Tag: edgematching

Three paths to pick from, part 2: Distant connections

~ munizao ~ Leave a comment

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!

Edgematching to the Stars

~ munizao ~ 1 Comment

The best known combinatorially complete set of edgematching tiles are the 24 squares with 3 edge colors discovered by Percy MacMahon. These can be rotated, but not flipped over. I showed an illustration of them when I first discussed edgematching puzzles on this blog. And now I’ll do it again!

Can we find an even more basic set? After all, 3×3 square edgematching puzzles were common as advertising promotions in the first half of the 20th century. A set of 24 tiles seems excessive. And surely 2 edge colors would be simpler than 3.

The problem is that with 2 colors, you get only 6 tiles, and they don’t make an interesting puzzle at all. But if you use tiles with fixed orientation, you get 16 tiles. Per David Singmaster’s Sources in Recreational Mathematics, this set was described by J. J. M. Verbakel in 1975. We’ll use a line pointing out of a side to represent one of the colors, and a blank edge to represent the other, as this convention will help us to make better visual sense of some of the later figures.

They still don’t make a very good puzzle, as the above solution is basically trivial, so even adding the restriction that only one color can touch the border adds no challenge. But as a starting point for further exploration, they lead in some very interesting directions. Christian Freeling shows some of these on his page on the “BackSlide” puzzle, which turns the pieces into a double-sided sliding 15-puzzle. (Freeling’s site also contains information about variations of these puzzles using hexagons as well as squares with diagonal connections.)

The binary nature of the colors allows us break out of the 4×4 square. Instead of both colors matching themselves, we can arrange the tiles into sets of polyominoes where one color matches itself, and the other matches an outer edge. Freeling calls these “transcendental” solutions. Finding these is still not much of a puzzle, but any unusual polyomino recreation is appreciated around here!

Breaking up the tiles further leads to the “starmaps” discovered (in the hexagonal context) by Martin Medema. Here, we place these tiles on an 8×8 board. A line pointing out of a piece must connect, at some distance, to one pointing out of another piece. An empty edge must see only empty space in its direction. Conveniently, there are 32 empty edges, exactly as many as edges in the perimeter of the board. Freeling states that it is “considered good form” for no tiles to be adjacent. If we don’t use good form, it is not hard to transform a regular transcendental solution into a starmap by breaking it apart until no empty edges see each other. Finding a good form starmap isn’t a trivial puzzle at all.

My impetus for this post was an idea I had for a new type of starmap. We might require one edge type to match an adjacent instance of itself, and have an additional edge type that must see itself from more than one square away, enforcing “good form” for that type. The third edge type must see only empty space as before. There would be 81 fixed orientation pieces with three edge types, which is way more than I want to deal with. Surely there must be some better set with three edge types.

Right, the MacMahon tiles! There are 32 instances of each edge type, so we can use an 8×8 board again. I found this puzzle to be quite challenging.

Puzzles with sparse (or sparse-but-clumpy) piece placements are unusual and visually distinctive, and I hope to encounter more like these.

Edge Pip Puzzles

~ munizao ~ 1 Comment

Recently* I was perusing the section on edge matching puzzles on Rob’s Puzzle Page. While puzzles using a 3×3 square grid are the most common, one that I found interesting was the 2×3 grid “Matador” puzzle, where matches form dominoes with doubled numbers of pips. Possibly to compensate for the reduction in size, an extra rule requires that not only must edges match, but one match must be present for each pip number between 0 and 6.

My first impulse was to find the simplest complete set that would work as a puzzle. There are 6 different cyclic permutations of a set of four different numbers. Why not try edge-matching with cards using these?

It turns out that this is a pretty easy and not terribly interesting puzzle.

However, in exploring the space of similar puzzles, I found what I think is a particularly elegant gem. If we use cards with edge values between 0 and 8, there are 17 different sums between 0 and 16, which is the same number as the number of internal edges in a 3×4 grid of 12 cards. Coincidentally, 12 is also the number of sets of different numbers between 0 and 8 that sum to 16, so those can be the sets of numbers on our 12 cards.

