Mind Wrap

MIND WRAP BRAIN OWNER’S MANUAL

MIND

WRAP

BRAIN OWNER’S MANUAL

Copyright© 2014 by Wiley.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

All rights reser ved. No part of this book may be reproduced in any form without written permission form the publisher.

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Library of Congress Cataloging-in-Publication Data: Mind Wrap: The Science Behind your Thinking.—1st ed.

ISBN 123-4-567-89000-1 Printed in the United States of America 10 9 8 7 6 5 4 3 2

TABLE OF CONTENTS

01 09 29 45 BRAIN POWER

VISUAL THINKING

NUMERICAL REASONING

WORDS WORKOUT

02 ⁄ Brain Potential 04 ⁄ Structure 06 ⁄ Left vs. Right

10 ⁄ Vision 12 ⁄ Visual system 14 ⁄ Hermann Grid 20 ⁄ Necker Cube 22 ⁄ Illusory Contours 26 ⁄ Hidden details

30 ⁄ Numerical Aptitude 32 ⁄ Triangle Ratio 38 ⁄ Broken Calculator 41 ⁄ Tips 42 ⁄ Monty Hall Problem

46 ⁄ Verbal Reasoning 48 ⁄ How We Read 50 ⁄ Ambigrams 52 ⁄ Remembering 58 ⁄ Colored words 60 ⁄ Personal Capacities

BRAIN POWER

1.0

BRAIN POTENTIAL Introduction

“The human brain, then, is the most complicated organization of matter that we know.”

—Isaac Asimov

The brain should need no introduction. You should know it intimately. After all, the brain is what makes you you. But it’s a paradox that the organ that lets you know and connect with the world understands so little about itself.

Now, thanks to stunning research building upon centuries of investigation, science is peeling away the layers of mystery to reveal how tree pounds of flesh create an entire universe inside your head.

Today’s fastest supercomputers can perform millions of mathematical calculations within a single second, they can send messages from person to person around the world, adjust the flight of rockets zipping at bulletlike speed to intercept other rockets, and checkmate grand masters at chess without breaking an electronic sweat. Yet no machine available today comes close to matching the computational ability of the human brain. Machines are not poets, architects, doctors, or artists. They do not think. And, perhaps surprisingly, they have great difficulty making even the most rudimentary sense of the world. The brain makes humans unique. While it duplicates many of the functions of other animal brains—including the analysis of stimuli from the five senses; the coordination of muscle movement; and the regulation of heart, lungs, and other organs—the human brain also creates consciousness. Human brains synthesize and internalize a version of the world and create awareness of one’s place in that world. Unlike animals, humans know that they know. And they choose to act in response to that knowledge.

Millions of messages are speeding through your nervous system at any given moment, enabling your brain to receive, process, and store information, and to send instructions all over the body.

Your brain is capable of so much more than you might give it credit for. Just take a moment to consider all the things made by human beings. From the earliest tool, such as a pickax, to the modern skyscraper, and from the largest dam to the smallest microchip—the human brain is where all of these objects were first conceived, Undoubtedly, the brain is the most powerful tool at mankind’s disposal.

Your brain works around the clock. It generates more electrical impulses each day than all the mobile phones in the world. You have billions of tiny brain nerve cells interacting with each other in permutations that have been estimated to equal 1 with 800 zeros behind it. (To make that graspable, the number of atoms in the world—one of the smallest material things we can get a fix on—is estimated to be 1.33 with 48 zeros after it.).

STRENGTHS AND WEAKNESSES So, if we have such a powerful brain, why aren’t we all good at everything? Why are some of us forgetful?

Why do some of us have trouble reading maps? Why do some of us lack a sense of rhythm? Surely with all that “electrical” activity going on inside our heads, we shouldn’t be faced with these difficulties?

Think of the brain as a busy fairground with an assortment of rides and attractions, each representing a different area of the brain, and think of the people as tiny nerve cells or “neurons”. Now, the popularity of the various attractions tends to differ from one fairground to another; a ride in one fairground may draw more people than the same ride in another.

In brain terms, the “popular rides” are the parts of the brain with lots of nerve cell activity and, hence, tend to be more developed. This development is aided significantly by the kind of education we receive as a child. One person can be proficient when it comes to reading maps, another might be more creative, and a third, more logical. Of course this is a crude analogy because the different areas of the brain differs from person to person. In short, it’s a question of education and genetics. So, don’t be too hard on yourself if you think you’re bad at math or terrible at languages. The chances are that you excel in another area.

However, this doesn’t mean you cannot develop a mental ability that you consider weaker than another. It’s wrong to think that just because you’re not naturally gifted in something, such as map or map-reading, that there’s no point in trying to improve it. Your brain is similar to any muscle in your body in that exercise will raise its potency.

You can always strive to improve and expand your current mental aptitude.

3

The average adult brain weights about 3 lbs and it is divided into two hemispheres: the left and the right.

These are linked by a central processing unit. Each half is split into more compartments: 01 02

03

04

At the very back is the occipital lobe, which handles much of your visual sense.

Just behind each ear are the temporal lobes, which are involved in the organization of sou­ nd, memory, speech, and emotional responses.

At the top of the brain are the parietal lobes, which handle sensations, such as touch, body awareness, pain, pressure, and body temperature.

They also help you with spatial orientation.

Behind the forehead are the frontal lobes, which are considered the home of our personality. The uppermost part of the frontal lobes is involved in solving problems, activating spontaneous responses, retrieving memories, applying judgment, and controlling impulses. It also modulates our social and sexual behavior. This area is more developed in humans than in any other animals.

LIMBIC SYSTEM Inside the ridges and grooves of each hemisphere lie a set of structures forming what is known as the limbic system. This system includes the amygdala, hypothalamus, thalamus, and hippocampus.

These parts activate our emotions, appetites, inst­ incts, pain and pleasure sensations, and other drives that are essential to survival. The amygdala activates emotional responses, such as fear or euphoria, while the hypothalamus is the control center for brain-to-body, body-to-brain messages, causing for example, blood pressure to rise when we are agitated.

The thalamus receives auditory and visual sensory signals and relays them to the outer layer of the brain, known as the cerebral cortex, where the information is processed. The hippocampus is critical to learning and remembering spatial layouts. At the very back of the brain lies the cerebellum, which handles movement and balance and, along with the brain stem, is the part of the brain that evolved first, inherited from our primeval ancestors.

