### Rules

- Normal sudoku rules apply
- Tile the grid (except for the shaded central cell) in pentominos
- Each pentomino shape is associated with a number from 1 to 9 (its pentominoku number)
- F: 6
- L: 2
- N: 3
- P: 4
- T: 5
- U: 8
- V: 7
- W: 8
- X: 9
- Y: 1
- Z: 9

- The pentominoku number must occur exactly twice in each pentomino
- Starred cells show where some of these occurrences are
- It is possible for other numbers to occur twice within a pentomino
- You do not need to use all pentomino shapes
- Pentominos of the same shape can be orthogonally adjacent

Examples of pentomino shapes together with how the numbers may occur are shown below:

### Links

### Background story

Like many people, I discovered sudoku variants through Cracking the Cryptic in 2020 when my third child was little. After my first puzzle was featured on the show, I was excited to create another new variant that combined the best of pentominous puzzles with sudoku. I eventually hit on the ruleset used for this puzzle, but was unable to create one by hand, or even show that such a puzzle existed. I realised that in order to create this puzzle I was going to have to write some software, as I did for my previous puzzle. The problem was I had no idea how to start to solve such a problem. Pentomino tiling seemed very different from sudoku and I didn't know how to solve both at the same time.

Fast forward three years. I had gotten distracted from sudoku to build a non-profit search engine, but after watching a few CtC episodes, was inspired to revisit my old idea. After a few more attempts to set it by hand, I started investigating how I could solve it computationally. Eventually I came across the exact cover problem and Donald Knuth's beautiful dancing links algorithm. It turned out that both pentomino tiling and sudoku solving could be described as an instance of this problem! I knew I would be able to solve it with this method.

As we were expecting our fourth child, Layla, it felt like I had come full circle since my first sudoku escapade, and decided to dedicate the puzzle to her. She was born a few days after I finished setting the puzzle. That was four days ago. I couldn't be more grateful for this beautiful gift that is sleeping on me now in the sling as I type.

### Solver code and demo

- Source code (MIT license)
- This code is deployed at pentominoku.com

I hope that others will be inspired to create puzzles with a similar ruleset. I personally find them quite fun to solve and I hope others do too! The solver should provide a good starting place to add other types of clues or constraints. Feel free to reach out on the CtC Discord if you want help with that, or just to let me know you enjoyed the puzzle!

### Acknowledgments

The solver uses Jonathon Taylor's excellent dlxlib which saved me having to actually write my own implementation of dancing links.

I thought up the name "pentominoku" myself, but perhaps unsurprisingly it has been used before in a puzzle by Laura Taalman in 2006. I hope she will forgive me appropriating that name.

Many thanks to jmerry and kendfrey on the CtC Discord for testing the puzzle and invaluable feedback.

**am 19. September 2023, 07:10 Uhr von ibn Muhyiddin**

Thanks BlitheSolver! So glad you enjoyed the puzzle.

Zuletzt geändert am 19. September 2023, 16:56 Uhr

**am 19. September 2023, 05:59 Uhr von BlitheSolver**

My first foray into Logic-Masters! I saw this puzzle referenced on Hacker News today. Difficult, but fair. I love how, at the outset, it looks insurmountable, but gradually yields its secrets. Well done, ibn Muhyiddin!