Tag: path puzzles

Stripe Club

~ munizao ~ Leave a comment

I posted last year about a path puzzle using polyiamond tiles. Those tiles were marked with a complete set of paths between cell edges on the perimeters of diamonds and triamonds. Recently I’ve been exploring a variation on tiles with marked paths. In these tile sets, the paths are constrained to straight lines aligned with the grid and connecting the midpoints of opposite cell edges. By this scheme, there are 16 ways to stripe the tetrominoes. I wasn’t able to come up with any elegant tiling using just these pieces, but with a set of unstriped trominoes, they can make a rectangle with four stripes. We follow the typical rule of path puzzles: the stripes must connect between pieces.

There are nine distinct ways to stripe the three trihexes. There is an arrangement of parallel stripes on the figure below that looks like it could have a solution, but it proved to have none when I checked it with a solver. Luckily, non-parallel stripe lines are perfectly acceptable — as long as their intersections occur outside the tiling!

Striping polyiamonds brings a new complication: the line connecting cell edge midpoints is not perpendicular to the cell edge. That means we can change the direction of paths at piece boundaries. The solution below takes advantage of this feature:

Fortuitously, the striped 2-, 3-, and 4-iamonds together contain 49 triangular cells, allowing us to tile a triangle of side length 7. The striped 4-iamonds alone contain 36 cells, but they are not able to tile the triangle of side length 6.

Where else can we go with stripe problems? Todor Tchervenkov, Roel Huisman, and Edo Timmermans looked at tetrominoes with diagonal stripes on the Puzzle Fun Facebook group. (There are 17, which makes them awkward for tiling with the full set, but there are workarounds.) We could try other stripe orientations on polyiamonds and polyhexes as well. Polytans (or polyominoes with tans added or subtracted) could have line bends at diagonal boundaries similar to what happened with the polyiamonds. Another variation I’m looking at is what can be done with multiple stripes per piece. Stay tuned for more stripe content! (Does that count as a stripe tease?)

Three paths to pick from, part 2: Distant connections

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I promised two more path puzzles in part 1, and their time has come. When I posted recently about “starmaps” as a variation on edgematching puzzles, my variation there was actually the second puzzle inspired by them that I had found recently. The first was this set of 2×2 square tiles with one cell being marked with 2 orthogonal or diagonal arrows. (The tiles can be flipped.)

Part of the inspiration to use arrows may have come from the game of Trippples, [siccc] which uses a complete set of fixed square pieces with arrows in three directions. I recently read about Trippples in issue 7 of Abstract Games magazine. Once I had these squares with arrows, a puzzle challenge seemed natural: connect the arrows into a single path, which may not enter any cell with arrows in any direction that does not correspond to an arrow.

Problem 61: Find a closed circuit using these pieces. I spent enough time finding the path above; I suspect that a closed circuit may be solvable if you have the patience of a Lewis Patterson, which I do not.

One element I like to consider in puzzle design is non-locality. A puzzle exhibits non-locality if, when you are placing a piece, you must consider pieces that are not immediately adjacent. Most polyform and edgematching puzzles are generally local. If half of a puzzle frame is filled, pieces in the interior of the filled region do not directly affect how new pieces can connect to the edge of that region. (Of course, I am eliding the fact that they reduce the set of remaining pieces that are available to place.) In the above puzzle, the empty space allows long distance connections, turning path-making into a non-local problem.

My Color Tubes puzzle from my Edge Collection Connection set of edgematching card puzzles was also a path puzzle with non-local considerations. I neglected to introduce it on the blog back when I produced the set, so let’s remedy that now.

The configuration shown is a solution to the challenge of placing the pieces so that each tube has three segments of three different colors. Segments can break in the middle of a card, or at a connection across a card boundary with non-matching colors. (Here, the cards cannot be flipped over; the back sides of the cards contain a second, related puzzle.) Other challenges for the cards are placing them so there are two differently colored segments, or four. This was definitely more of a “designed” puzzle than a “discovered” puzzle, which was a bit of a departure for me. I’ll have another excuse to muse about the distinction in a future post, but at this point I’ve hinted at more than one future post, so they can’t all be the next one.

With a couple of instances of non-locality under our belts, can we say anything useful about it as a puzzle design tool? In the case of Color Tubes, I think it gives it a little more depth than a typical 3×3 edgematching puzzle, which would seem to be welcome. In the arrow path puzzle, it adds difficulty and complexity, but the result is a little too much difficulty and complexity, at least for my tastes. It is a spice that should be judiciously applied. But then, so is hinting at coming posts, and that won’t stop me from teasing more material about non-local puzzles in the near future!

Three paths to pick from, part 1: A compact gem

~ munizao ~ Leave a comment

I’m going to be sharing a few different puzzles I’ve discovered that share the theme of path building. The first is a pretty polyiamond puzzle I recently prototyped; the other two have been split into a second post.

Path puzzles are a natural extension of edge matching. Instead of just marking edges in some way so that we can match like markings, we can choose and connect any pair of edges. Then we can make challenges involving the paths spanning multiple tiles that are formed. I’ve examined cell markings on polyforms a fair amount here, but I’ve left edge markings understudied. One reason is that the combinatorial explosion of possibilities can become overwhelming. The L-tetromino (to pick a simple asymmetric example) has 16 different cell markings if each cell can either be marked or not. Kadon sells this set in an 8×8 square frame as L-Sixteen. Since the L-tetromino has 10 edge segments, there are 2^10 = 1024 different markings if we do the same with the edges. If someone made this, it’d need a 64×64 square frame!

Cell markings also work for path puzzles! (Adapted from here.) Not shown: 4096 square unit monstrosity.

It’s probably little surprise then that most puzzles involving edge markings use simple squares or hexagons. But recently, when I was looking for something to do with diamonds and triamonds, I found a nice set for a path puzzle.

But first, an aside on why I was looking at diamonds and triamonds. My recent puzzles with row and column pip sum challenges used multiple copies of small polyominoes with different markings. The ability to exchange copies of the same shape usefully inflates the size of the search spaces for those puzzles. But tile placement remains an important aspect, even if the finding a tiling is easy, so it still has the feel of a polyform puzzle. In a sense, these puzzles “punch above their weight” in terms of the amount of puzzle you get for the size. I wanted to find other puzzles like this, and started to look at polyiamonds. For small polyiamonds, a hexagon of side length 2 seemed like the ideal frame.

There are a number of ways to divide up the 24 cells of that hexagon, but 3 diamonds and 6 triamonds was a top candidate. I tried magic pip sum puzzles first with mixed results, but then I checked the ways to connect edge segments and found that I got exactly 3 diamonds and 6 triamonds. And making paths with them is indeed a nice manual puzzle.

After I came up with this I was curious about which exits from the hexagon it is possible to get to from a given starting point. The darker triangles overlaid on that diagram give a clue: the number of transitions between dark and light squares must always be the same, however the pieces are placed. This means that the path must always enter and leave the hexagon at triangles of the same color, so only the positions marked 1, 4, 5, and 8 are possible exits if you start at the ★. Solutions do exist for each. (Positions marked with the same number give equivalent paths by symmetry.)

I later produced a lasercut acrylic prototype of the puzzle. Here it is:

One change I’d like to make for a production version would be to cut a circle in the frame around the hexagon so that the whole puzzle can be easily rotated, allowing the ends to match the symbols. As it is, you might find a solution that connects “wrong” edges, and it’s a pain to take the pieces out and put them back in the right orientation.

In future posts, I can promise not only more path puzzles, but also another “compact gem” of small polyform sets with big puzzling possibilities.