LOOKING CLOSER The brain’s basic unit is the neuron. It is a specialized cell designed to share information electrochemically with other neurons. Some chains of neurons send information to the brain from the body’s extremities.

They tell the brain to register the pain of a finger struck by a hammer, the noise of passing traffic as it falls upon the ears, and the sublime colors of Arizona sunset. Other chains send information from the brain to the body. They direct fingers to type, tongues and lips to form words, and eyes to swing right and left to focus on the ball at a tennis match. Other chains share data among themselves to construct subconscious or conscious thoughts and feelings.

Each neuron contains a cell body with a long, taillike fiber called an axon. The axon’s job is to send electrical impulses to other cells, thereby telling muscle cells to contract, relaying sensations from the body, and otherwise sharing information with other neurons. Some axons are short, extending only to adjacent cells in the brain. Others are much longer, carrying impulses down the spinal cord to move the arms, legs and feet. Axons may split and branch into as many as 10,000 knoblike endings that disperse impulses across many cells.

Each neuron also extends into networks of dendrites, which are thin, short fibers that transport electrical signals to the scope, axons and dendrites somewhat resemble the roots and branches that form the myriad tangles of a mangrove swamp. Only in this case, the branches of one tree reach out toward the roots of another, and vices versa.

The human brain contains perhaps 100 billion neurons. Each neuron links to so many others that the entire network forges literally trillions of connections, making the brain the most complicated in the known universe. And because humans and other animals can learn, these connections do not remain static. Every day, as the brain incorporates new experiences and new knowledge, neurons forge new connections. They can do so because neurons do not join one another like bricks mortared together in a wall or steel girders welded to form a bridge.

1.1 Instead, a small gap called a synapse lies between the axon of one neuron and the dendrite of another.

When a neuron sends an electrical discharge along the length of its axon, it halts at the synapse like a car at the edge of a cliff. There, the impulse activates electrically charged molecules stored in the neuron’s cell wall. These molecules, known as a neurotransmitters, leave the membrane of the first neuron, move across the synaptic cleft, and dock at a second neuron. The arrival of a neurotransmitter alters the electric charge at the edge of the new neuron and sparks a new electrical impulse. As impulses pass among complicated chains in the central nervous system, they form networks that specialize in performing particular functions, such as understanding language, remembering experiences from the past, and comprehending the outside world. They store, retrieve, and transmit information. All information processed by the brain is nothing more than electricity passing through neuron after neuron and pausing only to be converted into chemical energy as it leaps across each synapse.

DIFFERENT PARTS Neural networks lie in four main parts of the brain.

By evolutionary reckoning, the oldest portion of the brain is its stem, which begins as an extension of the spinal cord. The brain stem controls basic physical actions necessary for survival, such as heartbeat and respiration. It is home to many sensory and motor nerves, the latter named for their function of controlling movement in muscle tissue. Motor neurons also densely populate a second part of the brain, the cerebellum, at the back and bottom of the skull. The cerebellum coordinates precise, voluntary movements, such as tying a shoe or playing a violin, and also plays a role in emotion. A third component, known as the diencephalon, lies in the brain’s center.

It controls actions of the nervous system such as digestion, and relays sensory stimuli to other brain regions.

STRUCTURE What’s in your brain?

sue called the corpus callosum. The cerebral cortex is the outermost brain layer, folded and wrinkled and resembling a squishy pink walnut. The cortex is home to the higher functions that separate the human brain from those of other animals: reason, creative thinking, and language. The amount of neurological firepower necessary for such exploits is considerable: 76% of human brain mass lies in the cerebral cortex, a greater percentage than any other animal, and within the cortex lies about 75% of all neural connections.

The cerebral cortex processes information so that you may comprehend enough about the world to survive—and even to thrive. Evolution and experience have molded the cortex’s neural connections to favor sensory and cognitive functions that have proved successful over eons of human life.

CHALLENGE YOURSELF Anything that challenges your brain’s time-tested circuitry, such as the illusions and paradoxes of this book, opens a window onto hidden depths of self-knowledge.

Your brain’s ability to interpret what it experiences adds complexity to the end product of evolution that lies within your skull. A human brain, which can ask questions about what it sees and knows and then ponder what’s gained by the answers, must turn to the ultimate question: Just who is it, posing and solving these problems? A journey into your brain leads to yourself.

The fourth region, the cerebrum is what most people think of when they envision the brain. It lies in two hemisphere, left and right, connected by a band of tis5

Left Brain

LEFT VS. RIGHT

The sum of its parts

Analytical thinking Logic Organization Self-esteem

1.2

Attention to detail Processing information Receiving auditory input Language Word reading Math calculations Curious and impulsive actions Each hemisphere of the brain deals with different types of mental activity. The left side deals with logic, numbers, language, lists, and analysis—the so-called reasoning activities. The right side is more visual, and deals with imagination, color, spatial awareness, pattern, recognition, and making sense of the abstract.

Most people seem to have a dominant side. The crucial word here is “dominant”. It’s a natural preference, and not an absolute. What this means is that when you’re learning something new, your brain prefers to learn in a certain way. It is not so much that you are biologically right-brain- or left-brain-dominant, but that generally you’ve become comfortable with applying one side more than the other. The truth is that in practice you are always using both sides of the brain simply because most task demand it, so you shouldn’t get to hung up on this division.

Right Brain

Interpreting auditory input Creativity Comprehension Imagination Cautions and safe actions Unconscious actions Social skills Big-picture thinking Intuition Empathy Problem solving Interpreting information Non-verbal processing

7

VISUAL THINKING

2.0

VISION Interpreting the world around us

Sensory perception plays a crucial role in how the brain creates a simulacrum of the “real” world based on input from the five senses. Vision is nothing more than the creation of symbols in our head that represent what exists outside the body. Similarly, hearing decodes auditory stimuli in ways that give them meaning, and we react to that—when we hear a car backfire and we thing “gunshot”, for example.

The fact that you pick up a great deal of information from sight isn’t surprising since about 40% of your brain is dedicated to seeing and processing visual material. On average, people know the names of approximately 10,000 objects and can recognize them by their shapes alone.

Touch, smell, and taste interact with molecules outside the body to create their own sensations in the brain, leading to everything from pleasure to disgust.

In each instance, we react to the symbols, not to the reality upon which they are based.

Your visual sense is key to interacting with the world around you. By the time most children are six years old, it is estimated that they’ve already committed to memory the names of a fifth of the objects they will know in their lifetime.

