Posts Tagged ‘polyominoes’

Introducing Agincourt (to the Blog)

February 25th, 2010

Agincourt is one of the lasercut acrylic puzzles which I’m selling through the store. It’s the set of all of the ways to make 2-, 3-, and 4-ominoes with arrow shaped holes in each square pointing in the same direction. The symmetry of the arrows means that you can flip over pieces without changing the arrow directions, but you can’t rotate them. Most of the puzzles I have designed for the set ask for the solver to make all pieces point the same way, but the arrows naturally suggest a scoring system to handicap the puzzle for different levels of solvers — just count the number of pieces you had to put in the wrong direction, and try to improve on your score.

Here’s a solution to the puzzle that literally comes out of the box. (The puzzle comes in the box with 4 layers of pieces in 4 × 4 squares.)

Expect more Agincourt puzzles later.

2-coloring Pentomino Packings

January 24th, 2010

I like to collect pentomino coloring problems.

So it should come as little surprise that I was intrigued by the cover of Puzzle Fun 16. Puzzle Fun was a ‘zine produced by Rodolfo Marcelo Kurchan in the ’90s covering a variety of polyomino problems. I missed out on subscribing to it myself, and the Puzzle Fun website languished for a decade after new issues stopped appearing.

A few months ago, Kurchan put the content of all of the back issues of Puzzle Fun online.

Puzzle Fun 16 focused on pentomino packing problems. Packing problems differ from tiling problems in that empty space is allowed, and the goal is to minimize the amount of empty space required. Packing, usefully, makes some kinds of problems possible to solve that would not be solvable as tiling problems.

One such puzzle type is packing polyforms that are 2-colorable, (that is, one can use two colors to color every piece such that no piece touches another of the same color.) This is the puzzle type I saw on the cover of Puzzle Fun 16.

The problem itself was this: Place two sets of pentominoes in the smallest possible rectangle such that no pentomino touches another in the same set. [PF problem #549]

A solution was printed in Puzzle Fun 18:

(Solution by Hector San Segundo.)

I should note that this problem implies strict coloring: pieces are not allowed to touch even at corners. I am more interested in non-strict coloring, which is generally the default in coloring problems, and I am interested in colorings of a single set of pentominoes. (Which all of the problems on my pentomino coloring page are.)

#1: Place a 2-colorable (non-strict coloring) set of pentominoes in the smallest possible rectangle. My best attempt has 65 squares:

Filling out the matrix of variations gives two more problems:

#2: As #1, but with a strict coloring.

#3: As in PF #549, but with a non-strict coloring.

(The following problem in PF 16 (#550) was a variation on #549 minimizing perimeter rather than area, but this is less interesting to me.)

http://www.puzzlefun.com.ar/