The last detail is the matter of how we arrange those numbers. Making all of the cards have clockwise ascending edge values is simple enough, although it hurt my symmetry senses to have to pick a direction. And indeed, we don’t have to, because we can make the cards flippable, so that the other sides have values in counterclockwise ascending order. Luckily, the flippable cards are just what the puzzle needed to handle another issue: without them, the puzzle would be unpleasantly hard to solve.

In addition to the collect-all-the sums puzzle, simply matching the numbers makes a good puzzle with this set:

A third challenge is to make a 4×4 square with the corners removed such that every difference between values at the same edge between 1 and 8 occurs exactly twice. I believe I solved this at some point, but I didn’t record the solution, so I can’t show it to you right now.

As you may have noticed, the last puzzle set has higher production quality than the first two. That’s because I’ve had a prototype custom deck of cards made including several different puzzle sets. I intend to have a small run of these made to sell.

*By recently, I mean, whatever was recently last June, when I started writing this post. I don’t mean to only finish blog posts during the earlier part of the year, but it does seem to tend to work out that way.

Edgematching with Catalan number patterns

~ munizao ~ 2 Comments

Edgematching puzzles are a genre that I haven’t previously explored very deeply. A classic example is the MacMahon squares puzzle. If you subdivide a square diagonally into four triangles, there are 24 ways to color the triangles using three colors. Using a set of 24 squares with all of the possible colorings, it is possible to make a 6×4 rectangle where all of the edges match.

As I was pondering the possibility of designing my own edgematching puzzle, I considered that pieces with multiple colors are a bit of a pain to make with a laser cutter. However, engraving lines is relatively easy. And I found a promising set of line patterns related to the Catalan numbers. The Catalan number sequence (1, 2, 5, 14, 42, 132…) is one that comes up in a surprising variety of contexts. For example, Catalan(n) is the number of distinct ways that n pairs of matched parentheses can occur in a string. For n = 3, we have ()()(), (()()), ((())), ()(()), and (())().

Replacing the parentheses with diagonal lines, and stretching them until they connect, we can make the following figures:

We can place these figures around the edges of a square, and use them to do edgematching— but with a new twist as to what counts as a match. Looking at the patterns that are made when two of these figures meet at an edge, we see that one thing that distinguishes them is the number of loops they make. One, two, or three loops can be made. This gives us three different puzzle objectives: place the pieces of the puzzle so that all internal edges have the same number of loops, where that number can be one, two, or three. The fact that there are three loop numbers is promising because it means that, on average, the probability of an edge in these three puzzles being a match will be one-third, the same as in the MacMahon Squares puzzle. This table shows the possible edge patterns, colored according to the number of loops:

From the perspective of puzzle design, it’s convenient that the proportion of pattern matches of each type is different; this seems likely to result in puzzles that are correspondingly different in difficulty, which is desirable. The table also reveals symmetries in how the edge figures form the different types of patterns. If you swap both the rows and the columns of the mirror pair of figures, you get a table with the same loop numbers in the same places, as we would expect. Perhaps more surprisingly, this also works with the first and second rows and columns. Thus those two edge figures form a sort of hidden symmetric pair. If you swap all instances of one in a puzzle solution with instances of the other, you will get a second solution.

There’s still one task remaining before we have a puzzle design: choosing the pieces. The MacMahon squares puzzle used every possible piece. Since we have five edge types here, instead of three, that would give us more pieces than we’d really want for a good manual puzzle. There are probably several reasonable options here, but in the interest of prototyping, I wanted to start with a small piece set, so I decided to explore a set of nine pieces with bilateral symmetry. Here’s a solution with single loops:

And here’s a double loop solution for comparison:

I’ve already had prototypes made of this puzzle, and it feels promising. I hope to have at least one more post about variations on this design, but for now I’ll stop here.