LOOKING TO LEARN How much do you learn form your sense of sight?

Well, most experts agree that about 75% of your learning is though your visual sense. Take babies, for instance. With their inquisitive eyes they pick up behavior traits by observing the things that people do around them; they process and interpret facial expressions and physical gestures. From a single glance, babies can tell when their mothers are happy or angry with them. It’s not something that ever changes. Consider a first date. How much attention are they really paying to the conversation and how much attention are they spending on reading each other’s body language?

VISUAL SENSE

Studies have shown that visual stimulation helps brain development the most, and aids more sophisticated types of learning both when you’re growing up and during adulthood.

The ability to glean information from more abstract types of visuals, such as tables, graphs, webs, maps, and illustrations, is unique to the human race. By being able to interpret information from such sources, you are able to find meaning, reorganize and group similar things, and compare and analyze information.

“What the eyes see and the ears hear, the mind believes.”

—Harry Houdini

COMPLEXITY OF VISION Consider your sense of sight. As you read this sentence, the visual networks of your brain are taking in more than 100 million bits of information. Your eyes constantly flit from place to place, usually never landing longer than a split second on any one word.

Peripheral vision outside the dimensions of this page is a blur of color and shape; only a tiny region in the center of the eye called the fovea contains enough photo-receptors to see with great sharpness.

You may think you see the entire world as a sharp and seamless whole, but your retinas are segregating information into various categories, such as color, shape, and line, and permit acuity only on a small spot in the center of your field of vision. This screening process keeps the brain from getting overwhelmed by too much visual information. Instead, it collects what the brain needs to create a useful image of the world as you shift the focus of your attention.

As you perceive attributes of an object, whether it is a word on a page or a car going by on the street, your brain synthesizes the streams of information, matches them to images stored in memory, and makes the connection—and you recognize what you see. Because you reach these conclusions based on sketchy information, your brain fills in the blanks of perception.

For example, each eye has a blind spot at its connection point with the optic nerve, a point where

there are no photo-receptors. The overlap of binocular vision fills in that gap. You’re probably not aware of the blind spot with either monocular or binocular vision until someone else test for it by moving an object slowly across your field of vision.

Such perceptive synthesis fills in the gaps, for exa­ mple, when you spot a deer on the other side of a latticed fence. After you spot it, it startles and runs away: You are actually seeing a hundred bits of deer through the multiple gaps of the fence, but your brain integrates them into a whole animal.

For most people, the brain relies on information from two eyes. As the brain develops in a young child, it learns to process information from both eyes into one coherent image. The evolutionary benefit of this development is depth perception. If you close your left eye, then open it and close your right, you will note that the difference between the two images is far greater for objects close to your face than far away.

Judging distances is useful in the tactile work, such as threading a needle, and avoiding potential threats.

11

Anatomy

VISUAL SYSTEM

Right

2.1 Visual field

Much synthesis of visual information occurs in the retina, as signals from the cones and rods receptors get processed by three other cell types. The two optic nerves—one for each eye—relay visual information to each half of the brain, depending on which side of the retina the information is coming from. Each optic nerve splits the information it receives and routes some to each half of the brain. The right half of your brain sees the left side of your field of vision (which strikes the right side of your two retinas), and the left half of your brain sees the field of vision’s right side.

Visual information goes to the thalamus, just above the spinal cord, and is relayed to the visual cortex in the occipital lobe for further processing.

Left

Optic chiasma

Primary visual cortex

Right eye

Left eye Optic raidations

Optic nerve

Lateral geniculate nucleus

13

HERMANN GRID Illusions

Let your eyes linger on the grid of black squares separated by perpendicular white bars.

15

When you use your peripheral vision, do you notice anything unusual about the intersections of the white bars?

WHAT HAPPENED?

HISTORY

In the regions where the white bars intersect ghostly gray phantoms appear and then dissolve back into nothingness. These slippery ephemera seem real until you try to fix one in your sights by staring directly at it. Then it disappears.

German physiologist Ludimar Hermann discovered this phenomenon in 1870 while reading a book. The book’s author had arranged figures in a matrix on a page. When Hermann stared at the numbers, he saw gray spots at the intersections. He investigated the phenomenon, reported his analysis in scientific literature, an gave his name to the illusion: the Hermann grid. Others have used it for science and fun. A century after Hermann’s discovery, Professor Günter Baumgartner used the grid as an instrument with which to measure the size of the human retina’s receptive fields.

You can prove the phantoms are not on the page by covering two rows of black squares with white paper, or by using a sensitive light meter to take readings from the intersections and from the surroundings.

The explanation for the illusion lies in the organization and function of light-detecting neurons in the retina. Some of the retina’s most sensitive photo-­ receptors fire in response to light but lessen their response when nearby photo receptors fire at the same time. The process in which neurons suppress their neighbors’ actions is known as lateral inhibition. Where white and black come together, lateral inhibition causes the white to lose some of its brightness and to appear gray.

The gray disappears when you stare directly at it because the fovea—about the size of this letter o—is less sensitive to light than the rest of the retina and therefore is less susceptible to lateral inhibition. So, when you look directly at the ghostly gray shapes at the intersection of the grid, they likely disappear.

Three investigators—Keffer Hartline, Floyd Ratliff, and Henry Wagner—demonstrated the existence of lateral inhibition in 1954.

Work ing w ith the simple eyes of hor seshoe crabs, they found that photo-receptor’s electrical output decreases with stimulation of nearby photo-­ receptors. A nerve bundle called the lateral plexus connects the photo-receptors and allows them to influence each other.

The human eye, is much more complicated, but the principle is the same. The human retina contains five kinds of vision neurons: receptors (rods and cones), horizontal cells, bipolar cells, amacrine cells, and ganglion cells. When light hits rods and cones, they send signals that travel to the bipolar cells and then to the ganglion cells, which collect pulses for relay to the optic nerve. Horizontal and amacrine cells also interact with photo-receptors.

This arrangement lets one neuron communicate with many others. Convergence among rods heightens sensibility to light. Convergence among cones heightens the ability to see fine details. In particular, the phenomenon enhances the perception of edges.

Sometimes information gets added during neural convergence. Sometimes, strangely, the brain seems to toss it out. Why this is so springs in part from the evolutionary advantages of seeing edges, such as the lip of a chasm, as sharp and clear. If a light-colored object, such as a rock, lies next to a space, such as a shadowy canyon, lateral inhibition makes the canyon’s darkness seem darker and the rock seem brighter. This makes the rock´s edge to stand out, and you don’t tumble into space.

