Sequences of pieces with a distinguishing linear feature, such as a gate or a path, are like edge pieces. Once pieces with the feature are sorted into a pile, there are just one knob or blank to consider. These pieces might be slightly harder than edge pieces to assemble if they run diagonally in the picture, and if there are interruptions, such as people or gates, in the pattern.
Dealing with writing is still easier, since consulting the puzzle picture will give you the coordinates of any individual piece within the block of text. The beginnings and endings of lines are other clues. The only time when writing might be difficult is when almost the entire puzzle is writing, such as in a puzzle of a newspaper’s front page. In this case pictures within the page, and their borders, are easier to start with than the writing.
When dealing with writing different font sizes and styles can help also. In the newspaper example, one might start with the larger headline before working on the text of the page.
We mentioned above that some solvers like to start with internal blocks of the puzzle instead of edge pieces. We can now understand when this makes sense. If a color block is of size smaller than the number of edge pieces, e, then constructing the block will go faster. Not only is the size reduced, but edge pieces can only be filtered on knob versus blank in one dimension, while an internal block will have many spots with two adjoining pieces, which allows better filtering. A set of edge pieces with a single color will also be harder to solve than an internal block with a useful pattern.
Some solvers may choose to do edge pieces first out of neatness (since this defines the puzzle’s size on the solving board) or to keep from accumulating blocks of pieces not attached to the frame. This is an esthetic choice. No one requires us to do puzzles in the most efficient way.
VI Complexities and Heuristics
In our discussion of color blocks above, we assumed just one block per color. Puzzles, however, usually have multiple blocks of the same color, such as patches of grass or multiple balloons. How do we treat them?
Since each block will have a boundary, we can treat these as one block with multiple disconnected edges. If there are two blocks, b1 and b2, one with “edge” piece count e1 and the other e2, one with internal piece count n1 and the other n2, the solver can do the e1 edge pieces first, choose an additional edge piece at random, then complete the remaining e2 pieces. After this the solving is almost identical to that of a larger block of n1 + n2 pieces, except that since we get into the more highly constrained (3 surrounding pieces placed) we can solve more rapidly. This analysis can be extended to any number of blocks of the same color.
This also applies to the case of many very small blocks, for example lots of independently placed 4 or 5-piece balloons. The difficulty here is block clutter, which we will discuss below.
Piece Memory
While we have developed a way of figuring out how many pieces should be tried on average, most experienced solvers will complete a puzzle using far fewer moves. There are two reasons for this.
First, most filtering is done visually, without having to try a piece. This is why single color or few color puzzles (like the classic Springbok puzzle “Little Red Riding Hood’s Hood, which is all red) are considered so difficult. Even if colors or patterns are not pre-sorted, few pieces will be removed from the puzzle box and attempted.
Most solvers do this, with differences in solving ability depending on visual acuity. A bigger differentiator is what I will call piece memory. This is the ability of some solvers to remember where a needed piece is among hundreds.
I’m sure many non-expert solvers working with an expert solver have had the experience of searching for a piece only to have the expert reach out a hand to the middle of the piece depository and grab the wanted piece. Expert solvers have the experience of doing just this.
My experience is that I can do this more often for “interesting” pieces, that is pieces with odd colors, patterns, or shapes. I somehow build a map of where these pieces are located and can go right to them. Even when I don’t know exactly where the piece is, I remember what it looks like and can find it right away. I also remember the probable color and shape of missing pieces, and when I come across one in the search for another piece can place it even though I was not working in that area.
It would be interesting to study how piece memory operates. Perhaps it is like the way chess experts can memorize real board positions better than random ones.
Piece memory speeds solving in another way. What we do is to search for many pieces at once as we search through the piece repository, as opposed to one at a time. We turn our brain into a parallel processor. The more pieces you remember the faster your solving will go.
V. Putting the Pieces Together
Our analysis lets us construct an optimal solving strategy, in general. Details of the best way to solve depends on the puzzle being worked on. How fast the puzzle will be solved also depends on the piece memory of the solver and how good they are at filtering pieces to reduce the number of choices.
1. Examine the frame and color blocks. Determine if the frame is the place to begin (it has pieces differentiated by color or pattern) or if there is a single-color block or pattern easier to assemble.
2. Repeat decision for next color block. When the frame is the simplest to assemble, work on it. You might also work on a simple section of the frame or wait until assembled blocks next to the frame can be connected to individual frame pieces.
3. When working on a difficult single-color block or pieces between blocks where the pattern or color does not aid you much in placement, sort blocks by shape and look for holes in the puzzle with the smallest set of shape candidates. For example, a space in the puzzle might take a piece with two adjacent knobs.
4. When a block is complete, check to see where that block goes in the puzzle and whether it can be connected to an already completed block. If not, check the interface between blocks. Sometimes pieces to be placed here contain two colors and can be picked out from the pool of pieces without much effort. Connecting a block to the rest of the puzzle is satisfying. It also can reveal a row of missing pieces that is constrained enough to be filled in easily.
5. You will find that as you solve the puzzle goes from islands of blocks floating in the frame to islands of space floating in the mostly solved puzzle. At this point switch from finding and connecting blocks to sorting the remaining pieces, which probably do not have distinct colors and patterns, into the block of space they might go into. This is also a good time to find distinctive piece shapes matching those revealed as blocks are connected. If you have a small section without distinctive colors and patterns, your solving will go quickly. If you have a large section, such as a sky or bushes with random patterns, you will have more of a struggle. The Vienna rooftop puzzle I mentioned above was 90% like this.
6. Finally, enjoy the rush you get as the final pieces are discovered and connected into the almost complete puzzle. When I finish a puzzle, I take a picture of it to save. Some keep track of completed puzzles in a list. If it is a particularly nice puzzle you could glue it into place, but I do too many, and have too little free wall space, to do this.
7. Put the pieces back into the box and start on your next puzzle project.
VI Conclusion
I hope this has given you insight into the art of jigsaw puzzles. You probably do most of the things I recommend already, and so can feel justified. I hope I have also given you some reasons to try some new methods.
But if you don’t, that’s okay too. Solving jigsaw puzzles is fun, so whatever makes your puzzling experience the most fun is the strategy to use.
