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Impossible Folding Puzzles and Other Mathematical Paradoxes

Dover Publications, Inc.

Topology

Thirty spokes meet in the hub, but the empty space between them is the essence of the wheel. Pots are formed from clay, but the empty space between it is the essence of the pot. Walls with windows and doors form the house, but the empty space within it is the essence of the house.

—Lao Tse

Mathematically speaking, can we describe an object only by measuring all its visible aspects? Can we define an object or space only with the concept of distance or quantity? Objects are not static, in one way or another they change or deform.

Topology is a fascinating branch of mathematics that describes the properties of an object that remain unchanged under "smooth" deformations. If we imagine objects to be made of clay, a smooth deformation is any deformation that does not require the discontinuous action of a tear or the punching of a hole, such as bending, squeezing and shaping. From a topological point of view a cube, a pyramid and a sphere are all the same. In fact, in the world of topology, an object is classified by the number of holes it contains: cubes and pyramids are both "genus" 0—no holes. Donuts and coffee cups are "genus" 1, because each has 1 hole. A doughnut can be smoothly manipulated into the shape of a coffee mug, the doughnut hole becoming the coffee cup handle so that the number of holes is preserved.

Many puzzles are based on topological principles and understanding some very basic principles may help you analyze whether a puzzle is possible or not.

For example, the concept of what is closed or open is not always clearly evident. Can you say if the rings in the structure to the right are linked? If we apply a "continuous transformation" to them, you will find that they are not interlaced at all!

Topologists create and study strange objects, such as one-sided or three-dimensional objects where the inside and the outside are the same side! Probably the most famous of these is the Möbius strip, which you will discover in the following pages.

* The Tricky Hole

Trace a 2-cm (1/2 inch) diameter circle with a compass, or just draw an outline of a dime on the middle of a piece of paper. Cut out the circle you have just drawn (Fig. 1).

Your mission is to find a way to push a quarter (or any other similar coin) through the hole without ripping or making any cuts in the paper. There are two possible approches—using lateral thinking or, using your math skills.

Solution

To push a quarter through the hole, take the coin and set it on the table, insert your finger through the hole, and push the quarter across the table. Ok, this is a fun gag, but you have fulfilled the conditions of the puzzle: to push a quarter through the hole without ripping the paper sheet!

Here is the "real" (mathematical) solution: Fold the paper in half through the middle of the hole that you cut out (Fig. a). Cradle the quarter in the hole, so that it rests between the two folds of the paper. Then rotate the paper downward as you hold it on either side of the hole (Fig. b). If you play with it just a little, the quarter will slip through the hole.

When you rotate the two sides of the paper down, the hole becomes elliptical, increasing the size of the opening enough for the quarter to slip through.

* Elastic playing card

Is it possible to cut a hole into a playing card (Fig. 1) large enough for an average person to climb through?

Solution

Fold the playing card in half crosswise, and cut slits with a pair of scissors as shown in Fig. a. Cut down the original central fold, leaving the two end portions uncut. Finally, open up the hole in the card. It should unfold like an accordion (Fig. b). If you did it right, the hole is big enough for you to squeeze through!

* An intriguing three-dimensional puzzle

Can you reproduce this pyramid-like figure just by cutting and folding a single piece of strong rectangular paper? (You cannot cut the paper into two or more different pieces).

Solution

Just follow the visual steps shown in Figs. a to g to obtain the pyramid-like 3D shape.

Hold both ends of the paper as shown in Fig. a. Make a short crease in the middle of the sheet by "pinching" the central part only, and stopping about 2 cm from the left and right edges (Fig. b). Reopen the sheet and make a perpendicular cut from the middle of the bottom edge to the crease (Fig. c). Turn the sheet upside-down and make a diagonal cut from the middle of the bottom edge to one end of the crease (Fig. d). Then make a specular cut as shown in Fig. e. The cuts should look like an arrow pointing down ... Finally, turn the right flap of the sheet over by folding it forward 180 degrees (see Figs. f and g).

It is amazing how many people fail to find a solution to this simple puzzle! The mental block is caused by the fact that we perceive the paper as being a two-dimensional shape and we are not able to imagine an action in space, like turning a flap of the sheet to transform it into a three-dimensional structure.

More fun

The pictures below show some interesting variants of the puzzle.

* Flexible bottle-opener

Cut out the "bottle-opener" from a thick A4 Bristol sheet following the dimensions given in Fig. 1.

Bend and pass the end of the long paper strip (the stem or handle) through the rectangular opening (see Fig. 2) in order to form a loop.

Then fasten a paper ring around the strip as illustrated in Fig. 3.

Finally, tape the long end of the strip to a flat surface as shown below ...

