# The Paper Puzzle Book

The subtitle of the book : All you need is paper (and scissors and sometimes adhesive tape if you want to be picky), might be tricking you into an imaginary situation of a kindergarten with children producing some artwork for mom, dad, or one of their grand parents. This is a completely different kind of book. You need a well trained set of brains and a strong puzzler's attitude to solve the puzzles that are collected by some of the best.

Ilan Garibi is an Israeli origami specialist, David Goodman is a designer of (mechanical) puzzles, and Yossi Elran is a mathematician, head of the Davidson Institute Science Education Accelerator of the Weizmann Institute in Rehovot, and a big puzzle fan. When they met at a meeting of recreational mathematics and games, the idea for this book was born.

In the best of Martin Gardner's tradition 99 puzzles are collected. Some are classics, some are found in the literature, and others are new. The authors are kind enough to give the origin of the puzzles when appropriate. The number of 99 is just a rough indication because there may be 99 problems formulated, but their solutions, which are given at the end of the chapters, sometimes propose variations or end with an extra challenge left open for the reader.

It may seem not very easy to represent with a static image (or images) in a book, all the necessary operations of folding an cutting that have to be performed in 3D and that sometimes even result in a 3D object. However the different steps are represented using some pictoral vocabulary that is explained in the beginning and that is remarkably clear and easy to read.

The puzzles are grouped according to techniques and topics in ten chapters. Sometimes puzzles are sequential, i.e., you first need to solve puzzle x before you solve puzzle x+1 because solving x is a subproblem of x+1. The puzzles are also rated with one up to four stars. Sometimes the shape of the paper is important for the technique to work: it need to be square or A4, but in other cases it can be just rectangular, or it has to be a long strip. Here is a list of the chapters with some simple illustrative example:

1. Just folding. For example fold a square paper into an equilateral triangle with a follow-up problem to fold the largest possible equilateral triangle that is contained in the square.

2. Origami puzzles. These need so called Kami paper whose sides have different colours, for example black and white. A first exercise is to fold the paper such that the visible areas of black and white are equal. This chapter is rather extensive.

3. 3D folding puzzles. Given a strip of size 1 by 7, fold it into a cube with side 1.

4. Sequence folding. Here one is given for example a square paper with a 2x2 grid defining 4 squares that are marked with the numbers 1 to 4 in lexicographical order. The problem is to fold the paper until it has size 1x1, but such that the squares on the folded stack have the natural order 1,2,3,4. Many variations are possible, starting from different configurations, or allowing a few cuts, etc.

5. Strips of paper. Here of course the Möbius band plays a prominent role, but there are other puzzles to formulate with strips.

6. Flexagons. This is an invention of Artur Stone of 1939 and popularized by Martin Gardner and later picked up by several others. Paper is folded into a polygonal form in such a way that that it has a front and a back side, but it allows for an simple flipping operation such that it is so to speak turned inside-out, showing different faces. One could define it as a flat folded configuration that has more than two faces. As a simple example one could start from a particular configuration of 6 connected squares (neighbouring squares have exactly one edge in common). Both sides have two squares marked 1, two marked 2 and two marked 3. Counting both sides, there are thus four 1's, four 2's and four 3's. This has to be folded into a 2x2 square and the 'first' and 'last' square are taped together so that one gets a sort of Möbius ring object that will allow only a limited number of hinged flips. The 2x2 square has to show the four 1's on the front and the four 2's on the back. By 'flipping' it, one gets all 3's on one side and all 2's on the other. There are three faces that can be shown in turn by flipping.

7. Fold and cut. For example, you have to fold a piece of paper in a certain way and cut it with one straight cut to obtain a prescribed shape like a cross or a star.

8. Just cutting. A classic is to cut a hole in an A4 size paper, such that a person can step through the hole without tearing the paper.

9. Overlapping paper puzzles. It is clear that, given three paper squares, one may arrange them in a partially overlapping way such that all three are only partially visible. This is impossible with four squares. Problems based on this principle can be formulated putting restrictions of the number or size of papers you start with, or restrictions on the shape of the outer boundary of the stacked papers.

10. More fun with paper. This is the miscellaneous section with many diverse fun constructs like putting together a rotator or an helicopter, performing magic tricks, solve (seemingly) impossible bets, etc.

The examples I gave above are just to illustrate the idea of what kind of puzzles are possible. They are usually the first kick start puzzles for the chapter rated with one or two stars. Sometimes these innocent looking problems can be be surprisingly difficult to solve even if they get the lowest difficulty rating. Although the solution methods for the puzzles are reminiscent to geometry, no mathematics is required. It reminds me of the ancient Greek idea of constructions using only compass and straightedge, but this is definitely different and even more basic: there is no compass, and there is no ruler. It is for example difficult to divide an edge of a square in three (or in n if n is odd) equal parts. That is only possible using an iterative pinching procedure. Such basic techniques are explained in an appendix. There is also a (limited) list of books, papers, and websites for further reading.

This is a marvellous book. The diversity of possible puzzles that can be given with these very limited resources, which are basically some paper and scissors, is overwhelming, and the challenges are sometimes very tough. Even the two-star problems may be hard for an untrained puzzler. This is medicine against boredom on long rainy days, but be careful not to get addicted or it may suck up your less empty and sunny days as well.

**Submitted by Adhemar Bultheel |

**10 / May / 2018