So You Think You've Got Problems

by Alex Bellos

  If you slice a doughnut, like the one above, with one straightline cut, you’ll get two pieces wherever you make the cut.

  How do you slice a doughnut with two straight-line cuts to make five separate pieces, and with three straight-line cuts to make nine separate pieces?

  I mentioned the joy of geometric puzzles. Let’s not forget the pain. Puzzles that require you to cut an object into as many pieces as possible are surprisingly hard. You can find yourself staring at the page for what seems like an eternity and making no progress. These puzzles are almost twice as painful because they look like they should be easy. But there’s no trickery involved.

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  A STAR IS BORN

  The five-pointed star shown below contains five ‘disjoint’ triangles – they don’t overlap, nor is any one triangle contained inside another. Draw two straight lines across the five-pointed star in such a way that the resulting drawing contains 10 disjoint triangles. I’ve included an extra star for practice, since it’s unlikely you’ll get it first time.

  The Greek historian Herodotus wrote that the study of geometry originated when the ancient Egyptian Inland Revenue dispatched tax inspectors with rope to measure fields flooded by the Nile. Members of other traditional professions, however, were also practising geometry at that time, at least informally. Carpenters and tailors, for example, are always faced with the problem of how to cut a piece of wood, or cloth, in the most efficient way possible. How, for example, do you cut up two identically sized squares of material to make the biggest possible square?

  Here’s the most efficient way:

  What if you had three identically sized squares of material and wanted to make the biggest possible square? Here’s a nifty solution from the tenth century that divides the squares into nine pieces. (First cut two of the squares along the diagonals. Then position the triangles as below right. Cut the parts outside the dotted square and insert them into the gaps.)

  Persian astronomer Abul Wefa described this method in On Those Parts of Geometry Needed by Craftsmen, a treatise he wrote out of exasperation at the lack of communication between artisans and geometers. When it came to cutting up shapes and fitting them back together, the artisans made erroneous deductions, while the geometers had no experience in actually cutting real things. Perhaps the theoreticians and the practitioners should talk!

  ‘Dissection puzzle’ is the term for a problem in which a shape is cut up into pieces and reassembled into another shape. The following dissection puzzle dates from the sixteenth century.

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  SQUARING THE RECTANGLE

  A tailor has a rectangular piece of fabric that measures 16cm by 25cm. How does he cut the fabric into two pieces that can fit together to make a square?

  Readers who bought a 701 series IBM ThinkPad in 1995 will have no problem solving this puzzle.

  In the nineteenth century, both academic and recreational mathematicians had fun with dissections. The German mathematician David Hilbert proved that any polygon (a shape with straight edges) can be transformed into any other polygon of equal area by cutting it into a finite number of pieces and reassembling them. In 1900 he published a list of 23 important open problems, which was hugely influential in setting a direction for twentieth century mathematics. The third question on the list concerned the dissection of polyhedra (three-dimensional solids with flat sides). Given two polyhedra of equal volume, is it possible to cut the first into a finitely large number of polyhedral pieces that can be reassembled to yield the second? That same year, someone proved that it’s impossible, making it the first of Hilbert’s problems to be resolved.

  Dissection puzzles flourished in the late nineteenth and early twentieth centuries. For Sam Loyd and Henry Dudeney, the most prolific puzzle setters of the age, they were stock-in-trades. Loyd, an American, liked to bring them to life with quirky set-ups. ‘Speaking about modes of conveyance in China…,’ he wrote as an introduction to the following problem.

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  THE SEDAN CHAIR

  Cut the shape shown below – which looks like a cross-section of a sedan chair – into two pieces that will fit together to form a square.

  The set-up for another of Loyd’s puzzles began: ‘During a recent visit to the Crescent City Whist and Chess Club, my attention was called to the curious feature of a red spade which appears in one of the windows of the main reception room.’ Curious indeed.

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  FROM SPADE TO HEART

  Imagine that the spade shown below is red. Can you cut it into three pieces to make a heart, thus changing its suit?

  You must use all three pieces, and none must overlap – otherwise you could just snip the stem and be done with it.

  I like the following dissection puzzle, devised by the French writer Pierre Berloquin, because we must transform a shape with curved edges into a shape with only straight ones.

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  THE BROKEN VASE

  With two straight-line cuts, divide the vase into three pieces that can be reassembled to form a square.

  Henry Dudeney, a Brit, was particularly innovative and ingenious in his dissection puzzles. Hilbert had showed that any polygon could be transformed into another, but his proof relied on dividing the original polygon into a potentially huge number of smaller triangles. Dudeney, on the other hand, valued elegance, and developed an extraordinary capacity to achieve beautiful transformations with as few cuts as possible.

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  We saw earlier that Abul Wefa needed nine pieces to transform three small squares into a big square. Dudeney did it with six.