Though it seems impossible, try to find an elegant way to disentangle the paper ring from the puzzle, without removing the tape or tearing any paper element. Will you be able to meet this challenge?

Solution

Pull the opening along the paper handle (Fig. a) and turn the rectangular frame inside out as shown in Fig. b until the loop has completely disappeared. If you did it correctly the strip should now look like Fig. c. Finally, bend the rectangular frame in order to easily slide the paper ring off the strip (Fig. d).

More fun:

Cut out two bottle-opener shapes from thick Bristol sheets and assemble them as shown in Fig. e. Tie a piece of loose string around one of the paper loops. Is it possible to remove the string loop from the puzzle? I will let you discover it by yourself!

* King of Love

Reproduce and cut out the template containing the picture of a King of Spades.

Fold the template along the central fold line and glue the two blank sides together as shown in Fig. 1. Then cut out the puzzle along all the dotted lines and remove the central rectangular piece. The rectangular "frame" on the back should remain connected to the rest of the card by the central stem (leg).

Your puzzle is now ready. The aim of the game is to transform the King of Spades into a King of Hearts, as shown in Fig. 2, by folding the frame without tearing or cutting it. Believe me, it isn't so easy!

Solution

This type of puzzle belongs to the large family of topological "Trap Door" puzzles. To solve it gently pull the top of the outer frame through the inner frame from back to front ...

Continue until the whole of the outside has passed through the small frame ...

Adjust the small frame as shown ... ET VOILÀ!

* Magic Flexagon

A "flexagon" is a flat geometric model constructed by folding a strip of paper that can be flexed or folded to reveal faces besides the two that were originally on the back and front.

Reproduce, cut out, and assemble this flexagon featuring triangular portions of snowflakes by following the simple instructions below (Fig. 1). You will need a little practice to fold the flexagon inside out (see Fig. 2): pinch the triangles together, then pull the very center outwards until the puzzle flips inside out (it might need a bit of gentle encouragement at first, but after you have done it a few times it will work quite easily).

Question: how many distinct snow crystals can you form with this puzzle?

Solution

The flexagon forms six different snow crystals in a kaleidoscopic fashion!

* Key Tag Puzzle

Knot a string through the hole of a door key as shown in Figs. 1 and 2. Attach three different color beads to either loose end of the string before tying both ends together to form a closed loop (Fig. 3). Your challenge is to move the bead on the left (blue in the diagram) to the other side next to the bead on the right (red in the diagram). As you can see, the hole of the key is too small for a bead to pass through. Is it possible to achieve the challenge? Try this puzzle with your friends and family!

Solution

Pull the string to make the loop large enough to pass the BLUE bead through, as depicted in Fig. a.

Keep pulling the loop along the string, over the other two beads.

Eventually, pass the BLUE bead through the loop (Fig. b). The puzzle is solved! (see Fig. c)

The knots of the initial and final states of the puzzle look different, yet they are equivalent: the knot of the final state is the flipped projection of the initial knot. A puzzle based on "knot equivalence" can be solved only by equivalent projection.

* St. George's Cross

Reproduce and cut out the rectangle with the crossing red lines (Fig. 1). Try to fold it so that to form a perfect cross as shown in Fig. 2.

Yes, there is a trick ... you are allowed to cut a very short slit into the paper to achieve the challenge.

Solution

Fold the rectangular paper sheet across the width blank sides together (Fig. a). Cut a slit into it as shown in Fig. b. Fold along two different axes and collapse the flap down (Fig. c). Turn the puzzle over (Fig. d) and fold the flap along the diagonal red line (Fig. e) to obtain a perfect red cross (Fig. f) on both sides of your paper structure.

* One-side loops

Fold a sheet of paper along its long side twice (Fig. 1) and cut it along the creases to obtain four uniform strips of paper (Fig. 2). Paste or tape the strips together in pairs, to form two single long strips. Form a paper ring with these two strips, having one strip joining the two sides of the loop as shown in Fig. 3. Curiously enough, this particular paper structure is a one-sided surface! That means that you can draw a continuous line on both sides of the loop without leaving the paper surface.

Can you find an elegant method of transforming this particular paper structure into a perfect square? You can use scissors, but no paper off-cuts should be left and only two straight cuts are allowed. This is a great visual thinking challenge!

Solution

First, cut along the length of the loop (Fig. a) in order to yield two rings of paper. Pass one ring through the other as shown in Fig. b and turn it over (Fig. c). You now have what look like a pair of paper "handcuffs." (Fig. d). Finally, make a straight cut along the strip of paper between the two loops (Fig. e) and a perfect square appears (Fig. f)!

More fun:

If you cut the paper structure along the length of the S-shaped loop, you will get two peculiar structures (Fig. g). Can you predict what shape you will obtain if you cut one of these structures along its middle line (see Fig. h)?

