Origami Heaven

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

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Paperfolding Puzzles - Introduction
 
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 paperfolding 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.

As a result of this multiplicity of solutions I have not been able to include solutions to every part of every puzzle, or, where I know that multiple solutions exist, to include all the solutions I am aware of. Space simply does not permit this. It is also quite conceivable that I have not found the best solutions. They may still be waiting for you to discover them.

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. I have taken great care to try to state the conditions for each puzzle clearly, and you should take equal care to make sure you understand them before you start looking for the solution. This should not mean, however, that you should not feel free to think about unusual possibilities.

All the puzzles in this book start from squares but they require two different types of paper. Most require irogami, which is paper that is white one side and a single plain colour the other. The others require homogeneous paper, which is the same colour on both sides. Irogami is commonly sold as ‘origami paper’ in packs of pre-cut squares. Homogeneous paper usually comes in rectangles. The best source is photocopy paper. Pages 133 and 134 show you how to cut rectangles down to squares.

Some of the puzzles, and many more of the solutions, rely on the division of the width of the paper into three, four, five or more equal parts, or on grids that are created by dividing both the width and the height of the paper in this way. Sometimes estimating these divisions by eye will work perfectly well. It is, however, often preferable, and at times essential, to be able to do this division accurately. For this reason, three supplements showing you how to fold a 4x4 grid, and how to divide into thirds and fifths are included at the back of this book. You can adapt the methods shown there to allow you to divide a square into any other number of equal parts.

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 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. Most of the puzzles in this book are pure origami puzzles in this sense.

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. Merlin’s Mat, Nimue’s Mat, Foursquare and Juxtaposition are all examples of this type of puzzle.

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, all the pattern forming puzzles in this book 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, the exceptions being Equality, where the finished pattern can be of any shape, and Challenge 1 of Striptease, where the challenge is to create the desired pattern within the largest possible rectangle. All the target patterns in this book are flat but there is no reason why the challenge could not be, for instance, to create the pattern on all the surfaces of a cube using identical modules.

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. Construction creases do not count towards the total number of creases required to solve the puzzle.

The majority of the puzzles in this book are pattern forming puzzles, and of these the majority are single-sided patterns (i.e. the pattern is created on just one side of the paper). Double-sided pattern puzzles (where the pattern is created on both sides of the paper) are not necessarily harder to solve but they are more complicated to understand since there are several variants to consider. Examples of all four types of double-sided pattern are given on page 31. You may need to go and lie down after you have studied them.

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. Sequential is a simple puzzle of this kind.

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. There are two puzzles of this kind, Turnover and Inside Out, in this book.

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. Three such puzzles, Square Cut, Cross Cut and Staircase, are included in this book. 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 Centrepiece where the possibility for such a transformation seems unlikely. Transformation puzzles can be set up so that the aim is to find a route that avoids one obvious state. I have not included any of this type of puzzle in this book.

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. The Windmill Flexagon is a simple example of this kind of puzzle.

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 do not fit into the core concept of this book. However, the assembly aspect of modular origami is so much an integral part of modern day origami design that I have chosen to include two examples, Quartetra and 2 into 1, despite this consideration.

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It is perhaps worth briefly considering what it is that goes to make a good paperfolding puzzle. I think it is a combination of three things, simplicity, misdirection and surprise. A good origami puzzle should have a solution that is not immediately obvious, but is equally not beyond the reach of paperfolders who apply their fingers and imaginations to solving the challenge in a focused way. Ideally, there should be that 'Ah, yes ...' moment when the solution suddenly appears, and seems thereafter to have been obvious from the beginning. Solutions to simple puzzles often have this quality. The more complex the puzzle, the more likely it is that it can be solved logically rather than through a sudden flash of insight.

The first edition of this book was published in 1998. Except for Black and White, T-Time and the Flexatube, this second edition contains all the puzzles and solutions found in the original edition plus many additions and amendments. Unless otherwise stated all the puzzles and puzzle solutions in this book were originated by the author.

In a book of this kind it is always tempting to move straight from the puzzle to the solution. Please try to resist this temptation. I have had the pleasure of creating these puzzles, and working out their solutions, over many years and I would like you to share this same enjoyment. If you go straight from the puzzle to the solution not only will you miss out on the pleasure of finding at least one solution for yourself but also be unable to evaluate how easy or difficult the challenges are, or to appreciate the combination of mental and physical skills that are needed to solve them.

David Mitchell