马丁·加德纳Martin Gardner

马丁·加德纳的父亲是一位地质学家，拥有地质学博士学位，在创办自己一家很小的石油公司之前，曾为史密森尼学会挖掘化石。还在为史密森尼学会工作期间，他经常带着年幼的儿子加德纳去挖掘。加德纳的母亲在结婚前是列克星敦的一名小学教师，但孩子出生后，她就留在家里照顾他们——不过，她仍然保持着自己绘画的爱好。加德纳是他父母三个孩子中最大的，有一个弟弟Jim和一个妹妹Judith。家里很富裕，石油生意利润丰厚，到加德纳大到可以开始打网球时，他们甚至有了自己的网球场。在上小学之前，加德纳已经学会了阅读[2]：-

我小时候，母亲给我读《绿野仙踪》，她读的时候我从她肩膀上方看过去。我就这样学会了阅读。

加德纳以数学谜题闻名，许多人会说他是世界上最著名的现代数学谜题作家，而这种对谜题的兴趣在他早年就产生了，当时他父亲给了他一本森姆·莱特的Cyclopedia of Puzzles。他早期的兴趣之一是魔术，这导致他于1930年5月在The Sphinx——一本魔术杂志——上发表了第一篇作品New Color Divination，当时他还是个高中生。他在高中的成绩参差不齐[2]：-

我在高中时数学非常好。事实上，数学和物理是我唯一取得好成绩的科目。其他课程让我无聊得要死。我有一门拉丁语课不及格，不得不重修。我就是对语言没有好耳朵。

特别是，他的高中数学老师Pauline Baker培养了他对演绎推理的热爱，而他认为他的物理老师M E Hurst是学校里最鼓舞人心的老师。高中毕业后，加德纳想进入加州理工学院学习物理。然而，入学要求是在大学学习两年，因此加德纳进入了芝加哥大学，打算两年后转到加州理工学院。然而，他迷上了哲学，并留在芝加哥主修该学科。他于1936年从芝加哥大学毕业，获得学士学位，但这是找工作的困难时期，因为正值大萧条的高峰[2]：-

我做过各种工作。我曾在芝加哥救济管理局担任个案工作者。我不得不定期走访被称为黑人区的140个家庭。我还做过几份零工：服务员、冷饮柜台店员等。

他还曾一度担任Tulsa Tribune的记者，但在1941年12月美国参加第二次世界大战之前，他的最后一份工作是在芝加哥大学担任公共关系官员。他入伍加入海军，在威斯康星州麦迪逊待了一年，在一所无线电培训学校工作，随后在USS Pope号上度过了三年，这是一艘护送大西洋船队的驱逐舰。战争结束后，他回到芝加哥，并将他的第一篇短篇小说卖给了Esquire Magazine。意识到可以通过写作养活自己，他没有回到公共关系的工作（尽管这本来是可能的）。然而，此时他仍然对哲学非常感兴趣，并利用退伍军人法案在芝加哥大学花了一年时间攻读研究生课程，特别是Rudolph Carnap讲授的科学哲学课程。除了为Esquire Magazine写作外，他还为儿童杂志Humpty Dumpty撰稿。

加德纳于1947年搬到纽约，继续通过撰写文章谋生，并成为Humpty Dumpty的编辑。他于1952年与Charlotte Greenwald结婚；他们有两个儿子，Jim和Tom。在1952年同年，他出版了他的第一本书In the Name of Science，该书于1956年以Fads and Fallacies in the Name of Science为标题作为平装本重新出版。他的另一本书Mathematics, Magic and Mystery于1956年出版。加德纳在序言中写道：-

W W Rouse Ball的经典参考著作，即《数学消遣与随笔》，包含了许多早期的数学魔术例子。……本书是首次尝试综述现代数学魔术的整个领域。……其原理可以迅速掌握，无需高等数学的训练。

1956年12月，他的第一篇文章，关于六边形折纸，由Scientific American发表。出版商非常喜欢这篇文章，并问加德纳是否认为可以开设一个定期专栏。Gradner很快同意撰写“数学游戏”，辞去了在Humpty Dumpty的编辑职务，并在1957年1月的Scientific American上开设了他的第一个专栏。从那时起，他为每月“数学游戏”专栏撰写了二十五年。