GRAY VERSUS GRAY

“The eye sees only what the mind is prepared to comprehend.”

—Henri-Louis Bergson

The two central gray squares reflect the same amount of light. Due to the effect of simultaneous contrast caused by lateral inhibition during retinal processing of the different backgrounds, however, the gray on the light background appears darker.

Lateral inhibition pools sensations of dim lights so the human eye can detect them. By sharpening edges between regions of darkness and light, it helps you distinguish between sparks of fireworks and the blackness of surrounding sky.

TAKEAWAY Neurons in the retina influence each other through lateral inhibition. This sharing of information lets rod cells pool faint signals and detect a weak light on a dark night. It allows cone cells to compile a dense mosaic of sharp, colorful detail in daytime. It also helps the brain detect the outlines of objects more readily. A minor side effect of these abilities is the existence of many optical illusions such as the Hermann grid.

17

NECKER CUBE Ambiguity

Take a long look at this cube.

Which side is closer to you?

HISTORY The Necker cube was designed inn 1832 by Swiss crystallographer Louis Necker. In this well-known illusion, the two-dimensional lines of the figure are interpreted by the brain as the projection of a three-dimensional object, the cube.

WHAT HAPPENS? Because of an absence of depth clues, the figure remains ambiguous, i.e., capable of being seen from two different perspectives. Keep staring at the illusion and you will experience your brain shifting between the two perceptual interpretations.

Timing differences in the firing rates of various netw足o rks of neurons are likely responsible for the fluctuations we experience in our interpretation of many illusions.

While it is possible after some practice to alternate rapidly back and forth between the two interpretations, you cannot perceive them simultaneously: the brain is forced to interpret the ambiguous drawing only one way at a time.

RESEARCH The Necker cube has shed light on the human visual system. The phenomenon has served as evidence of the human brain being a neural network with two distinct equally possible interchangeable states.

T he Necker c ube i s u sed i n epi s temolog y (t he study of knowledge) and provides a counter-attack against na誰ve realism. Na誰ve realism (also known as direct or common-sense realism) states that the way we perceive the world is the way the world act ually is. The Necker cube seems to disprove this claim because we see one or the other of two cubes, but really, there is no cube there at all: only a two-dimensional drawing of twelve lines.

21

Do you see a number in this figure?

WHAT HAPPENS? When we look at things, the cells in the striate cortex respond to lines of specific orientation, size, and location; edges and motion. And each of the striate cells is very particular: change a line’s orientation by only a few degrees one way or the other and the cell will cease to respond.

Since we don’t see the world in terms of lines, edges, or ientation, a nd motion, our bra ins must br ing add iti­o n a l pa r ts of the v isua l cor tex into play.

Activation of the cells immediately surrounding the striate cortex leads to increasing detail but also leaves us susceptible to visual illusions such the one in the following page based on the Gaetano Kanizsa’s famous figures.

We see a number one although no number is actually there. The missing contours of the circles, trick the brain into perceiving the outline of a “1”, although it is not there, it’s just par t of the background. Thus, “edge neurons” in the striate cortex can’t really be detecting the edges of the “1” because there aren’t any real edges to detect! This phantom edges, which are perceived even though they’re not really there are called illusor y contours. They are created based upon the brain’s tendency to perceive a line as continuing in what seems to be its apparent direction.

Visual misperceptions also occur if some par t of the figure is hidden or obscured by another figure.

Think of the brain as a gambler who wagers on the most likely of two or more alternative explanations for an ambiguous figure. As with any gambler, the brain isn’t always right.

1 ILLUSORY CONTOURS Seeing edges

23

25

An impossible object is a type of optical illusion. It consists of a two-dimensional figure which is instantly and subconsciously interpreted by the visual system as representing a projection of a three-dimensional object.

HIDDEN DETAILS Visual patterns

FINDING PATTERNS This puzzle strengthens your ability to scan for visual patterns and details.

The hidden details puzzles focus on the two-dimensional pattern of shading and shape in a picture, rather than on naming the objects in a picture. As with solving a jigsaw puzzle, you must look for cues in a small fragment that tell you where it fits in the larger picture on the right.

If you are an artist of visual designer, this exercise will seem familiar. If words are your preferred mode of thinking, then you will probably approach this problem by verbalizing the features of each detail, then searching the picture for those features. The more you do this puzzle, the quieter your internal dialogue will become, and the more directly you will be thinking visually increasing your visual abilities.

Find where each of the small circular details fits in the big picture.

27

NUMERICAL REASONING

3.0

NUMERICAL APTITUDE Interpreting the world around us

Numbers are everywhere! But mention math­e matics and many of us cower. That may be surprising since even babies and animals can register some kind of rudimentar y counting mechanism. Ever yone has an innate degree of numerical aptitude. It’s built into our nature. We are always handling numbers and performing mental exercises with them. Think about it. When we wake up, it is usually because our alarm goes of f at a set time—a time that we interpret through reading numbers. When we make our favorite dish following a recipe in a book, we use numbers to get the proportions of the ingredients right. Numerical reasoning forms the cornerstone of logic, rationality, argument, and proof. Yet, when many of us are asked whether we are any good at math, we tend to answer in the negative because the word dredges up memories of struggling with formulas and fractions, geometry and trigonometry.

NUMEROPHOBIA Some people have difficulty dealing with numbers from a young age. Whether it’s caused by a fear that developed at school or is some kind of mental block, they cannot cope. If you’re one of them, you might be someone who suf fers from numerophobia: literally, the fear of numbers—an irrational belief that your brain cannot process mathematical problems (although math is about applying logic and rationalit y, it is, paradoxically a f fected by emotion).

The truth is that even those of you who suf fer from the phobia still apply mathematical skills unconsciously throughout your daily life. Overcoming the anxiety requires an ongoing commitment to learning, to acknowledging fears and working through them. You’ll be surprised how q uickly the brain learns new responses to enduring fears.

Everyone has an innate degree of numerical aptitude.

It’s built into our nature. We are always handling numbers and performing mental exercises with them.

