Origami Heaven

Origami Heaven is the website of paperfolding designer, author and illustrator David Mitchell

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An Introduction to Paperfolding Puzzles
 
Because I am a professional paperfolding designer and author you will not be surprised to learn that most of my puzzles, though not all, are set and solved by folding paper. I have collected many of them into a book, inventively titled Paperfolding Puzzles, which is available from the Water Trade Shop on this site or through major book retailers worldwide. Much of the information on this page is adapted from the introduction to this book.

There are many kinds of paperfolding puzzles, but the ones I am particularly drawn to. and like to invent, are the kind that can be solved by a mixture of trial and error and sudden insight. A good puzzle should always have an 'Ah, yes!' moment. Puzzles of this kind will seem extremely simple if you have that crucial insight to see how they can be solved and devilishly difficult if you don't.

Paperfolding puzzles are puzzles that are solved by folding paper. They are not just origami versions of puzzles which were originally created in some other way. In order to qualify as a paperfolding puzzle folding paper must be integral to the solving of the puzzle not just to its construction. So, for instance, a set of Tangram pieces folded from paper is not a paperfolding puzzle. Nor is a set of Soma Cubes made out of folded paper modules. Similarly a puzzle that is set by folding paper but is solved by the use of mathematics does not qualify as a paperfolding puzzle. Paperfolding may also be used to create, or set, the puzzle, but that is not integral to the definition.

It follows that paperfolding puzzles are not inherently difficult to solve. Their solutions are easily accessible through the intelligent and persistent application of trial and error. Play around with the possibilities long enough and a solution will emerge. It is worth bearing in mind, however, that many paperfolding puzzles have multiple solutions, and that the first solution you come across will not necessarily be the simplest or the best.

Puzzles have rules, or conditions. In puzzles set using more robust materials, such as wood, the conditions are often entirely physical. Will the pieces go together or not? In paperfolding puzzles, however, the conditions tend to be rather more subtle and it is necessary to take great care to state the conditions for the puzzle clearly. You should always take equal care to make sure you understand the puzzle before you start looking for the solution. This should not mean, however, that you should not feel free to think about unusual possibilities along the way.

Paperfolding puzzles do not easily fall into neat compartments within some overall scheme. There are several different ways to divide them into categories, all of which are useful and illuminating.

One such distinction can be drawn between unfolded sheet, grid and apparatus puzzles. As the name suggests, unfolded sheet puzzles start from one or more unfolded sheets of paper, the challenge being to fold them freestyle, and perhaps also assemble them, until the solution is achieved. Grid puzzles start from a sheet of paper that has already been folded into a grid of creases. Only the creases in the grid may be used to achieve the solution. Apparatus puzzles start from paper that has been cut and glued together into some kind of simple apparatus, such as a flexagon, which is then folded, or manipulated, until the solution is achieved.

Paperfolding puzzles can also, using established origami language, be characterised as pure or, presumably, impure origami puzzles, a pure origami puzzle being one that can be set and solved without the use of cuts, decoration or adhesives.

Alternatively, and perhaps more usefully, paperfolding puzzles can be divided into eight broad categories related to the object of the puzzle. These are shape forming puzzles, pattern forming puzzles, layering puzzles, table-top puzzles, fold and cut puzzles, transformation puzzles, discovery puzzles and assembly puzzles. These categories are not, however, always mutually exclusive.

Shape forming puzzles: In shape forming puzzles the object is to fold the paper to match the shape of a flat motif.

Pattern forming puzzles: In pattern forming puzzles the object is to fold a piece of irogami (paper which is white one side and coloured the other) to match a specified target pattern. For convenience, many pattern forming puzzles start from an unfolded square, but there is no compelling reason why this should be the case. In many cases the target pattern is also square, but in other cases the puzzle may specify that the finished pattern can be of any shape.

The aim of pattern forming puzzles is generally not only to achieve the target pattern but also to achieve it in the smallest possible number of folds. This requires some explanation. Technically a fold is a change of direction in the paper. When you flatten a fold you get a crease. So folding is a process and a crease is the result of this process. Counting folds and creases is not always the same. If you lay two sheets of paper together and fold them in half you could argue that you have made one fold but two creases. You can also make a fold without making a crease. Sometimes you have to begin to solve a pattern making puzzle by making one or more construction folds (folds that are not used in a solution but help locate other folds that are). For instance if you want to fold one corner of a square into the centre you need to know where the centre is. You can find this centre by creasing in both diagonals. The centre is where they cross. I do not personally count construction creases towards the total number of creases required to solve the puzzle since they can be avoided if necessary by using other ways to construct the necessary location points.

The aim of pattern forming puzzles may be to create single-sided patterns (i.e. the pattern is created on just one side of the paper) or double-sided pattern puzzles (where the pattern is created on both sides of the paper). Double sided patterns are not necessarily harder to solve but they are more complicated to understand since there are more variables to consider.

Layering puzzles:  The object of layering puzzles is to fold a sheet of paper so that the corners, or other clearly identifiable parts, such as certain squares within a larger grid, lie on top of each other in a given order.

Table-top puzzles: A table-top puzzle is a folding puzzle where some part of one or other surface of the paper must remain in contact with the top of a table (or a similar hard surface) while each of the folds (or unfolds) that lead to a solution is being made.

Fold and Cut puzzles: The challenge of a Fold and Cut puzzle is to fold a sheet of paper in such a way that the target shape can be cut from the paper using just a single straight cut. It is worth noting that, despite the name, Fold and Cut puzzles are pure origami puzzles. The solution is achieved just by folding the paper. The cut simply confirms that the solution has been found.

Transformation puzzles: The object of transformation puzzles is to change one state of a puzzle into another. There is a sense in which every paperfolding puzzle is a transformation puzzle, but I usually reserve the term for apparatus puzzles such as my Centrepiece where the possibility for such a transformation seems unlikely. Transformation puzzles can also be set up so that the aim is to find a route that avoids one obvious state.

Discovery puzzles: A discovery puzzle is a kind of transformation puzzle which includes a state that is difficult to access or can only be found as the result of some lateral thinking.

Assembly puzzles: The object of assembly puzzles is to find a way to put folded paper modules together to create a target shape. Since they are not solved by folding paper, but rather by assembling pre-folded or partly pre-folded modules, there is a sense in which these puzzles are not precisely paperfolding puzzles. However, the assembly aspect of modular origami is so much an integral part of modern day origami design that I have chosen to ignore this small consideration.