Martin gardner where is the missing face

Intuitively, the idea of independence is when events do not "affect" each other. What I do when I flip a "fair" coin does not typically affect the result of the next flip of that coin. There are people who practice flipping coins so that they always get heads. With practice one can accomplish this. But there are situations where what happens first affects what happens later. This involves the notion of conditional probability. Suppose I have a bag with 3 balls, 2 of which are black and one is white.

Suppose I draw one ball after another from the bag without replacing the earlier selected balls. I can ask what is the probability that the first of the two balls I draw is black?

Now suppose I look at the first ball and see that the ball is black. What is the chance now I get two black balls? If the first ball had been white, what is the chance that the first two balls are black? The answer is that this cannot happen, so the result must have probability 0. The way these ideas are treated is as follows: If X is an event we say that the probability of X is P X. So, how might we deal with the probability that event X will occur given that event Y has occurred?

And what about the probability that Y will occur given X has occurred? Clearly these are sometimes different! So let us "invent a notation" for such "conditional" probability occurrences.

How can one calculate P X Y? We will make the following "definition. Events A and B are represented by the circles, and outcomes that have properties A and B are shown by the circle's overlap.

If we want to know the probability of B occurring given that A has occurred, P B A , we can restrict our attention to things inside A since we know A occurred. Now to find out what chance there is that B occurred given A occurred means we want to see "what part" of A is represented by the overlap of A with B.

If A and B are independent, that is, they do not affect each other's occurrence, then we should have that P A B should be equal to P A.

This is the multiplication rule for independent event probabilities. Using the notations developed above, here are the answers to the questions above, as well as some other probabilities, about probabilities of drawing 2 balls one after another from a bag with 3 balls, 2 of which are black. The "birthday surprise" is but one of many situations in which people's intuition for the nature of probability has to be "trained.

Martin Gardner had a playful streak and often had columns which had whimsical goals. In his Scientific American column of April he showed a plane map which purportedly required 5 colors to color the faces. The rule is that if two regions share an edge they should get different colors. The face-coloring problem for maps asks: What is the minimum number of colors with which one can color the regions of a map drawn in the plane so that regions faces which share an edge receive different colors?

This problem, which has a long history, resulted in one of the most famous "false proofs" in the history of mathematics. Alfred Kempe, a lawyer by profession but with a significant record of accomplishments as a mathematician, claimed to have proved what came to be called the 4-color conjecture.

This conjecture asserted that any plane map could be colored with 4 or fewer colors. Kempe's proof had a subtle error and that error was pointed out by Heawood, who showed that one could prove that plane maps could always be 5-colored.

The proof of Kenneth Appel and Wolfgang Haken, that any planar map could be 4-colored, was apparently completed in and did not appear in print until a year later. So Gardner's column occupied the niche of high interest in the status of the conjecture that pre-dated the appearance of a proof. Some controversy accompanied the proof because it involved the use of a computer in a way that precluded humans being able to follow all of the steps.

While many mathematicians were not bothered by this aspect of what Haken and Appel did, there was considerable discussion of the matter by mathematicians and others. However, this proof, while more streamlined than the earlier one, still involved the use of a computer. So yes, Gardner's column was an April Fools' joke. Here is a coloring of the challenge map above, designed by William McGregor, which was found by Stan Wagon.

While all plane maps are 4-colorable, finding a 4-coloring for a particular map is often a challenge.

I was in the audience to hear Martin Gardner but I never met him in person. However, in a variety of ways our paths crossed. Gardner first came to my attention when as a teen on a trip to Florida, an automobile accident stranded me and my parents in Winter Haven, Florida, while our car was being repaired. My "lifesaver" for this period was a local library within walking distance of where we were staying, which was where I got my first introduction to Scientific American magazine and the columns of Martin Gardner.

On returning to NY I started a subscription to Scientific American , and my subscription has never lapsed. Eventually, when I had large enough living quarters I started to acquire Gardner's books.

HIs books were initially easier to consult than trying to go back to his Scientific American columns. However, there was also the fact that when his Scientific American columns appeared in book form, Gardner updated information about the mathematics in the columns with additional details about what was new about the problem since it had initially been published. One wonderful aspect of Gardner's Scientific American columns was that he and an "army" of appreciative, motivated and talented people including some who were professional mathematicians thought about the questions that he raised.

In some cases his columns dealt with issues which, at the time he wrote about them, were not fully understood. So his books allowed him the opportunity to "update" columns with new ideas and results when the columns were collected for reprinting.

I also had an indirect "meeting" with Martin Gardner. While I was a graduate student at the University of Wisconsin in Madison, I became interested in what have come to be called Eberhard Type Theorems , in honor of the blind 19th century geometer Victor Eberhard.

In conjunction with this I became interested in ways to tile convex polygons with various shapes, in particular with equilateral triangles and squares. The diagram below shows a "patch" of tiles of a portion of the plane which uses only equilateral triangles and squares. Can you pick out in this tiling corners for convex polygons with 3, 4, 5, and 6 sides? What other values of n are there so that there is a convex polygon with n sides which can be tiled with equilateral triangles and squares all having the same side length?

In thinking about this one can have vertices of the tiling polygons that are not vertices of the convex polygon. For example, above you can find a convex pentagon which is tiled with two squares and a triangle. The answer to this question is a nice application of some simple ideas from elementary geometry so I submitted it as a problem to Mathematics Magazine.

Here is what appeared in November Proposed by Joseph Malkewitch, University of Wisconsin. For what values of k is there a convex polygon with k sides which can be dissected into squares and equilateral triangles which have the same length of side? The solution by Michael Goldberg appeared in May, There are still some open questions related to the shapes of those n -gons that can occur for some values of n.

