Grid puzzle nomenclature

[Scott Blomquist]

Hey, Puzzle Theory fans!

One thing I've noticed over time is that a ton of different puzzle types fit onto a grid. Just quickly flipping through the March 2010 issue of Games World of Puzzles magazine, I see approximately 38-and-two-halves puzzles that can be solved by packing symbols (mostly letters and numbers) onto a rectangular grid, and 19 puzzles that cannot. In pretty much any other month, you can probably expect to see a very similar distribution. Similarly, the New York Times, P&A Magazine, and many other puzzle publications tend to emphasize the grid-of-symbols puzzle formats.

This is probably due to side effects of how language works or how people think. If one of you feels like doing some thinking or research on psychology or sociology on this, we'd love to hear what you learn. But that's not what I want to talk about here.

I want to add a few observations about some of the common properties that these puzzle forms have, and propose nomenclature for them.

First, let's take the most common grid-of-symbols puzzle type, the crossword puzzle. Obviously, a solved crossword is a Grid of Symbols (Letters in this case), but the Letters themselves belong to Groups (i.e. Words). Words can be said to consist of a Path of Letters. (Let's define Path as a sequenced collection of items (items == Letters in this case).) Additionally, in a typical crossword Grid, there are two Layers of Paths--the across Layer and the down Layer. Each Layer in this case represents a collection of Paths that comprise a complete Covering of the underlying Grid. Within a Layer, the complete set of paths do not Overlap.

Secondly, let's contrast this with how we might use this nomenclature to describe Sudoku. A solved Sudoku instance also consists of a Grid of Symbols (Numbers in this case). These Numbers, like a crossword's Letters, also belong to Groups. In this case, the Groups are not Paths, because they're not sequenced. A sudoku puzzle has three Layers that each comprise a complete non-Overlapping Covering of the underlying Grid--the Layer of rows, the Layer of columns, and the Layer of 3x3 boxes.

The final grid-of-symbols puzzle that I'll discuss here is the standard word search. A word search also consists of a Grid of Symbols. Unlike the previous two where the Groups (Paths) are given and the Symbols must be solved for, in a word search the Symbols are given and the Paths must be solved for. There are Constraints on what correct paths look like that can vary in different types of word searches. Ultimately, the set of Paths Overlap arbitrarily, and do not necessarily form a complete Covering. I don't see anything obviously worth calling a Layer here (other, perhaps, than the underlying Grid of Symbols). Sometimes, in word searches, the Mask of which Symbols are unused by the complete set of Paths may also encode some Signal.

Exercises

1. How do the bits of nomenclature I used above (each Capitalized word) resonate with you? 
2. What other observations about these puzzles can we make?
3. To what other types of puzzles can this analysis be applied?
4. What other words do we need to define here?

[Larry Hosken]

A typical simple jigsaw puzzle kinda fits this model. Each puzzle
piece is a symbol defined by the squiggles of its edges. Each symbol
occurs just once in the puzzle. You gotta figure out where it goes in
the grid. You figure out paths as you put pieces together.

I'm trying to fit challenges of the form "can you tile this grid with
*ominoes" into this model and I can't quite make it happen. Maybe
that means that they don't fit the model. Maybe that means that more
creativity is called for.

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[Scott Blomquist]

Coincidentally, ominoe-tiling is one of the things that crossed my mind when I was thinking about crossword variants, and it totally works. There's a crossword variant that Patrick Berry regularly contributes to Games Magazine as "Some Assembly Required".

In this type of puzzle, there are no blocked out Cells--that is, each Cell contains a letter. Clues are given for sets of words that are concatenated to form each row. Additionally, clues are given along with funny shaped pieces (all technically N-ominoes). These Pieces are then tiled onto the grid as a non-Overlapping Layer. The letters in the pieces, of course, match the letters in the underlying Rows.

If you can't visualize what I'm talking about, I've attached my 3 minute crappy example.