VISUALIZING MATH

NUMBER WORKOUT

Numerical reasoning becomes easier when you visualize mathematical concepts. Einstein once cla imed th at th is th in k ing process took place th rough v isu a lization a nd th at he ver y ra rely thought in word s at a ll. Cr ucia lly, bra in sca n s show that during calculations activity is not confined to the lef t hemisphere, but is also present in the visual, auditor y, and motor areas of the brain.

F u r ther more, geometr y a nd read ing g raph s by their nature require you to use your visual skills to understand complex numerical data, which immediately involves regions of the right temporal lobe.

What we do know is that when a math problem is presented visually, it becomes clearer and more accessible, and the brain is more capable of recalling the knowledge later on.

Did you know that doing numerical exercises gives your brain a workout similar to that which your body receives from a weight session at the g ym?

Here’s how it works. The ner vous system of your brain contains neurons and, within them, axons, which are the ner ve fibers that transfer impulses between neurons. The speed of this transfer determines how ef ficient your brain is at processing information. Doing addition is one of the easiest ways of protecting the axons because the activity increases the insulation around them (also helped by diet), which fortifies the connections between the neurons. Mental arithmetic helps speed and accuracy, while more sophisticated math boosts your problem-solving ability. Turn over to begin some numerical workout.

31

TRIANGLE RATIO Visual Math

A circle has an eq uilateral triangle touching its circumference on the outside and another eq uilateral triangle touching its circumference on the inside, as pictured.

What is the ratio of the areas of these two triangles?

VISUALIZATION IS KEY Different parts of your brain become active once you start making math visual, which leads to a more holistic brain workout. In addition, you learn to understand the language of mathematics by finding ways o visualize its logical meaning. The truth is, many people are instantly put off by a numerical problem whit it is presented with large numbers and arcane symbols. So it stands to reason that adding a visual component to learning math makes it more engaging from the start.

Here’s an example to get you started. If you simply read the problem, you might become confused but the problem becomes much easier when you study the diagram and visualize it.

DOES FASTER EQUAL SMARTER? If you’re able to think more quickly does that make you a smarter person than someone who takes more time? In general terms, we’d have to say it’s debatable. For example, an artist may take years working on a masterpiece but does that reflect on his or her intelligence? In numerical terms, however, the answer is “yes”. The speed in which you managed to complete these exercises is an indicator of your current numerical aptitude. The ability to process information rapidly indicates there is more neuron activity in those areas of the brain. However, you should always take enough time to ensure that the answers are correct. It’s not very smart to make mistakes through sheer carelessness.

>> 33

Solution: You don’t have to do any complicated mathematics.

If you rotate the inner triangle by 180 degrees it should become obvious quickly that the ratio is 1:4.

CAREER WITH NO MATH? So y ou m ight be w onder i ng ho w t hese m at h exer­c ises could relate to your ever yday life. The familiar stor y is that we leave school or college and consign our math books to the attic, or even throw them away, thankful that we won’t have to do another calculation for the rest of our days. Think again! Do you know how many jobs or careers exist where you won’t have to use what math you have learned? On average, less than 10% throughout the world (considering a very conservative estimate).

The truth is that everybody needs to use numerical skills, whether in personal or working lives, and the more well-honed yours are, the better.

35

BROKEN CALCULATOR Visual Math

This calculator fell out of your bag and into a puddle, and is now experiencing a major malfunction. Only the buttons highlighted in the illustration on the right actually work.

Your task is to compute the numbers 1 to 15 with the limited means the calculator offers.

Hint: For instance, 0.5x2 will give you 1 —the first of the 15 digits.

Turn the page to find the complete solution.

>>

4 5 2 3 0 . + = 39

>> Solution:

1

=0.5x2

USEFUL SHORTCUTS

2

=0.5x4

3

=0.5x5+0.5

4

=2x2

15 percent tip: if you need to leave a 15% tip af ter a meal at a restaurant, here’s an easy way to do it.

Work out 10% (divide the number by 10)—then add that number to half its value and you have your answer in a heartbeat.

5

=2.5x2

6

=2x3

7

=4+3

8

=4x2

9

=4+5

10

=5x2

11

=2+4+5

12

=3x4

13

=5x2+3

14

=5x2+4

15

=5x3

15% of $35= (10% of 35) + ((10% of 35) / 2) $3.50 + $1.75=$5.25 Dividing by 5: dividing a large number by 5 is actually very simple. All you do is multiply the number by 2 and move the decimal point: 2978/5 Step 1: 2978x2=5956 Step 2: 595.6 Multiplying by 9: if you have to multiply a number in your head by 9, let’s say 168: multiply it by 10 (1,680) and take away 168, giving you the answer: 1,512.

Adding big numbers: if you have to add some fairly tricky numbers in your head, for example 329 and 457, round one of the numbers up (329 to 330), making a total of 787 easier to calculate, then subtract 1 to get the answer of 786.

Percentages: find 7% of 300. Sounds tricky? First of all, think about the words “percent”—it means per hundred. So, it follows that 7 percent of 100 is 7, 35.73% of 100= 35.73. But how is that usef ul?

Back to the 7% of 300 question. &% of the 1st hundred is 7. 7% of the 3rd hundred is also 7, and yes, 7% of the 3rd hundred is also 7. So 7+7+7=21. If 8% of 100 is 8, it follows that 8% of 50 is half of 8, or 4.

3.1 1

TIPS Improving numeracy

Take your time, especially if you’re out of practice.

Think of it as a process to improve your aptitude rather than a test you need to pass in the quickest time. Speed will come with practice and understanding the mathematical process.

2 3

4

Be imaginative with math. Tr y to see a problem in dif ferent ways. This will allow you to use a range of dif ferent methods to arrive at the answer. Opposite are a few shortcuts that will be helpf ul if you repeat the quick-fire test.

To develop your ability to perform quick calculations, include numerical testers into your daily life. Add up grocer y bills in your heads as you go around the store. If you drive, calculate in your head how much you’ll have to pay for a quarter, half, three-quarters, and f ull tank of f uel. Next time you’re in a restaurant with friends, don’t use the calculator on your mobile phone; instead, use mental arithmetic to figure out how much each of you will owe.

Once you understand a concept, keep practicing.

This is important since the more you practice, the more the concept will transfer from your working brain to your long-term memor y.

5

Visualize! It is natural f unction of the mind and should be applied to many mathematical tasks and problems. Key concepts such as division or place value (100s, 10s, and units) are of ten made clearer by using pictorial explanations such as graphs and tables.

