cyrebjr (cyrebjr) wrote,

Fillomino meanderings

In unrelated news, I'm doing a work-assessment program in which I'm demonstrating my office skills. We're hoping this will lead to finding a good placement for me at some company.

Kinda been relaxing on the puzzle construction front, though not so much on the solving front. There's been a cluster of Fillomino work (see here, puzzles 4 and 11 in particular) that sent me on a short mental voyage around the block. Here's what I brought back:

1. Magic Fillomino. All rows/columns and the two main diagonals have the same sum. Might add by cells or regions. Obviously, easier to construct with just selected line having the same sum; here's one where the rows (and possibly some other lines) have the same sum.
   _       _
-> |2| | | |
-> | |1|4| |
-> | |3|1| |
-> | | | |4|

Yet another possibility would be to take a square grid with an even side length and require that in the filled grid, every quadruplet of cells placed with 90-degree symmetry yields the same sum.

2. No-L Fillomino. Polyominoes are allowed to be full rectangles, but not rectangles with a smaller rectangle taken from one corner. (For example, a pentomino can be any shape except L, P, or V.) If you'd rather not do a Christmas theme, the rule is equivalently "no hexagonal polyominoes."

3. Prime/Composite Fillomino. As with the recent tweak in Even/Odd Fillomino, all prime-size polyominoes are in one contiguous group, as well as all composite-size polyominoes. Size-1s have no special restriction. (Plenty other possible decompositions, perhaps modulo-3 with all three classes grouped.)

4. +/-1 Fillomino. Each polyomino is given a label that's off by one. Touching is allowed or not based on what label the polyomino has, rather than its actual size. (Any other partially-consistent relabeling, such as modulo-4 or some rigorous shape class, might give fascinating interplay.)

5. Alternative grids. Triangles, hexagons, irregular polygons, mixes, non-euclidean, 3-space. Probably already exist, but having a little trouble finding examples.
Tags: open, puzzle
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