  Dudeney’s most famous geometric discovery was the four-piece dissection that transforms an equilateral triangle into a square, shown below. If you link the pieces by their corners, as if they’re connected by hinges, folding them one way produces the triangle and folding them the other way produces the square. Dudeney was so proud of his discovery that he made a physical model of it with mahogany pieces and brass hinges, which he presented at a meeting of the Royal Society in 1905.

  Dudeney’s clever dissection puzzles inspired huge interest in this kind of problem throughout the twentieth century. Finding a ‘minimal dissection’ – that is, a dissection from one shape to another that uses the fewest number of pieces – was a perfect recreational challenge, since it required no deep knowledge of geometry. There is no general procedure to find a minimal dissection. What you need is creativity, intuition and patience. Before the computer age, an enthusiastic amateur might be able to outperform a professional, and many did.

  The next few puzzles are also about cutting up shapes in interesting ways.

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  SQUARING THE SQUARE

  The image below shows how to divide a square into four smaller squares.

  Show how to divide a square into:

  [1] Six smaller squares.

  [2] Seven smaller squares.

  [3] Eight smaller squares.

  In each case the squares can be of different sizes.

  That was the warm up. Now get out the scissors. The next puzzle is from Henry Dudeney’s Amusements in Mathematics (1917).

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  MRS PERKINS’S QUILT

  The patchwork quilt shown below is made up of 169 square patches. Divide the quilt into the smallest possible number of squares by cutting only along the sides of patches.

  To put it another way, you’re trying to find the smallest number of square pieces that can be sewn together to make the quilt.

  Mrs Perkins’s Quilt was the first appearance in mathematical literature of the concept of a ‘squared square’ – that is, a big square cut up into smaller ones. Like many of Dudeney’s puzzles, it captured the interest of academics.

  In the solution to Mrs Perkins’s Quilt, some of the squares are the same size. In the 1930s, mathematicians at Trinity College, Cambridge, set out to find a ‘perfect’ squared square: a square divided into other squares, each of which is a different size. (A group of Polish mathematicians around the same
time also looked into the problem.) In 1939, the German mathematician Roland Sprague was the first to publish a solution: a 4205 x 4205 square divided into 55 smaller squares with side lengths of different whole numbers.

  For the Dutch computer scientist A. J. W. Duijvestijn, finding squared squares was a lifelong obsession, the subject of his 1962 thesis and of much work over subsequent decades. He wanted to find the smallest possible ‘simple’ perfect squared square, meaning one in which no subset of the squares fits together in a rectangle or square. In 1978 his computer discovered the 112 x 112 squared square shown below, which is made up of only 21 squares. It is the smallest possible simple perfect squared square, and one of the most famous images in mathematics. (The number in each square describes its side length)

  An uncontroversial observation from the image in Problem 60 is that four identical squares placed together make a (bigger) square. Likewise, four identical Ls can be placed together to make a (bigger) L, as illustrated below. Shapes that can be placed with identical copies of themselves to make a larger version of the same shape are called ‘reptiles’, because they are tiles that replicate. Equivalently, reptiles are also shapes that can be subdivided into smaller identical versions of themselves.

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  THE SPHINX AND OTHER REPTILES

  Divide each of the shapes shown below into four smaller copies of itself.

  If it helps, the second and third shapes are composed on an underlying grid of equilateral triangles. The third shape is known as a ‘sphinx’, a term coined by T. H. O’Beirne, the New Scientist’s puzzle columnist in the early 1960s.

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  A reptile will tile a flat surface without gaps or overlaps, since you can arrange the tiles to make larger and larger versions of the basic shape. Here’s another type of reptile that will also tile a flat surface with no gaps or overlaps (although it will not replicate the original shape in a larger form).

  The lizard is based on a tessellation in Reptiles, a 1943 lithograph by the Dutch artist M. C. Escher. (A tessellation occurs when a single tile fits perfectly with identical tiles.) Escher created many tessellating tiles in the shapes of living creatures. His interlocking reptiles, fish and birds make up some of the most recognisable mathematical art of the twentieth century: striking, playful and utterly ingenious. To create a single tile that looks like a convincing representation of an animal, and which also fits together with identical tiles to leave no gaps or overlaps, is a hard challenge. Escher inspired many others to try the same. The number one tessellation artist working today is the Frenchman Alain Nicolas, who created the tiles in the following puzzle.

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  ALAIN’S AMAZING ANIMALS

  Divide the following shapes into the number of pieces indicated. The pieces are in the shapes of creatures.

  [1] Two identical pieces.

  [2] Two identical pieces, one of which is flipped over.

  [3] Three identical pieces, one of which is flipped over.

  One particularly addictive type of geometric puzzle shows a shape, or shapes, and asks, ‘What’s the area?’ Here are two such problems, one with squares and one with a triangle.

