 4 squares in the sea) as well as E4 and then one more tile to connect the two parts of the island. This lets us place sea around the rest of the island.
- After this, the top left falls easily, and a few deductions about sea connectedness on the right hand of the board let us finish the last few islands.

⭐ | ⭐ | ||||||||

⭐ | 5 | ⭐ | 9 | ||||||

⭐ | ⭐ | ||||||||

6 | 4 | ⭐ | ⭐ | ||||||

⭐ | ⭐ | ||||||||

⭐ | 6 | ⭐ | |||||||

⭐ | ⭐ | 6 | |||||||

4 | ⭐ | ⭐ | |||||||

⭐ | ⭐ | 6 | |||||||

5 | ⭐ | ⭐ |

⭐ | ⭐ | ||||||||

⭐ | 5 | ⭐ | 9 | ||||||

⭐ | ⭐ | ||||||||

6 | 4 | ⭐ | ⭐ | ||||||

⭐ | ⭐ | ||||||||

⭐ | 6 | ⭐ | |||||||

⭐ | ⭐ | 6 | |||||||

4 | ⭐ | ⭐ | |||||||

⭐ | ⭐ | 6 | |||||||

5 | ⭐ | ⭐ |

- A lot of standard Slitherlink deduction can't be used here since we're not forming a single loop. This makes lots of weird patterns possible, so be careful.
- If a 2 is diagonally adjacent to a 0, then either its edges form a corner next to the 0 or opposite the zero.
- A key late deduction is the top-leftmost region and its shape. The 2 2s stacked on top of each other have only 2 possible shapes thanks to the zeroes nearby, and eliminating one of them solidifies the shape of this area.

⭐ | 0 | 1 | ⭐ | 2 | 1 | 1 | |||

1 | 2 | 2 | ⭐ | 3 | 2 | 0 | ⭐ | ||

2 | 2 | ⭐ | ⭐ | 2 | |||||

0 | ⭐ | 1 | ⭐ | 1 | 0 | 2 | |||

2 | 2 | 1 | ⭐ | 3 | ⭐ | ||||

⭐ | 1 | ⭐ | 2 | 2 | 3 | 3 | 2 | ||

3 | 2 | ⭐ | 0 | 2 | 2 | ⭐ | 2 | ||

⭐ | 3 | 2 | 2 | ⭐ | 0 | 1 | 2 | ||

1 | 1 | 0 | ⭐ | 2 | 1 | ⭐ | |||

1 | ⭐ | 1 | 1 | 1 | ⭐ | 0 |

⭐ | 0 | 1 | ⭐ | 2 | 1 | 1 | |||

1 | 2 | 2 | ⭐ | 3 | 2 | 0 | ⭐ | ||

2 | 2 | ⭐ | ⭐ | 2 | |||||

0 | ⭐ | 1 | ⭐ | 1 | 0 | 2 | |||

2 | 2 | 1 | ⭐ | 3 | ⭐ | ||||

⭐ | 1 | ⭐ | 2 | 2 | 3 | 3 | 2 | ||

3 | 2 | ⭐ | 0 | 2 | 2 | ⭐ | 2 | ||

⭐ | 3 | 2 | 2 | ⭐ | 0 | 1 | 2 | ||

1 | 1 | 0 | ⭐ | 2 | 1 | ⭐ | |||

1 | ⭐ | 1 | 1 | 1 | ⭐ | 0 |

After solving the puzzles, we note that each star is in a unique position on a 10x10 grid.

We fill in the "number" on each star - that is, the number that would be in the given position if a clue were in the star's position.

Finally, we use the checksums to solve for the values on the remaining 20 squares.

1 | 6 | 5 | 3 | 9 | 0 | 4 | 9 | 1 | 9 | 47 |

6 | 5 | 12 | 2 | 2 | 5 | 5 | 12 | 9 | 1 | 59 |

8 | 9 | 9 | 0 | 13 | 2 | 14 | 2 | 2 | 9 | 68 |

1 | 2 | 10 | 1 | 5 | 4 | 6 | 0 | 8 | 15 | 52 |

18 | 6 | 8 | 4 | 19 | 11 | 3 | 1 | 1 | 6 | 77 |

2 | 4 | 3 | 4 | 5 | 2 | 6 | 7 | 6 | 12 | 51 |

4 | 4 | 6 | 8 | 2 | 1 | 20 | 1 | 0 | 5 | 51 |

9 | 2 | 8 | 20 | 6 | 9 | 2 | 7 | 6 | 3 | 72 |

1 | 4 | 21 | 8 | 2 | 2 | 5 | 4 | 3 | 1 | 51 |

8 | 5 | 0 | 4 | 5 | 19 | 1 | 2 | 4 | 6 | 54 |

58 | 47 | 82 | 54 | 68 | 55 | 66 | 45 | 40 | 67 |

Using A1-Z26 to convert to letters gives us our answer phrase, ** CALL IN HORSE LATITUDES**.

3 | 1 | 12 | 12 | 9 | 14 | 8 | 15 | 18 | 19 | 5 | 12 | 1 | 20 | 9 | 20 | 21 | 4 | 5 | 19 |

C | A | L | L | I | N | H | O | R | S | E | L | A | T | I | T | U | D | E | S |

This puzzle is hard and I was worried that it wouldn't be hard enough. You'd think that the fact that it took me, the puzzle author, hours upon hours of work to solve these puzzles and prove their correctness would have been enough, but I didn't really realize how hard I had made it until I sent it to testsolvers (who showed that my proofs were wrong and the puzzles were, in fact, incorrect). Turns out that writing logic puzzles is hard: every single one of these minis as well as the extraction has had correctness errors.

I was motivated in this puzzle by a few things: I wanted to put a twist on a common logic puzzle, I wanted to make a 5/5 difficulty logic puzzle which revolved more around key insights and theorem proving than endless bifurcation, and I wanted it to not overstay its welcome.