Articles
 

Visual Brainstorms 2*

by Martin Gardner
 




e are living in a time when good, fundamentally sound education is essential to our future. The onset of the information age presents entirely new challenges to us as a society in the 21st century. Both children and adults are being asked to adapt to new technologies, to learn new skills, and to think in new and smart ways.

Clever brainteasers such as those found in the Visual Brainstorms cards are designed to introduce you to smart ways to think. Kids will have fun learning while chuckling at the amusing way the questions are illustrated; teachers will enjoy the variety of problems to present to inquiring minds; and adults will discover that they are often on common ground with their younger counterparts as they struggle to organize their thoughts and arrive at a solution to the question.

As you will find, Visual Brainstorms 2** will spark your own curious mind to think in new ways and to join in the celebration of learning.

Here is an overview of all the places this deck will take you on your journey of learning, as well as a few comments on the connections the cards make to today's evolving world of math and science.

Mistakes...
Several of the cards in this deck are marked with an icon of a magnifying glass. This means you must search for a mistake in the illustration. One such "mistake," in the card titled "Celestial Delete," is the picture of a star within the arms of a crescent moon. For our young audience the answer may teach, for the first time, that a moon's crescent is the sun-illuminated edge of an invisible globe, and that a star is enormously farther from the earth than our moon.

Physics...
"Dueling Pendulums" is a problem that teaches a bit of basic physics. "High Tide Hoopla" introduces some little-known facts about the moon's role in causing tides, and why high tides form at the same time on opposite sides of the earth.

Topology...
Learning how to trace a geometric figure without taking your pencil off the paper, and without going over part of the path twice or crossing a line, is an entertaining introduction to topology, a flourishing branch of mathematics that has important applications in physics, cosmology, and biology.

What is topology? It is the study of properties of a figure that remain unaltered when the figure is distorted. For example, tracing networks such as those in "Park Prowl," "Something Fishy," and "Ancient Walls" all have exactly the same solutions if they are drawn on a rubber sheet and the sheet is stretched and folded as you please. For this reason topology is sometimes called "rubber sheet geometry."

Combinatorics...
Another branch of modern mathematics is called combinatorics because it studies the ways in which numbers and other mathematical objects can be combined. Combinatorial problems in this box include river crossing tasks, the problem of matching socks, and liquid pouring puzzles. "Time for Toast" is another combinatorial problem as well as a simple instance of what is called operations research-finding ways to increase the efficiency of factory and business operations. Discovering a set of numbers on the target in "Lizzie's Darts" is a simple example of what is called a "subset-sum problem."

Symmetry...
Symmetry is a concept of enormous importance both in mathematics and science. If you can perform a certain operation on a figure, and the figure remains the same, it is said to have symmetry with respect to that operation. For example, the letter "H" has 180-degree rotational symmetry because after you invert it, it looks the same. "H" also has mirror reflection symmetry because it is unchanged when viewed in a mirror.

A fun project for kids is make a list of letters, both upper- and lowercase, that remain unchanged when held up to a mirror, either with paper upright or upside down. Try to form words that are unchanged by mirror reflections. Some words are unchanged when written vertically:
 

T
I
M
O
T
H
Y

O
H
I
O

M
A
M
A

T
O
M
A
T
O

A
U
T
O
M
A
T

T
O
Y
O
T
A

T
O
O

H
O
T

T
O

H
O
O
T

 

"TOO HOT TO HOOT" is also a palindrome (it reads the same from top to bottom as it does from bottom to top). How many things can you think of that are not the same when seen in a mirror? Your left ear and left hand, for example, change into a right ear and right hand in the mirror. If you clasp your hands or fold your arms, are they the same in the mirror? Is a pair of scissors mirror symmetric? An overhand knot? A moebius strip? (See Answer
As Alice said in Through the Looking Glass, in a mirror things seem to "go the other way."

Algebra and Geometry...
A number of puzzles in this box find their roots in algebra. The balancing pans in "Balance Board Blues" area model of an algebraic equation. "The Cyclists and the Fly" and half a dozen other puzzles are easily solved by algebra.

"Continuous Line" is one of my all-time favorites. You are asked to connect nine dots by four continuous straight lines. It's a wonderful instance of how a sudden "aha!" insight comes with realizing that the lines can extend outside the square array.

