Perfect Production & Foundational Implications

(last updated on the same day to add an example for the first type)

 On rare occasions, a dream of mine can end with cracking open a puzzle book and finding something I don't recognize as an existing genre. If I'm lucky like I was last night, then I might be able to leaf through a few pages and glance at multiple examples of clue types and suggested input types. Then when I wake up, it becomes an interesting challenge to try to invent a fun ruleset that could apply to anything I can remember. The fact that I can only barely remember certain aspects makes it more lateral than instructionless puzzles typically are.

This first type was inspired by a puzzle I thought allowed multiple loops with really large numbers as clues, but I specifically remember looking at some of the lengths of rows used by a portion of the grid to verify that they were factors of the number in the portion. Since I decided to go for an edge input instead, the end result feels like Lohkous, but I think the logic is distinct enough when it comes to factorization and limits.

Perfect Production: 

Divide the grid into regions so that every region has one number, which reveals the product of the lengths of all its internal rows and columns.

(5x5)

(7x7)

For this second one, I specifically remember a square grid portioned neatly into maybe 36 8x8 squares with columns of shaded cells that were supported by the bottom and reached different heights, which initially reminded me of an equalizer. However, as I was coming up with the ruleset, I tried to make sure it would lead to an interesting puzzle with the region layout in the dream. I think the result is completely novel and a little off-kilter.

Foundational Implications: 

Shade some cells.

• If a column within a region contains shaded cells, the bottom of the column must be shaded and any other shaded cells in the column within the region must be directly above another shaded cell; these formations will be called pillars. 
• The relative height of a pillar will refer to how many rows it spans and the absolute height of a pillar will refer to the row of its topmost shaded cell. 
• Within each region, no more than one column lacks a pillar, and no two pillars have the same relative height.
• If a cell in a region is directly above a shaded cell in another region, the relative height of the pillar the shaded cell belongs to describes the amount of distinct absolute heights reached by the pillars in the upper region. 
• If a cell with an arrow is shaded, the cell the arrow points to is also shaded.

(7x7)

(9x9)