他的书也对数学的普及产生了巨大影响。他写了六十多本精装书，以及许多大约50页的小册子。我们当然不想列出六十多部作品的标题，所以我们将给出一个选录：Logic Machines and Diagrams (1958)；The Annotated Alice (1960)；Relativity for the Million (1962)；The Ambidextrous Universe: Mirror Asymmetry and Time-Reversed Worlds (1964)；Mathematical Carnival: A New Round-up of Tantalizers and Puzzles from "Scientific American" (1975)；The Incredible Dr Matrix (1976)；Aha! Insight (1978)；Science: Good, Bad, and Bogus (1981)；Aha! Gotcha: Paradoxes to Puzzle and Delight (1982)；The Whys of a Philosophical Scrivener (1983)；Codes, Ciphers and Secret Writing (1984)；Entertaining Mathematical Puzzles (1986)；Time Travel and Other Mathematical Bewilderments (1987)；Perplexing Puzzles and Tantalizing Teasers (1988)；Fractal Music, Hypercards and More (1991)；My Best Mathematical and Logic Puzzles (1994)；Classic Brainteasers (1995)；Calculus Made Easy (1998)；A Gardner's Workout: Training the Mind and Entertaining the Spirit (2001)；Mathematical Puzzle Tales (2001)；以及Bamboozlers (2008)。

1981年，为庆祝加德纳六十五岁生日，The Mathematical Gardner一书出版。编者弗洛伦斯·南丁格尔·大卫 A Klarner在序言中写道：-

本卷献给加德纳，以庆祝他1979年10月21日的65岁生日。他与数学界的关系使他独一无二：加德纳在《科学美国人》上的每月专栏已经出现了二十多年。这些专栏连同他的一些书（并非全部限于数学）通俗地介绍了现代数学的一些最新发展。他不仅仅是记者或普及者，与读者的通信使他成为一位“数学园丁”。这个短语描述了他对思想的培育和传播——多丽丝·沙特施奈德在本卷中的文章《为业余爱好者喝彩》就是一个例子。在能够理解和欣赏数学之美的一般人群中，很少有人真正有机会这样做。加德纳的写作极大地提高了该主题的可及性。本书的作者以及更广泛的粉丝群体赞扬并感谢他。

多年来，加德纳一家住在纽约州Hastings-on-希尔达·菲比·哈德森的欧几里得大道（每个人都评论这个地址很贴切），但在1979年他们搬到了北卡罗来纳州的亨德森维尔。此时Gerdner六十五岁，认为该退休了。他继续为Scientific American制作“数学游戏”专栏，直到1981年他从该角色退休。然而，他仍然非常积极地出版书籍，事实上他放弃专栏的主要动机是给他更多时间专注于写书。2000年，他的妻子Charlotte去世，两年后他搬到俄克拉荷马州的诺曼，靠近他的儿子Jim，Jim是俄克拉荷马大学的教育学教授。

他为Scientific American撰写的专栏已出版为十五本书。让我们引用加德纳自己在A Gardner's Workout: Training the Mind and Entertaining the Spirit（2001年）序言中的话：-

25年来，我有幸在⟦N1⟧的⟦E1⟧杂志上撰写“数学游戏”专栏，并乐在其中。所有这些专栏文章现已重新刊印，并加以更新，共十五卷，始于[1959年]，终于[⟦E2⟧，1997年]。停止撰写专栏后，我时常为学术期刊和大众杂志撰写有关数学的文章和书评。这里收录了其中的四十一篇。最具争议的是最后一篇评论，我在其中批评了当前一种被称为“新新数学”的教学时尚。到本书出版时，我猜测并希望新新数学将几乎与旧新数学一样迅速地消失。我可能错了。无论如何，我们的公共教育可能需要几十年才能吸引到合格的教师，他们懂得如何向大学前的学生教授数学而不让他们昏昏欲睡。当然，有许多教师值得称赞。本书正是献给他们。

