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This is a terminology/nomenclature question in the context of grid based games, specifically the game of Pentomino. I'm not a native English speaker, so I feel like I would easily miss something.

In Pentomino, each play piece covers exactly five cells of a grid. These cells/tiles then belong to the same play piece; they are "connected".

I am looking for a name for a set of tiles that is known to be connected without knowing which piece they belong to. For instance, four connected tiles in a row may belong to an L-, Y- or I-piece. So these tiles form an incomplete piece. How should I call this set of tiles?

As an example from a different game: In the game of Go, stones placed on the board grid by the same player that are (transitively) adjacent are "connected". They are often called a "group" or (more science-y) a "string".

So far, my best guess is to call a connected set of tiles in Pentomino a "string". Would this be understandable? Are there better options?

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In David Parlett's The Oxford History of Board Games, pages 165-166, when discussing variants on Pentominoes, the author states

The other [variation], which has just occurred to me, is a linear version of Pentominoes played with Cuisenaire rods, which are used for teaching numbers and lengths to young children.The linear equivalent of Pentominoes would be a set of rods each consisting of a number of squares joined in a straight line, there being one of five squares, two of four, three of three, four of two, and five of one.

I therefore suggest that you adopt the term rod to identify any straight sequence of squares within a pentomino.

Reference : History of Board Games

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If the tiles are known to all be part of the same piece, without forming the entirety of the piece, then that would be a subpiece.

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It's hard to prove a negative, but I don't think there's an unambiguous term for this, especially one that is readily understood in multiple different boardgames.

If you call it a "string" after explaining that concept to your audience, it'll work fine. Others may use two words, e.g. "connected group" and that would work too. In the case of Pentomino, you might call it an "incomplete piece"; that obviously doesn't apply to go.

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