41

MONTY HALL PROBLEM Visual Math

1

3

2

This problem is a brain teaser in the for m of a probability puzzle, loosely based on the American television game show Let’s Make a Deal and named af ter its original host, Monty Hall.

You are a contestant on Monty Hall’s Let’s make a Deal. There are three doors onstage. Hidden behind one door is a new car; behind the other two are goats.

You choose door 1. Now the Monty Hall paradox: Monty opens door 3, revealing a goat. He of fers you a deal: Keep your door, or trade it in for the other unopened door.

Should you trade?

Surprisingly, it is better to trade. When you choose door 1, there was 1 /3 chance that the car would be behind it. This leads to two possibilities.

Possibility 1: The car is behind door 1, in which case the host can choose to open either of the other two doors. In this case, which occurs 1 /3 of the time, it is better to stick with door 1 an not trade, since the car is behind door 1.

Possibility 2: The car is not behind door 1, in which case the host has no choice but to open the one closed door with the goat. In this case, which occurs 2/3 of the time, the host’s choice of door tells you that the car is behind the door that he did not open, so it is better to trade doors.

In other words, 1/3 of the time is best to stay and 2/3 of the time is better to trade.

The Monty Hall problem is deceptive because it seems that the host’s choice of door doesn’t tell you anything usef ul, but 2/3 of the times it does.

43

WORDS WORKOUT

4.0

VERBAL REASONING The power of language

Over your lifetime, your brain may store as many as one quadrillion (a million billion) bits of information in long-term memory.

There is a direct correlation between verbal aptitude and success in life. We’re not just talking about the ability to complete crosswords, unravel anagrams, or figure out antonyms, although all of those activities are great for exercising your verbal aptitude. We’re talking more generally about the ability to use words, to manipulate language so that you can communicate ideas, thoughts, opinions, and feelings cogently.

Arguably, politicians and lawyers utilize this skill best, as do rap artists and talk show hosts, who are all adept at engaging a mass audience with the power of words, of ten using them to influence an audience’s way of thinking. In short, the better your verbal intelligence, the more confident you will be at asserting your needs and wants. You will be better understood and will be able to form closer relationships. Whatever the path you take in life, improving your verbal aptitude will have a marked ef fect on your social progress and prosperity.

LANGUAGE AND THE VISUAL Scientists believe that by the age of five you may already have about 2,000 to 3,000 words in your vocabular y, but that does not mean you know the exact meaning of these words. For example, a child seeing a ball might say “ball” but he might also say “ball” pointing at a balloon, a chocolate egg, or a

pebble. What this suggests is that on an instinctive level, the visual sense has an enormous influence on how language develops. For instance, consider the first alphabet book a kid looks at. An image is used to qualif y and give meaning to a character in an alphabet. For example, “A” for Apple, “B” for Bear, and so on, so it stands to reason that a young child uses the same word for similar-looking shapes until his vocabular y grows. And while you may think that this reliance upon the “visual” is something you grow out of by the time you get through school (having accumulated a vocabulary of about 50,000 words), consider the use of analogies, metaphors and similes. Visual concepts influence lang uage throughout your life. For instance, public speakers and those in positions of power such as politicians know that they stand a greater chance of keeping you engaged if they use words to tell a stor y that conjures up a “big picture.”

Words may evaporate, but use them to convey an image and the idea behind it will be more memorable. The great orators have always relied on the “visual” to fashion speeches. Consider Martin Luther King’s famous address to the nation. The words “I have a dream...” instantly open a window to his vision of the f uture!

HOW IT WORKS The ability to use words and tap into the vast possibilities of the spoken and written language boosts the brain’s processing power by opening up additional neuron pathways. Scans have traced activity throughout the brain and not just the left side, indicating that verbal reasoning is an extremely complex process.

When you engage in a conversation, a whole series of cognitive functions take place even before a sentence reaches the tip of your tongue. A thought lights up in your head, your brain then refines it suing all the sensory associations, sends this information to two key areas of the brain, which then select the necessary words to convey its meaning, and finally place the words into a grammatical framework. Only then are you ready to speak.

THE LANGUAGE POWERHOUSES The two main powerhouses of the brain’s linguistic system are called Wernicke’s area and Broca’s area, named after the two scientists who discovered these regions in the 1800s. Broca’s area, which is located in the frontal lobe cortex, is responsible for language production—putting together sentences, using the correct syntax, and so on. Wernicke’s area, which is located in the temporal lobe, is responsible for la ng u age processing—u n scra mbling others’ sentences, analyzing them for syntax and

in f lection, and ex tracting meaning f rom them.

A connecting neural pathway called the arcuate fasciculus runs between the two so that the areas are always working together. This system taps into other areas of the brain, allowing you not only to talk and to understand speech, but also to read and write, and even make speech-associated gestures.

It also gives you the power to understand complex thoughts, and acquire new knowledge.

VERBAL FLUENCY Improv ing you r vocabu la r y ra ises you r intelligence, plain and simple. The average person’s spoken vocabular y is about 1,000 words and the number of words available to feed the brain is over three million. So there’s a vast scope for improvement. The broader your vocabulary, the more it will stimulate the brain by firing cell interaction during conversation, reading, and writing. A broad vocabular y gives you an advantage in school, business, and social situations. This is because you are able to think about more complex things precisely. Verbal f luenc y w ill g ive you the double advantage of thinking more quickly under pressure and talking more composedly under duress.

47

HOW WE READ Making connections

Read the words on the facing page.

WHAT HAPPENED? The text you just read is a hoax. A compelling, scalp-scratching hoax, but a hoax just the same.

The text contains shor t words that were never scrambled and many words that can be g uessed from context. Matt Davis of Cambridge’s Cognition and Brain Sciences Unit believes that letter order, word shape, and context all play important roles in reading.

WORD CONTOURS Your brain begins to amass a storehouse of printed words by learning letters and then linking them to create meaning. But as you read faster, you seek clues such as shape and letter order to identif y words on a line of type.

shape CONNECTIONS You do not read letter by letter. Instead, par ts of a word link together in the visual and reading circuits of your brain. It actually picks out letters more quickly in the middle of a word than in isolation.

TAKEAWAY Reading is a complex activity integrating unconscious motor skills to move eyes in discrete jumps, sensory skills to recognize words by their letters, and cognitive skills to decode meaning. How complicated, yet how sublime, that reading creates sounds and pictures in the mind.