American Mathematical Society于1987年在盐湖城举行的夏季会议上将Leroy P 凯瑟琳·斯蒂尔奖授予加德纳[1]：

……因为他撰写了大量数学书籍和文章，尤其是他在⟦N1⟧《科学美国人》上的专栏“数学游戏”。

引文继续写道[1]：-

加德纳 将一代又一代读者引入数学与数学思维方式所带来的智力激荡、惊奇、多样性和纯粹的乐趣。凭借 ⟦E1⟧ 博士及其其他朋友，他揭露了从数字命理学 to 经济学等领域中的虚假思维和数学滥用。加德纳 抓住了读者的注意力，获得了他们的积极参与，并将他们的思维拓展到令我们所有从事教学的人羡慕的程度。

在回复中，加德纳说[1]：

如果我在芝加哥大学读本科时没有对哲学产生强烈兴趣，我可能会主修数学，成为专业人士，也许会对该领域做出一些贡献。碰巧的是，我没有接受过正式的数学训练，只有对数学奇迹的业余热情，以及对数学领袖的钦佩和敬畏。我认为自己就像一个热爱古典音乐的人，但他的才能从未超越在音乐锯上演奏简单曲调。自学数学没有比写关于它的文章更好的方法了。我为《科学美国人》完成的每一篇专栏都是一次学习经历，给了我极大的乐趣。如果我能够向他人传达数学的某些魅力，那是因为我不知道足够多的知识来在技术层面上写它。……被授予斯蒂尔奖是我能想象自己获得的最大荣誉。

让我们以注意到加德纳与顶尖数学家合作发表了许多数学论文来结束。他说[3]：-

我非常喜欢数学，因为它有一种奇特的超凡之美。在思考一个优雅的证明时，有一种难以描述的强烈愉悦感，而在发现一个前所未知的证明时，愉悦感更甚。在较低层次上，我经历过四次这样的愉悦。
(1) 我发现了正方形可以被分割成的最少锐角三角形数量。（哈罗德·斯科特·麦克唐纳·考克斯特在其经典著作《几何导论》中包含了这一分割。）
(2) 我找到了连接棋盘所有角的最小雅各布·施泰纳树网络。
(3) 我构造了一个双色证明，证明每个90度的序列等角多边形——一个所有角均为直角，边长按1、2、3……序列变化的多边形——其边数必须是8的倍数。
(4) 我设计了一种新颖的方法来图解命题演算。

加德纳在此处(2)下提到的结果出现在论文“金芳蓉, 加德纳 and 葛立恒, 雅各布·施泰纳 trees on a checkerboard, Math. Mag. 62 (1989)”中。堵丁柱写道：-

这是一篇非常有趣的文章。它告诉我们棋盘应该具有什么样的雅各布·施泰纳最小树。证明作为开放问题留给读者。雅各布·施泰纳最小树是连接给定点的最短网络。一个阶为$n$的棋盘是形成$n \times n$矩形阵列的$n^{2}$个点的集合。棋盘雅各布·施泰纳最小树的连接模式相当复杂。然而，作者用许多精美的图片非常清晰地描述了它。令人惊讶的是，证明似乎如此困难。除了$n = 2$或3之外，每种情况都只是猜想。希望这篇文章能引发一系列的努力。

加德纳在(3)下提到的结果出现在“Lee Sallows, 加德纳, Richard Guy and 高德纳, Serial isogons of 90 degrees, Math. Mag. 64 (1991)”中。作者们写道：-

1988年，Sallows 设计了一个计算机程序，在单位正方形网格上搜索具有以下性质的闭合路径。路径从一条格线开始，先走一段单位长度的线段，向任一方向转 90 度，继续走 2 个单位，再向任一方向转，继续走 3 个单位，依此类推。换句话说，路径的各段按顺序为 $1, 2, 3, .. N$，在每一段的末端转直角。由 $N$ 段组成的路径——这个数当然与转弯数或角数相同——称为 $N$ 阶路径。如果路径回到起点，并与它的第一段成直角，我们称它为 90 度串行等角线。我们证明，对于任何 90 度串行等角线，$N$ 必须是 8 的倍数。