"Rep-tiles"...
"Five Easy Pieces" is another of my favorites. The first two examples of how to cut a figure into four congruent shapes provide misdirection that makes the problem of the square seem much more difficult that it really is. The solution, by the way, is the only one possible, although proving this is extremely difficult.

The first two dissections are examples of what are called "rep-tiles" because the congruent pieces have the same shape as the outside figure.

Word...
Our set of cards does not neglect word puzzles. They, too, can serve as teaching devices. Solving anagrams - rearranging the letters of a word or phrase to make a different word or phrase - points up the importance of knowing how to spell. The same goes for spotting palindromes and solving rebuses, like those in "Border Patrol," that require adding and subtracting letters.

Logic...
"Spirit Search," "Tricky Twins," and several other puzzles are problems in pure logic. They can be solved by applying techniques of formal logic, but are simple enough so you can reason them out any way you like.

Number...
Number puzzles are as ancient as civilization. Perhaps the oldest of all combinatorial number puzzles, going back to ancient China, is the task of placing digits 1 through 9 in a square of nine cells so that every row, column, and the two main diagonals add up to the same sum. Can you do it? ( Hint: the magic sum is 15. And 5 must go in the center.) There is only one solution, not counting rotations and reflections of the pattern as different. (See Answer)

Polyominoes and Polycubes...
These are figures formed by joining identical squares along their edges. The twelve shapes - pentominoes - are all the ways that five squares can be joined.

When I introduced them in my Scientific American column, they aroused so much interest that hundreds of articles and several books have since been published about them, including one entitled Polyominoes by Professor Golomb who named and studied these shapes.

Identical cubes joined along their faces produce solids known as polycubes.

Recreational Mathematics...
Many of the worlds greatest mathematicians have been fond of what is called "recreational mathematics." It covers any kind of math that is touched by the spirit of play. The great 17th century mathematician and philosopher Wilhelm Leibniz wrote about the pleasure he had in solving problems involving a popular mechanical puzzle of the time, peg solitaire. Today, many world famous mathematicians and scientists not only are fond of recreational mathematics, but are also puzzle inventors. One of the great pleasures of writing the "Mathematical Games" column in Scientific American was making friends with so many of these notables.

During the past few decades there has been an increasing recognition by teachers that recreational math - puzzles, games, paradoxes, magic tricks - is one of the best ways to get children and high school students interested in serious mathematics. More and more papers on recreational topics are appearing in scholarly math journals. Magazines devoted solely to games and puzzles, such as the quarterly Journal of Recreational Mathematics, are flourishing here and abroad. Dozens of new books on puzzles have been published in recent years. You can now search the internet for puzzle, math, and science enthusiasts.

At the same time a growing number of ingenious mechanical puzzles and mathematical games are appearing on the market. Stores are springing up here and there which carry only board games and puzzles and scientific toys. Boxed kits of materials to be used for entertaining science experiments are now on the market.

Popular interest in recreational math and science has not been this strong since almost a century ago when Sam Loyd was America's foremost puzzle inventor, and Henry Ernest Dudeney was his British counterpart. Their books are rich sources of traditional puzzles. Today there is no single greatest puzzle maker. Instead, contributions are coming from hundreds of amateur and professional mathematicians. There are even conventions of puzzle inventors, collectors, and enthusiasts from all over the world who meet annually in various cities.

Charles Bombaugh, in an 1874 book Gleanings for the Curious, had a section on puzzles in which he likened them to sports that exercise the body. Such physical games, he said "are not work but play. They prepare the body, and make it alert and active for anything it may be called upon to perform." In a similar fashion," Bombaugh continued, solving brain-teasers "gives quickness of thought and facility in turning about a problem every way, and viewing it in every possible light."

Bombaugh couldn't have said it better. Visual Brainstorms 2 is a second selection of choice brainteasers intended to exercise your brain. You will find them amusing, entertaining, and, above all, educational. Enjoy!

----
* A brief version of the Introduction to Visual Brainstorms 2.

Visual Brainstorms - Original Smart Thinking Game from Binary Arts ** If you are interested in buying Visual Brainstorms, please ask about them in your local toy or book store, or simply visit ThinkFun.

 

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Last Update
March 29, 2005

 
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