* Figure-eight shaped pastry

Can you alter this figure-eight-shaped pastry in order to thread the stick into its second loop? You cannot unthread the stick from the pastry nor cut the pastry in any way!

Solution

You can solve this puzzle using a topological method called "continuous transformation" as depicted in the picture. Topology deals with the ways that surfaces can be twisted, bent, pulled, or otherwise deformed from one shape to another without tearing or cutting.

* Power or Love: an amusing revolving structure

Reproduce and cut out the template representing a structure with revolving paper shutters. Fold the template along the central fold line and glue the blank sides together as shown in Fig. 1. Then cut the puzzle along all the dotted lines to form 4 paper shutters.

The goal of the puzzle is to rotate the shutters in order so that the same symbol is shown on both sides of the puzzle (heart or power symbol, see Fig. 2).

Solution

This three-dimensional folding puzzle is not difficult to solve—you just need a little bit of patience. The most intriguing thing here is the twisting structure. If you follow Figs. a through d on the next page, you will obtain a heart on both sides of the puzzle (Fig. e). It is important that you move the shutters in groups.

* Paradrom'ic rings

Fold a sheet of paper on its long side twice (Fig. 1) and cut it along the creases to obtain four uniform strips of paper (Fig. 2). Paste the strips together in pairs to form two single long strips. Give a half twist (180 degrees, see Fig. 3) to the first paper strip before joining its ends together to form a loop. Take the other paper strip and give it a whole twist (360 degrees) before joining its ends together to form another loop.

Can you predict what happens if you:

A. Start drawing a line around either of your paper rings?

B. Start cutting each paper ring along the middle line (see Fig. 4),?

Solution

A. If you start drawing a line around the first paper ring (the strip with a half twist), you will end up both the outside and the inside of the ring. Actually, this peculiar paper ring does not have a top or a bottom since the twist in the strip connects what was the top to the bottom, making one unique side. A strip with an odd-number of half-twists will always have only one surface and one boundary; whereas a strip twisted an even number of times will have two opposite surfaces and two boundaries. A paper ring with just one half-twist is called a "Möbius strip" after the German mathematician August Ferdinand Möbius who first described it.

B. If you cut the first paper ring you don't get two rings! You end up with a much larger, thinner paper ring (see Fig. a). Yet if you cut the other paper ring, you will obtain two linked paper rings (Fig. b).

More fun:

Now, try to cut the Möbius strip (the strip with a half twist) along about a third of the way in from the edge as shown in Fig. c, what did you get?—Two linked strips!

Make a paper ring with THREE half-twists, and cut it lengthwise. Are you surprised by the result?

(A paper strip tied in a trefoil knot! See Fig. d)

The lesson here is that cutting a surface does not always produce the expected result!

* An impossible knot?

Hold a flexible piece of thick rope about 120–150 cm (4–5 feet) long between the thumb and forefinger of each hand as shown in Fig. 1. During this experiment, you must at no time RELEASE your hold on the ends of the rope. Now, bring your right hand inside your left wrist and pull the rope out over the wrist and down behind (Fig. 2). Push the end in your right hand in through the left-hand part of the loop, and back through the right-hand part (Fig. 3). Move your hands apart until they are 20–30 cm (1 foot) away from each other—the loops should look like Fig. 4. Finally, toss the rope off your wrist. Et voilà, there is your knot! If at first you didn't succeed, try it again ...

Solution

I know this is a cruel trick. Actually, it is topologically IMPOSSIBLE to tie a simple overhand knot without letting go of one of the ends of a string! In fact, your arms and the piece of rope hanging from your hands form an "unknotted closed loop" and the only way to make a real knot in it is with the intercession of the 4th dimension.

You can however—thanks to a secret move—achieve the challenge. Before tossing the rope off your wrist, secretly grip the end of the rope just below the small loop with your little finger of your right hand (Fig. a). Surreptitiously let go the end of the rope, and grip it again with the right forefinger and thumb (Fig. b). Now, you will get a knot when you toss the loops off your wrists.

You can perform this trick for your friends and ask one of them to imitate you and, of course, as he/she doesn't know the secret, c he/she will never get a knot in the rope. So, you can offer to help and before he/she tosses the rope off his/her wrists take the ends of the rope (Fig. c) and invite him/her to withdraw his/her hands delicately. As a result, a wonderful knot will magically appear (Fig. d) with the consequent exasperation of your friend!

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Excerpted from Impossible Folding Puzzles and Other Mathematical Paradoxes by Gianni A. Sarcone, Marie-Jo Waeber. Copyright © 2013 Gianni A. Sarcone and Marie-Jo Waeber. Excerpted by permission of Dover Publications, Inc..
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