加德纳 发表在 学院数学杂志 上的三篇论文是 Modeling mathematics with playing cards（2000 年）、Some new results on magic hexagrams（2000 年）和 L-tromino tiling of mutilated chessboards（2009 年）。

加德纳因其卓越贡献获得了多项荣誉，包括1978年巴克内尔大学的名誉博士学位以及1983年美国物理联合会年度科学作家奖。

Martin Gardner's father was a geologist, with a Ph.D. in geology, who dug for fossils for the Smithsonian Institution before starting up his own very small oil company. While he was still working for the Smithsonian Institution, he often took his young son Martin on his digs. Martin's mother was a primary school teacher in Lexington before her marriage but, once the children were born, she stayed at home to look after them - she did, however, keep up her hobby of painting. Martin was the oldest of his parents' three children having a younger brother Jim and sister Judith. The family were well-off, the oil business being highly lucrative, and they even had their own tennis court by the time Martin was old enough to begin playing. Before he entered primary school, Martin had learnt to read [2]:-

My mother read 'The Wizard of Oz' to me when I was a little boy, and I looked over her shoulder as she read it. I learned how to read that way.

Today Gardner is famed for mathematical puzzles, many would say that he is the most famous modern writer of mathematical puzzles in the world, and this interest in puzzles came early in his life when his father gave him a copy of Sam Loyd's Cyclopedia of Puzzles. One of his early interests was magic and this led to his first publication New Color Divination in The Sphinx, a magic magazine, in May 1930 when he was still a high school student. His achievements at high school were mixed [2]:-

I was very good at math in high school. In fact, it and physics were the only subjects in which I got good grades. I was bored to death by the other classes. I flunked a class in Latin and had to take it over. I just don't have a good ear for languages.

In particular, his high school mathematics teacher, Pauline Baker, gave him a love for deductive reasoning while he thought that his physics teacher, M E Hurst, was most inspiring teacher in the school. After graduating from high school, Gardner wanted to study physics at the California Institute of Technology. However, the entrance requirements were two years at College so Gardner went to the University of Chicago with the intention of moving to Cal. Tech. after two years. However, he became fascinated with philosophy and stayed at Chicago to major in that subject. He graduated from Chicago with a B.A. in 1936 but this was a difficult time to get a job since it was the height of the Depression [2]:-

I had various jobs. I worked as a caseworker for the Chicago Relief Administration. I had to visit 140 families regularly in what was called the Black Belt. I also had several odd jobs: waiter, soda jerk, etc.

He also worked for a while as a reporter for the Tulsa Tribune but his last job before the United States entered World War II in December 1941 was as a public relations officer at the University of Chicago. He enlisted in the Navy and spent a year at Madison, Wisconsin, working at a radio training school before spending three years on the USS Pope, a destroyer escorting convoys across the Atlantic. When the war ended he returned to Chicago and sold his first short story to Esquire Magazine. Realising he could support himself by writing, he did not go back to his public relations job (although this would have been possible). At this time, however, he was still greatly interested in philosophy and made use of the G.I. Bill to spend a year taking graduate courses at the University of Chicago, in particular one on the philosophy of science given by Rudolph Carnap. As well as writing for Esquire Magazine he also wrote for the children's magazine Humpty Dumpty.

Gardner moved to New York in 1947 and continued to earn a living contributing articles and becoming an editor of Humpty Dumpty. He married Charlotte Greenwald in 1952; they had two sons, Jim and Tom. In the same year of 1952 he published his first book In the Name of Science which was republished as a paperback in 1956 under the title Fads and Fallacies in the Name of Science. Another of his books Mathematics, Magic and Mystery was published in 1956. Gardner writes in the Preface:-

The classic reference work of W W Rouse Ball, namely 'Mathematical Recreations and Essays', contains many early examples of mathematical conjuring. ... The present book represents the first attempt to survey the entire field of modern mathematical magic. ... Its principles can be grasped quickly, without training in higher mathematics.