I was very pleased when Martin Gardner picked up on this problem and used it in his book Knotted Doughnuts and the later compendium Colossal Book of Short Puzzles and Problems. However, Joseph Malkewitch is really me! Vould I lie to you? Gardner fans were disappointed when Gardner no longer continued to write his column for Scientific American.

The magazine tried to find a "replacement" for Gardner by having various authors and columns that tried to capture the spirit of Mathematical Games. Eventually, however, Scientific American resigned itself to the fact that Gardner really could not be "replaced! Fortunately, the spirit of what Gardner started with his columns and books continues via a series of events to remember him and his name. This is done via the conferences and workshops in different locations that have come to be known as Gatherings for Gardner.

What is the smallest number of eggs that Elspeth could have got? Stewart is an enormously popular writer in mathematics, with books ranging from challenging Galois Theory to fun Math Hysteria.

The other was an inspiring teacher, Gordon Radford. I read the column every month, and often copied ideas from it into a notebook I was keeping on math that went beyond the school curriculum. There were often deeper ideas behind the games and puzzles. The overwhelming message that I received was: New mathematics is being created all the time.

I decided I wanted to become one of the creators. When I was doing my Ph. We typed it, duplicated it, and stapled it ourselves. It ran for 20 issues over 12 years. Early on, we sent Gardner a copy, and he subscribed to it and occasionally wrote us with encouraging comments. It even got a mention in one of his books. The impression we received was that he was kind and generous. As an example: In the April issue of Scientific American, he ran an April Fool spoof, one item being an alleged disproof of the famous and at that time unsolved four-color theorem.

To his surprise, a lot of people took it seriously. Gardner, what have you done? Porter, what shall I do? Instead of being upset, he reprinted it. Through a strange series of events, involving the French translation of Scientific American and a series of science comic books, I eventually became the fourth person to write the column, after Gardner, Douglas Hofstadter and Kee Dewdney. But I did take inspiration from his clear, comprehensible style.

Many people get a completely false impression of math at school, finding it boring, uncreative and useless. In reality it is exactly the opposite, and most of the things we take for granted in our daily lives would not have been invented, and could not work, without it. He proved to me that it can be done. I find that I have to understand an idea clearly myself, and be enthusiastic about it, before I am able to write a popular book or article about it. We also have a bonus challenge this week especially for Martin Gardner fans: an identification puzzle featuring the mischievous and magical art of Lucian McLellan.

Click here for a full-resolution version. There are 10 symbols that simply illustrate subjects featured in his writings. The that appears in the top left corner, for instance, refers to the binary number system, which uses only ones and zeroes. Add the first two and you get the third, the way you construct a Fibonacci sequence. However, simply spotting that this portrait holds a visual reference to binary counts as a win! There are nine others like that, but in addition there are six more visual references that are themselves impossibilities in some way.

The most glaring example of this is also on the left, an impossible cube just above that tiny figure of a girl. Most people will recognize this cube as something that features in the graphic work of M. Escher, he of the famous endless staircases and interlocking tiles where fish transform into birds as you read across the picture. There are five more impossibilities that are not so easy to put your finger on. I asked Lucian to say a bit about himself and his work, and he replied:.

I get utterly absorbed in concentration while drawing. The more you do it, the better you get. The resulting chimera of skills bears fruit as a book, a graphic guide to gravity which I am writing and drawing, and as graphic puzzles, one of which is the challenge presented here this week.

Stewart conducted by Howard Burton of Ideas Roadshow. In a very real sense, these were the first "card columns" aimed at the general public. To paraphrase recent comments from Dr. Coincidentally or not, as the case may be , his first publications from the s were also magical.

Basically, what Martin did—and did extraordinarily well— was take complex concepts and creations gleaned from experts and sometimes talented amateurs in areas as diverse as mathematics, magic and the mysteries of physics, logic and the then fledgling field of computer science, and present them with astonishing clarity of thought and economy of words.

Over time, his groundbreaking Scientific American articles reached an even larger audience — including teens overseas, as we and numerous others can attest — via 15 individual books of collected columns, which can now be enjoyed in text searchable form in a single place on a CD-ROM from the MAA called Martin Gardner's Mathematical Games. This month, with the permission of the MAA, we sample some of the most memorable of Martin's expositions of mathemagical card tricks from that collection of collections, using exclusively his own incomparable crystal clear words.

I'm strictly a journalist. I just write about what other people are doing in the field. He was neither a mathematician nor magician by trade—something we keep having to remind ourselves of—yet he left his own stamp on almost everything he wrote about in those arenas, including what follows below.

At the end, we challenge to readers to explain a curious trick which was recently aired in the New York Times. Chapter 10 Mathematical Card Tricks of The Scientific American Book of Mathematical Puzzles and Diversions originally , later republished as Hexaflexagons and Other Mathematical Diversions opens with this cautionary quoted exchange and commentary, before moving on to an expository tour de force:.

Having introduced the ideas of congruence and remainder modulo 9, Martin writes:. After observing that "Obtaining the digital root is simply the ancient process of casting out 9's ," and that IBM computers "use the technique as one of their built-in methods of self-checking for accuracy" , a few pages later he continues:.

Chapter 9 Victor Eigen: Mathemagician of New Mathematical Diversions from Scientific American originally , but this column dates back to opens thus:. There we leave Martin and Victor, who move on to explore other topics.

In an Addendum to this chapter, Martin gives many references to tricks which use the so-called first Gilbreath Principle, and also gives this explanation as to why it works:. In an Addendum to that chapter, Martin credits the legendary Bob Hummer whom he knew with "the earliest published trick I know of that exploits this formula" in