cyrebjr (cyrebjr) wrote,
cyrebjr
cyrebjr

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
Subscribe
  • Post a new comment

    Error

    default userpic

    Your IP address will be recorded 

    When you submit the form an invisible reCAPTCHA check will be performed.
    You must follow the Privacy Policy and Google Terms of use.
  • 0 comments