In December 1956 he had his first article, which was on hexaflexagons, published by Scientific American. The publisher liked the article a lot and asked Gardner if he thought a regular column would be possible. Quickly Gradner agreed to produce 'Mathematical Games', resigned from his editorial role with Humpty Dumpty and had his first column in the January 1957 issue of Scientific American. From that point on he produced the monthly 'Mathematical Games' column for twenty-five years.

His books have also had a huge impact on popularising mathematics. He has written over sixty hardback books as well as numerous pamphlets of around 50 pages. We certainly do not want to even list the titles of over sixty works so we will give a selection: Logic Machines and Diagrams (1958); The Annotated Alice (1960); Relativity for the Million (1962); The Ambidextrous Universe: Mirror Asymmetry and Time-Reversed Worlds (1964); Mathematical Carnival: A New Round-up of Tantalizers and Puzzles from "Scientific American" (1975); The Incredible Dr Matrix (1976); Aha! Insight (1978); Science: Good, Bad, and Bogus (1981); Aha! Gotcha: Paradoxes to Puzzle and Delight (1982); The Whys of a Philosophical Scrivener (1983); Codes, Ciphers and Secret Writing (1984); Entertaining Mathematical Puzzles (1986); Time Travel and Other Mathematical Bewilderments (1987); Perplexing Puzzles and Tantalizing Teasers (1988); Fractal Music, Hypercards and More (1991); My Best Mathematical and Logic Puzzles (1994); Classic Brainteasers (1995); Calculus Made Easy (1998); A Gardner's Workout: Training the Mind and Entertaining the Spirit (2001); Mathematical Puzzle Tales (2001); and Bamboozlers (2008).

In 1981, to celebrate Gardner's sixty-fifth birthday, the book The Mathematical Gardner was published. The editor, David A Klarner, writes in the Preface:-

This volume is dedicated to Martin Gardner for his 65th birthday, 21 October 1979. His relationship to the mathematical community makes him unique: Gardner's monthly column in 'Scientific American' has been appearing for more than two decades. These columns together with some of his books (not all of them limited to mathematics) have presented a popular account of some of the recent developments in modern mathematics. More than just a reporter or popularizer, his correspondence with his readers makes him a 'mathematical Gardener'. This phrase describes his cultivation and propagation of ideas - an activity exemplified by Doris Schattschneider's article in this volume, 'In praise of amateurs'. Very few people in the general population who are capable of understanding and enjoying the beauty of mathematics actually get the opportunity to do so. Martin Gardner's writing has contributed enormously to improve the accessibility of the subject. The authors of this book together with a larger group of fans praise and thank him.

For many years the Gardner family lived on Euclid Avenue (everyone remarks on the apt address) in Hastings-on-Hudson, New York, but in 1979 they moved to Hendersonville, North Carolina. At this time Gerdner was sixty-five years old and considered that it was time to retire. He continued to produce the 'Mathematical Games' column in Scientific American until 1981 when he retired from that role. However, he remained extremely active in producing books, in fact his main motivation in giving up the column was to give him more time to devote to writing books. In 2000, his wife Charlotte died and two years later he moved to Norman, Oklahoma to be near his son Jim who was Professor of Education at the University of Oklahoma.

The columns he wrote for Scientific American have been published as fifteen books. Let us quote Gardner's own words in the Preface to A Gardner's Workout: Training the Mind and Entertaining the Spirit (2001):-

For 25 years I had the honour and pleasure of writing the 'Mathematical Games' column in 'Scientific American'. All those columns have now been reprinted, with updating, in fifteen volumes, starting [in 1959] and ending with [The last recreations, 1997]. Since I stopped writing the column I have from time to time contributed articles and book reviews about mathematics to both academic journals and popular magazines. Forty-one of these pieces are gathered here. The most controversial is the final review in which I criticize a current teaching fad known as the 'new new math'. By the time this book is published I would guess and hope that new new math is being abandoned almost as rapidly as the old new math faded. I could be wrong. In any case, it may be decades before our public education is able to attract competent teachers who have learned how to teach math to pre-college students without putting them to sleep. There are, of course, many teachers who deserve nothing but praise. It is to them I have dedicated this book.

