Generally they entail the arrangement of puzzles to make specific shapes in two or three dimensions. Some of the world’s most famous and popular puzzles fall into this category and a variety of subsets within it, including Piet Hein's Soma Cube, the Pentomino by Solomon Golomb and of course the Tangram.
Modern tools, such a laser cutters and 3D printers have pushed the boundaries of assembly puzzle design, allowing for exceptionally complex creations made from wood, and acrylic plastic. 3D printers have opened up completely new possibilities!
Computers have also aided the development of puzzles design and solution; they allow for an exhaustive search of how puzzles can be designed so that they provide the fewest solution possibilities or a solution requiring the most steps possible.
The consequence of this is that solving the puzzle can become exceptionally difficult, sometimes almost impossible for humans. In the niche of assembly puzzles, we have arrived at a place where we have literally an infinite stream of possibilities. With the continued fusion between puzzle design and technology the boundaries of complexity will continue to be pushed – for this reason new forms of assembly puzzles are created with frequency as puzzle designers continuously strive to outdo each other.
Consider though, that the complexity of puzzle doesn’t directly correlate with fun and enjoyment. It is widely regarded that the technically most appealing varieties of assembly puzzles designs are the ones that achieve their frustrations with the fewest number of pieces.
This of course, is at the center of angst for many a puzzler, a puzzle that seems easy, is in fact a mind boggling challenge. The classic Pyramid Puzzle, is the archetype of this – with only two pieces they are amazingly difficult for most solvers.
The Pyramid Puzzle: which comes with a rather wonderful history as an enduring puzzle design, originally crafted on the pastoral frontier countryside - by pioneering fathers as toys for their children – in many ways the pyramid puzzles, and simplistic assembly puzzles represent an important piece of human life – an untainted, simple aesthetic. It did evolve to incorporate more pieces – but you will still find the classic two-piece Pyramid Puzzles in toy stores and Christmas crackers.
Ball Pyramid – are evolution from the Pyramid Puzzles, which are usually comprised on loosely joining spheres that form layers. In a Ball Pyramid the balls conjoin to form a lattice. A triangular prism – beehive pyramid are another groovy examples of this - which is shaped like a triangular honeycomb.
Usually Assembly Puzzles can be juxtaposed into many different configurations, but only a few configurations can actually make valid solution – often just one. Indeed the best variety of these puzzles often permit seemingly valid constructions – allowing for all puzzles to fit apart from the very final piece... where you have to start over.
Interlocking Puzzles this is where the puzzles interlock to form a free standing constructions that can remain stable - for example, Kumiki Puzzles will fall into this category. Interlocking Puzzles to start with the traditional six-piece burr puzzle although it has developed to take on many functions and fashions, like the diagonal burr, a style that has been rounded out as an interlocking burr puzzle with spherical outer.
Packing Puzzles: Simply stated, the challenge of a packing problem is to fit a given set of a pieces into a set parameter. The boundaries are either enforced by walls and a lid, or sometimes just lids. The container might also be in a tray (single layer) if the pieces don’t stack in 3 dimensions.
Often the container of Assembly Puzzles is simply implied instead of existing and being part of the puzzle. If there Is a physical container, and some pieces to go into it this is a packing problem; in rare instances the container is apart of the pieces themselves.
A subsect of this category is 3D packing puzzles, these can consist of identical or similar piece packing puzzles and dissimilar puzzle pieces. This means exactly what it says – a different piece 3D assembly would be something like the Filling Up – where all the pieces are entirely different, a similar piece assembly puzzle would be similar to Challenge Box – where there is a standard shape that needs to be grouped together to make a whole.
Single layer packing problems: like Tangram puzzles or the snake pool puzzle – essentially two dimensions as opposed to three. Single Layer packing puzzles are inclusive to but not exclusive to varieties of polyaminoes. Polyaminoes are a practical assembly puzzles that has a history that can be traced back over two and a half centuries. Les Amusmens – published in 1979 included a cross puzzle made from a handful of pentominoes – that quickly became exceptionally popular. A pentomino is a plane made up from 5 equal squares, edge to edge. There are 12 possible pentomino shapes, all of which represent a letter of the alphabet. Pentominoes form an unusual mathematical problem – more of which you can read about here.
Sequential Assembly Puzzles: They do exactly what they say on the tin, not only do you need to assemble them – which is challenging enough, they need to be assembled in a certain order to be fully completed. Big City Challenge, for example puzzles to put together in a specific order, if you fail to align the sequence correctly you will not be able to complete the puzzle fully.
Dissection Puzzles: There are various forms of dissection puzzle, all of which involve a figure that has been cut up (or dissected) – the most famous examples of this are probably the Tangram puzzle – although they are multiple examples of other geometric dissections including hear Tangram of Pentastar aka the Charlestown Fortress.