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  THE OVERLAPPING SQUARES

  The small square has a side of length 2, and the large square has a side of length 3. The left vertex of the large square is at the centre of the small square. The side of the large square cuts the side of the small one two-thirds of the way along.

  What’s the shaded area?

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  THE CUT-UP TRIANGLE

  A triangle is divided into four parts, such that the three numbered parts have areas of 3, 7 and 7.

  What’s the shaded area?

  For those of you who have forgotten, the area of a triangle is equal to half the base times the height, where the height is the perpendicular distance from the base.

  ‘What’s the area’ puzzles are both ancient and hyper-modern. Ancient because all you need are the rules of geometry set down by the Greeks more than two thousand years ago. Modern because they are eye-catching and shareable, perfect puzzles for the internet age.

  Indeed, Catriona Shearer, a maths teacher at a school in Essex, has gained thousands of Twitter followers by posting beautiful problems of this type, coloured-in with felt-tip pens. ‘I really like puzzles where a bit of clever thinking can sidestep a whole page of algebra,’ she says. ‘Plus I enjoy colouring them in.’ Here are two. The first one concerns circles, so you need to know that the area of a circle is πr2, where r is the circle’s radius. An arbelos, from the Greek word for ‘cobbler’s knife’, is an area bounded by three semicircles, as shown below.

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  CATRIONA’S ARBELOS

  The vertical line, of length 2, is perpendicular to the bases of the three semicircles.

  What’s the total shaded area?

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  CATRIONA’S CROSS

  All four shaded triangles are equilateral. What fraction of the rectangle do they cover?

  The next question asks you to find an angle, rather than an area. For the remaining problems in this chapter we’re in three dimensions. Hold on tight!

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  CUBE ANGLE

  What angle is made by the two bold lines on the sides of the cube?

  Prince Rupert of the Rhine is remembered by historians for his dog, and by mathematicians for his cube. A Royalist commander during the English Civil War, he was accompanied in battle by a large white poodle that his enemies, the Parliamentarians, believed had supernatural powers.

  When the war was over he became a noted artist and gentleman scientist, helping found the Royal Society. He was also the first person to notice a fascinating property of the cube that defies our geometric intuition. Take two identically sized cubes made of wood. It’s possible to cut a hole in one of the cubes and slide the other cube through it.

  The anthropophagic ability of a cube to eat itself arises because a cube’s width varies depending on where you measure it. For example, place a cube flat on a table. If you make a horizontal slice, the cross-section is a square. However, if you balance the cube on one of its vertices, so that the opposing vertex is vertically above it, a horizontal slice through the middle of the cube produces a hexagon. (It’s hard to visualise but consider it this way: when the cube is balancing on a point, a horizontal slice through the middle must cut every face. There are six faces, so the cross-section must have six sides, and because of the symmetry of a cube each side must have the same length.)

  The hexagonal cut can also be achieved by a slice that goes through the midpoints of six edges, as shown below.

  The square cross-section of a cube has a smaller area than this hexagonal cross-section does. In fact, the square cross-section can be made to fit entirely inside the hexagonal cross-section. In other words, if you make a hexagonal slice through a wooden 1m cube and then drill a 1m square-shaped hole into the slice, you will be left with a thin wooden ring that another 1m cube can slide through.

  Prince Rupert’s observation led to a further question: what is the largest possible cube that can fit through a hole in another cube with a side length of 1 unit? The problem was not solved for another hundred years, when Dutch mathematician Pieter Nieuwland showed that a cube with a side length of 1.06 (to two decimal places) can fit through a cube with side length 1. This cube of side length 1.06 is known as ‘Prince Rupert’s cube’. (In this optimal case, Prince Rupert’s cube does not go through the holed-out cube at an angle perpendicular to the hexagonal slice.)

  The next question is about hexagonally slicing a particularly interesting cube called the ‘Menger sponge’, a fractal object first described by the Austrian-American mathematician Karl Menger in 1926. The Menger sponge is a cube with smaller cubes extracted from it, and it is constructed as follows: Step A: take a cube. Step B: divide it into 27 smaller ‘sub-cubes’, so it looks just like a Rubik’s Cube. Step C: remove the middle sub-cube in each side, as well as the sub-cube at the centre of the original cube, s
o that if you looked through any hole you would see right through it. Step D: repeat steps A to C for each of the remaining sub-cubes – that is, imagine that each sub-cube is made from 27 even smaller cubes, and remove the middle one from each side as well as the central one.

  If you repeat steps for A to C for each of these 27 sub-sub-cubes, you get the cube illustrated opposite. (It’s called a ‘level three’ Menger Sponge since the iterative process has been carried out three times. You could, if you wanted, carry on the process ad infinitum, on smaller and smaller sub-cubes.)

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  THE MENGER SLICE

  Here is a Menger sponge. If you cut the object in two with a diagonal slice, what does the hexagonal cross-section look like?