The American Mathematical Society awarded Gardner the Leroy P Steele prize at the summer meeting in Salt Lake City in 1987 [1]:-

... for his many books and articles on mathematics and particularly for his column "Mathematical Games" in 'Scientific American'.

The citation continues [1]:-

Martin Gardner has introduced generations of readers to the intellectual excitement, the wonder, the variety, and the sheer fun of mathematics and mathematical ways of thought. With Dr Matrix and his other friends, he has exposed spurious thinking and misuse of mathematics in areas from numerology to economics. Martin Gardner has captured the attention of his readers, obtained their active involvement, and stretched their minds to an extent which is the envy of all of us who teach.

In his response, Gardner said [1]:-

Had I not developed a strong interest in philosophy when I was an undergraduate at the University of Chicago, I might have majored in mathematics, become a professional, and perhaps made some contributions to the field. As it happens, I had no formal training in mathematics, only an amateur's passion for its marvels, and admiration and awe for its leaders. I think of myself as like a person who loves classical music, but whose talents never advanced beyond playing simple tunes on a musical saw. There is no better way to teach oneself mathematics than to write about it. Every column I completed for 'Scientific American' was a learning experience that gave me intense pleasure. If I have been able to convey to others something of the fascination of mathematics, it is because I did not know enough to write about it on a technical level. ... To be given the Steele Prize is the greatest honour I can imagine myself receiving.

Let us end by noting that Gardner has produced a number of mathematical papers, written with leading mathematicians. He said [3]:-

I enjoy mathematics so much because it has a strange kind of unearthly beauty. There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known. On a low level I have experienced such a pleasure four times.
(1) I discovered the minimum number of acute triangles into which a square can be dissected. (Coxeter includes the dissection in his classic, 'Introduction to Geometry'.)
(2) I found a minimal network of Steiner trees that join all the corners of a chessboard.
(3) I constructed a bicolour proof that every serial isogon of 90 degrees - a polygon with all right angles, and sides in 1, 2, 3... sequence - must have a number of sides that is a multiple of 8.
(4) I devised a novel way to diagram the propositional calculus.

The results Gardner mentions here under (2) appear in the paper "Fan Chung, Martin Gardner and Ron Graham, Steiner trees on a checkerboard, Math. Mag. 62 (1989)". Ding Zhu Du writes:-

This is a very interesting article. It tells us what kind of Steiner minimum tree a checkerboard should have. The proof is left to the reader as an open problem. The Steiner minimum tree is the shortest network interconnecting the given points. A checkerboard of order $n$ is the set of $n^{2}$ points which form an $n \times n$ rectangular array. The connection pattern of the Steiner minimum tree for a checkerboard is quite complicated. However, the authors describe it very clearly with many nice pictures. It is amazing that the proof seems so hard. Every case except $n = 2$ or 3 is only conjectured. Hopefully, a sequence of efforts will result from this article.

The result Gardner mentions under (3) appears in "Lee Sallows, Martin Gardner, Richard Guy and Donald Knuth, Serial isogons of 90 degrees, Math. Mag. 64 (1991)". The authors write:-

In 1988 Sallows devised a computer program to search a unit-square grid for closed paths with the following properties. The path starts along a lattice line with a segment of unit length, turns 90 degrees in either direction, continues for 2 units, turns again in either direction, continues for 3 units, and so on. In other words, the segments of the path are in serial order $1, 2, 3, .. N$, with a right angle turn at the end of each segment. A path of $N$ segments - the number is of course the same as the number of turns or corners - is said to be a path of order $N$. If the path returns to its starting point, making a right-angle with its first segment, we call it a serial isogon of 90 degrees. We prove that, for any 90-degree serial isogon, $N$ must be a multiple of 8.

Three papers by Gardner which appear in The College Mathematical Journal are Modeling mathematics with playing cards (2000), Some new results on magic hexagrams (2000), and L-tromino tiling of mutilated chessboards (2009).

Gardner has received a number of honours for his remarkable contributions including an honorary doctorate from Bucknell University in 1978 and the American Institute of Physics science writer of the year award for 1983.