Aoccdrnig to a rscheearch sudty at Cmabrigde Uinervtisy, it deosn´t mttaer in waht oredr the ltteers in a wrod are; the olny iprmoetnt tihng is taht the frist and lsat ltteer be in the rghit pclae. The rset can be a toatl mses, and you can sitll raed it wouthit a porbelm.

Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the wrod as a wlohe.

49

An ambigram is a word, art form or other symbolic representation, whose elements retain meaning when viewed or interpreted from a dif ferent direction, perspective, or orientation. The meaning of the ambigram may either change, or remain the same, when viewed or interpreted from dif ferent perspectives.

SO 360Ëš

51 A natural ambigram is a word that possesses one or more of the symmetries when written in its natural state, requiring no typographic styling. For example, the words “dollop”, “suns” and “pod” form natural rotational ambig rams. In some fonts, the word “swims” forms a natural rotational ambigram. The word “bud” forms a natural mirror ambigram when reflected over a vertical axis. The words “CHOICE” and “OXIDE”, in all capitals, form natural mirror ambigrams when reflected over a horizontal axis.

The longest such word is CHECKBOOK. The word “TOOTH”, in all capitals, forms a natural mirror ambigram when its letters are stacked ver tically and reflected over a vertical axis.

>>

OS

REMEMBERING False memories

1

Carefully study the words in each list. Start with column 1 and then move to column 2.

Thread Pin Sharp Point Sewing Eye Thimble Prick Thorn Hurt Haystack Injection Syringe Cloth Knitting

Then turn the page.

Bed Tired Rest Awake Wake Dream Snooze Blanket Doze Slumber Snore Nap Drowsy Peace Yawn

2

>> 53

Which of the following words appeared in the list?

Awake Door Candy Needle Sewing Sleep Think about it and turn the page to see if you were right.

LOOK CAREFULLY False memor ies of unseen and un heard words arise from associations. Your mind associates all the words from the first list with needle and all the word from the second list with sleep.

>>

WHAT HAPPENED?

THE POWER OF ASSOCIATION

This experiment is known as a DRM test. That’s shor t for the names of the three researchers most closely a ssociated w ith the procedu re: Ja mes Deese, who developed the test in the 1950s, and psychologists Henry L. Roedinger III and Kathleen McDermott, who rediscovered it and gave it to college students in the 1990s.

Canadian psychologist A llan Paivio has found that a hierarchy of associations heightens memor y. The brain remembers objects more strongly than it remembers pictures of the same objects. It also remembers pictures of objects better than it remembers words for objects, which af ter all have no concrete connection to the objects themselves.

[Why is a rose called a rose?]

Roed iger a nd McDer mot t found what they ca ll “remarkable levels of false recall and false recognition” among undergraduates asked to identif y words they were told to remember from lists of 15 related terms. So, for words such as thread, sewing, and knitting, as many as half the students remembered, incorrectly, that the list contained the word needle. Similarly, they swore that the second list of words, which is included blanket, doze and pillow, contained the word sleep. Neither word appears on the previous page.

Harvard psychologist Danial Schater reports false memor y rates in such experiments as high as 80 to 90 percent among hundreds of his students .

According to McDermott, urging people to be on their g uard does little to change the results.“Warning them about the possibility of illusor y memories does not permit people to control their thought processes and avoid having them”, she said. “It’s clear that people have difficulty suppressing false memories. The key questions now are how and when are these mistaken memories generated and can they be avoided somehow?”.

“We believe that the concept of ‘sleep’ is aroused through activation spreading through a semantic network while people listen to the list of words”, Roediger said. Activation occurs both externally, as test subjects hear the words, and internally as they associate the words into a coherent structure.

The power of memor y increases when you link words to images. That’s because the encoding of such memories occurs across a variety of neural networks, including those for language and vision.

This explains the ef ficacy of a memor y technique that dates to ancient times. Called the Roman room for its description by Roman poet Cicero, it works by linking words and phrases to images. If you create a concrete image for a word and place it in a particular location, you are more likely to remember it.

Schater demonstrated that visual associations boost the accuracy of DRM tests. When he showed pictures while subjects heard words from a DRM list the association proved decisive. Subjects correctly said they had heard a par ticular word only when they could recall seeing its image.

TAKEAWAY Hearing or seeing lists of words activates neural networks of perception and cognition. Your brain waves the two together to make coherent associations. As a result, you may develop false memories.

Inside your brain, false memories quickly become just as true, in your estimation, as real ones. If that’s the case for something as simple as 15 words on paper, consider how likely it is for more complicated events, such as crimes or car accidents.

55

yellow black blue

red

red

green

purple orange orange black yellow orange

COLORED WORDS Switching tasks

In the fastest time possible, say aloud the color the word is printed in, trying not to read the word.

WHAT HAPPENED? It was tricky not to read the words and just focus on the color, right?

This exercise works both hemispheres of the brain, w ith neurons zapping bet ween the verbal and visual sites as you tr y to manage your attention, you are inhibiting one response in order to say something else.

MENTAL NOTE A lthough the right and lef t halves of our brain share some functions, not all functions are divided eq ually. On the lef t usually are the centers for understanding and using language, including reading and writing, and manipulating details. On the right usually are the centers for spatial awareness and information integration.

59

PERSONAL CAPACITIES What is your style of thinking?

Ever ybody has uniq ue learning styles. Students gain strong benef its when their teachers recognize their streng ths and weaknesses as learners.

Howard Gardner, a psychologist and professor of neuroscience at Har vard, developed one theor y in 1983. Gardner defines “intelligence” not as an IQ but, rather, as the skills that enable anyone to gain new knowledge and solve problems. Gardner proposed that there are several dif ferent types of intelligences, or learning styles.

01

Verbal-Linguistic (Word Smart): People who possess this skill learn best through reading, writing, listening, and speaking. Verbal students absorb information by engaging with reading materials and by discussing and debating ideas.

02

Logical-Mathematical (Logic Smart): Those who exhibit this type of intelligence learn by classifying,categorizing, and thinking abstractly about patterns, relationships, and numbers.

Combinations of the different types of intelligence abound. All of these learning styles indicate different ways of interacting with the world. Ever yone has some degree of each, but each person favors certain learning styles. This is significant because when you prefers one learning style over another, it af fects your success.

03

Interpersonal (People Smart): Those who are people smart learn through relating to others by sharing, comparing, and cooperating.

Interpersonal learners can make excellent group leaders and team players.

04

Intrapersonal (Self Smart): People that learn best by working alone and setting individual goals.

