The game of Pentomino is a tiling puzzle game played on a grid. A Pentomino piece is a shape of five non-diagonally connected tiles. There are 12 unique pieces (ignoring rotation/reflection). The goal of the game is to fill the board with the available pieces without overlapping.

Pentomino pieces and names

I'm interested in which strategies I can use to find good pathes to a solution faster. So far I primarily researched non-rectangular boards, possibly with 'holes' inside, that have exactly one solution. Later, I want to also try boards with multiple solutions.

I've identified a few ideas that were quite helpful:

  • Do not search where to place a piece, but search the grid for connected spaces. In other words: instead of asking "where can I put this piece?", ask "what piece could be put here?".
  • This search should start in cells with few options (i.e. narrow alleys). This often leads to long, forced sequences, which is good.
  • If unclear, assume that two cells are separated (not part of the same piece). Then try to refute this assumption by showing that the board cannot be solved anymore. This means they must be connected. Rinse and repeat.
  • If a subspace is created (that is not connected to other open areas), check if its number of cells is a multiple of 5. If not, it is impossible to solve.
  • Point symmetric subspaces of 10 cells cannot be solved.
  • (Some others that are more complex)

These came to my mind while just playing the game for many hours (I was a test subject in a study about Pentomino). Are there any resources, links, books, etc where I can learn about more advanced strategies?

  • 1
    Focusing on narrow spaces, checking if any piece goes and stopping if any gap isn't a multiple of five are the strategies that I use too.
    – tttppp
    Jun 15, 2020 at 6:23


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