L-topia, a Puzzle in 12 Brain-Bending Pieces

by Alexandre Owen Muñiz

L-topia was inspired by a puzzle by Ed Pegg posted on mathpuzzle.com based on the Fill-Agree set produced by Kadon. In 2002, I decided that my puzzle idea was cool enough that I ought to have real copies made, and I hired Kadon to produce a number of sets. You can purchase one at my puzzle shop site, PuzzleZapper.com.

This set of 12 L-shaped pieces can form numerous figures. The puzzles using them require making special patterns with the holes in the pieces. Can you solve them?

The pieces have a shape composed of 4 squares known as the l-tetromino. The 12 pieces contain all possible arrangements of a circular and a square hole in the centers of two of these squares. The set is made from transparent lasercut acrylic, and came with small pieces in contrasting colors to fill the holes, a sheet with instructions for puzzles using the set, and a cloth drawstring bag for storage. The original instruction sheet that came with the set is available here, in PDF format. (The current instruction sheet is a somewhat pared down version of this, but the original may appeal to hardcore puzzlers.) When formed into a 7 unit square with a one unit hole in the middle, the total size of the set is 6 1/2" on a side.

The Puzzles:

Most of these puzzles involve tiling a particular shape so that the pattern of the placement of the holes has certain properties.

There are many different shapes that these pieces can tile. Here are a few:

All of the puzzles below were developed for tilings of the 6×8 rectangle, but some of them can be adapted to other shapes. Be creative; half of the fun is in finding your own variations to solve.

1.* Place the pieces so that no set of pieces smaller than the whole forms a rectangle, and there is no corner where 4 pieces meet. (In this puzzle, the hole placement doesn't matter.) Solution.

Connection Puzzles:

In these puzzles two holes are considered to be adjacent if they are one step away from each other horizontally, vertically, or diagonally. Place the pieces so that:

2.** None of the circles are adjacent to each other, and the squares are all in a single connected group, i.e. it would be possible to get from any square hole to any other by a series of steps to adjacent square holes. Solution.

3.*** The squares are all adjacent to an even number of holes, and the circles are all adjacent to an odd number of holes.

4.** The circles all have either another hole or the right edge of the tiling directly to their right, and the squares have either another hole or the bottom of the tiling directly below. Solution.

Location Puzzles:

Puzzles where the locations of the holes are important. Place the pieces so that:

5.** All of the circles are only in every other row, and all of the squares are only in every other column. Solution.

(For puzzles #6, #7, and #8, consider the tiling as a grid of 2×2 squares.)

6.* Each 2×2 square contains exactly one square and one circular hole. Solution.

7.*** The same as #6, but the square and the circle in each 2×2 square must belong to different pieces.

8) **** Counting rotations and reflections as different, there are 12 ways to arrange one of each hole type in each square. Find a tiling where each of these arrangements is represented in one of the 12 2×2 squares.

Difficulty Ratings:

*: Moderate Difficulty.
**: Hard.
***: Very Hard. (I haven't solved it myself, but it looks doable.)
****: Not Recommended for Humans. (My best guess is that there should be solutions possible, but it may be nearly impossible to find one without searching with a computer.)

If you have questions, comments, solutions to unsolved problems, or ideas for new, related problems, please email me.

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