Intrapersonal learners are not necessarily shy; they are independent and organized.

05

Auditory-Musical (Music Smart): Students who are music smart learn using rhythm or melody, especially by singing or listening to music.

06

Visual-Spatial (Picture Smart): These people learn best by drawing or visualizing things using the mind’s eye. Visual people learn the most from pictures, diagrams, and other visuals.

07

Naturalistic (Nature Smart): Naturalistics learn by working with nature. They enjoy learning about living things and natural events. They may excel in the sciences and be very passionate about environmental issues.

08

Bodily-Kinesthetic (Body Smart): This individuals learn best through touch and movement. These people are best at processing information through the body. Sometimes they work best standing up and moving rather than sitting still.

Imagine that you are tackling a written essay or report. Consider different methods by taking advantage of your strongest learning style: Logical-Mathematical—Use a graphic organizer such as a web or stor y map to categorize and organize thoughts before writing. An outline is a written version of a graphic organizer.

Visual-Spatial—Draw the subject of the piece, and then write or create the written draft. Details in the drawing will lead to details in the writing.

Auditor y-Musical—Listen to background music to block out other, distracting sounds. Hum. Chant.

I have a sneaking suspicion that Dr. Seuss, with his talent for rhy thm and meter and rhyme, was an auditor y learner.

Finding, recognizing, and valuing dif ferent combinations of multiple intelligences is a key to applying these skills ef fectively. Sometimes an intrapersonal learner and an interpersonal learner working together will be in conflict. But when both step back and consider their dif fering outlooks, they may find that they’re both headed for the same result; they’re just taking dif ferent paths to arrive at the goal. Professionals such as these two learners might team up later on to create or advance a new, successf ul idea!

Multiple Intelligences Average test results for males and females, aged 19 or over, living in United States.

If you want to learn more and take the test yourself visit the web http://www.bgfl.org

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BIBLIOGRAPHY

BOOKS

WEB

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Harrison, James, and Mike Hobbs. Brain Training: The Complete Visual Program. New York, N Y: DK, 2010. Print.

“G e s t a lt P s y ch olo g y.” Wi k ip e d i a . Wi k i m e d i a Foundation, 21 July 2014. Web.

Ratey, John J., and Alber t M. Galaburda. A User’s Guide to the Brain: Perception, Attention, and the Four Theaters of the Brain. New York: Pantheon, 2001.

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Restak, Richard M., and Scott Kim. The Playful Brain: The Surprising Science of How Puzzles Improve Your Mind. New York: Riverhead, 2010. Print.

Sweeney, Michael S. Brain: The Complete Mind: How It Develops, How It Works, and How to Keep It Sharp.

Washington, D.C.: National Geographic, 2009. Print.

Sweeney, Michael S. Brainworks: The Mind-bending Science of How You See, What You Think, and Who You Are. Washing ton, D.C.: National Geographic, 2011. Print.

“ I mp o s s ib l e Ob je c t .” Wi k ip e d i a . Wi k i m e d i a Foundation, 17 July 2014. Web. “Mont y Ha ll P roblem .” Wik iped ia . Wik imed ia Foundation, 17 July 2014. Web. “Multistable Perception.” Wikipedia. Wikimedia Foundation, 18 July 2014. Web. “Necker Cube.” Wikipedia. Wikimedia Foundation, 22 July 2014. Web. “Perception.” Wikipedia. Wikimedia Foundation, 15 July 2014. Web. “Visual Cortex.” Wikipedia. Wikimedia Foundation, 22 July 2014. Web. “ V i s u a l I l l u s i o n s .” W i k i p e d i a . W i k i m e d i a Foundation, 15 July 2014. Web. “ V i s u a l S y s t e m .” W i k i p e d i a . W i k i m e d i a Foundation, 19 July 2014. Web. “ V i s u a l T h i n k i n g .” W i k ip e d i a . W i k i m e d i a Foundation, 14 July 2014. Web.

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Boileau, J.p.h. “Brain Weight And Brain Power.” The Lancet 120.3082 (1882): 485. Web.

Magpie Studios. Arrows. Digital image. Magpie Studio. N.p., n.d. Web. 10 July 2014. <http://magpie-studio.com /projects.php?projectid=145>.

Wordpress. Skyscraper New York. Digital image.

N.p., n.d. Web. 5 July 2014. <http://new yorkphotos.wordpress.com /2008/09/27/new-york-at-night/ new-skyscraper-new-york-2008/>.

A n ony m o u s . Penro s e Square. D ig it a l i m a g e.

N.p., n .d . Web. <ht t p: //en .w i k iped i a .or g / w i k i / Penrose_triangle>.

Brain, Lord. “Some Reflections On Brain And Mind.”

Brain 86.3 (1963): 381-402. Web. Dichr istina, Mar iet te. “Word Power.” Scientific American Mind 19.5 (2008): 1. Web. Hines, Terence. “Left Brain/Right Brain Mythology and Implications for Management and Training.”

The Academy of Management Review 12.4 (1987): 600. Web. Indicated, No Authorship. “Brain Power: Unlock the Power of Your Mind.” PsycCRITIQUES 36.2 (1991): n. pag. Web. Mckinlay, William W. “Achieving Generalization of Memor y Training.” Brain Injury 6.2 (1992): 10708. Web. Scull, A. “Lef t Brain, Right Brain: One Brain, Two Brains.” Brain 133.10 (2010): 3153-156. Web.

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COLOPHON This book was designed by Laia Pose as a student project for the Typography 3 class at the Academy of A r t Un iversit y. It h a s been typeset using the typefaces DIN Next Lt Pro for the headings and Kulturista for the boilerplate text.

It has been printed on Staples Photo Supreme Matte Paper Double Sided 61lb and bound by California Of fice Ser vice in San Francisco, August 2014.

MEET YOUR BRAIN

This book w ill test you in a ser ies of puzzles that will both challenge and entertain you. Your brain will be stimulated, and ultimately amazed, as you discover the fascinating science behind why these experiments work and how your mind wraps itself around them. Along the way, youâ&#x20AC;&#x2122;ll gain insights on how the brain handles dif ferent functions, and some tips on how to exercise it and improve its capabilities.

Mind Wrap celebrates the amazing flexibility and creativity of the human mind, leaving you with a richer appreciation of the mental mar vels we take for granted.

U.S. $85.00 ISBN 123-4-567-89000-1