by Jon Kujawa
On October twenty-first was Martin Gardner's 100th birthday. In celebration nearly one hundred Celebration of Mind (CoM) events are being held around the world. There are a few still yet to come: check their events listing to find one near you . The CoM is an annual affair in which people celebrate all the things Martin loved: magic, art, music, mathematics, science, and the sheer joy of curiosity and discovery.
You're the sort of person who should go to a CoM if you think a computer constructed from 10,000 dominoes is really, really cool. Or a paper cutout dragon which appears to turn its head as you move. Or if you can't help but be intrigued by a question like “If you heat up a metal washer, does the hole in the center get larger or smaller?”
Who was this Martin Gardner fellow who inspired so many people? Born and raised in Tulsa, Oklahoma, he is right up there with Will Rogers and Woody Guthrie in the pantheon of influential Okies. Gardner earned a degree in Philosophy from the University of Chicago and thereafter became a writer.
In 1956 he wrote an article for Scientific American about hexaflexagons: flat shapes you can construct from single sheet of paper which can flexed to reveal more than the two sides you expect. There's a nice video by James Grime in honor of Martin Gardner which shows how to construct the six sided version.
The article was so popular that Martin Gardner was invited to write a monthly column for Scientific American on “recreational mathematics”. He did so for twenty-five years and forever set the bar on writing mathematics for a general audience. His columns introduced the world to Conway's Game of Life and cellular automata, Penrose's infinite tiling of the plane with a pattern which never repeats despite using only two different tiles, the now-widely-used RSA encryption scheme, and the Mandelbrot set we ran into last month.
Despite the fact (or, more likely, because) he had no training in math, Martin Gardner intuitively understood how to communicate the excitement, wonder, and fun inherent in mathematics. As he wrote in the introduction to “Mathematical Carnival”:
The best way, it has always seemed to me, to make mathematics interesting to students and laymen is to approach it in a spirit of play. On upper levels, especially when mathematics is applied to practical problems, it can and should be deadly serious. But on lower levels, no student is motivated to learn advanced group theory, for example, by telling him that he will find it beautiful and stimulating, or even useful, if he becomes a particle physicist. Surely the best way to wake up a student is to present him with an intriguing mathematical game, puzzle, magic trick, joke, paradox, model, limerick, or any of a score of other things that dull teachers tend to avoid because they seem frivolous.
Countless people, including many professional mathematicians, trace their interest in mathematics to Gardner's column. A Bulgarian colleague of mine speaks with great excitement about reading Gardner's books while he was a student behind the Iron Curtain. I, though, was only dimly aware of his writings until after I completed graduate school . I did however have the good fortune to meet Martin before his death. His son is on the faculty at my university and Martin lived here for last few years of his life. It made us something of a pilgrimage site for people who came to visit Martin. In fact, I first learned he lived in town when I saw a photo in the local newspaper of the famed TV magician David Blaine visiting Martin Gardner!
Fortunately for you and me, his 300 or so columns were collected into a series of fifteen books which are still available from your local bookseller. Many can be found online via Google books and the like, but they are also well worth owning (or giving as gifts!). In particular, Gardner worked with the Mathematical Association of America to publish an updated version of his books. Four were released before his death and hopefully the MAA plans to print the rest as well.
In addition to his column on recreational mathematics, Martin Gardner wrote books on magic, skepticism and pseudoscience, and a fantastic annotated version of Alice in Wonderland. Over 100 books in all! His latest, an autobiography, just came out this fall. If you'd like a sampling of his work, I can recommend nothing better than Colm Mulcahy's recent “The Top 10 Martin Gardner Scientific American Articles”. Dr. Mulcahy also recently wrote a very nice essay about Martin Gardner and their friendship.
Flipping through Gardner's essays at random, I found the following one which I quite like. It's about the two person game “Sprouts” invented by John Conway and Michael Paterson after tea on Tuesday, February 21, 1967 in the Cambridge math department lounge. The rules are straightforward:
- You start with some number of dots drawn on a sheet of paper.
- On a person's turn, the player draws a line connecting one dot to another. You can draw the line however you like, but are not allowed to draw a line which crosses itself, another line, or a dot. The player also draws a new dot somewhere along this new line.
- Each dot can have at most three lines connected to it. Note that a new dot drawn in the middle of a line is considered to already have two lines connected to it.
- The first person who can't play loses.
Here's a game which starts with two dots :
At first it looks like it might be possible to play forever. Conway, however, found a simple argument which shows that if you start with n dots, the game must end in 3n-1 moves: if you think of the three lines which are allowed to touch each dot that dot's “lives”, then the game begins with 3n lives. After one player moves, they have connected two dots by a line and so killed two lives. The new dot drawn in the middle of the new line already has two lines connected to it, so it only creates one new life. This means that at the end of the player's move the total number of lives has gone down by one. Starting with 3n lives this means there are at most 3n-1 possible moves before the game must end. Gardner also writes that it's easy to see that every game will take at least 2n moves. I'll leave it to you to figure out why.
Even thought the rules are simple and the game ends relatively quickly, Sprouts turns out to be quite complicated to study. In a 2010 paper by Julien Lemoine and Simon Viennot the authors were able to use computers to analyze all Sprouts games which start with 32 dots or less. They weren't able to do the 33 dot game, however. Interestingly, their computations support the Sprouts Conjecture: the first player can always win the game if the number of dots you start with has remainder 3, 4, or 5 when you divide by 6, and otherwise the second player always wins. As far as I know, a complete analysis of Sprouts is still unavailable.
Let's have a virtual CoM here at 3QD in honor of Martin's centennial. Please share your favorite puzzle, brain-teaser, or conundrum in the comments. Gardner's book “Mathematical Magic Show” he has a delightful chapter entitled “Ridiculous Questions” which is in this spirit. He warns us that “None of the following short problems requires a knowledge of advanced mathematics. Most of them have unexpected or 'catch' answers, and are not meant to be taken seriously.”
Here's a sampling:
- In a certain village there live 800 women. Three percent of them are wearing one earring. Of the other 97 percent, half are wearing two earrings, half are wearing none. How many earrings all together are being worn by the women?
- Among the assertions made in this problem there are three errors. What are they?
- A secretary types four letters to four people and addresses the four envelopes. If the secretary inserts the letters at random, each in a different envelope, what is the probability that exactly three letters will go into the right envelopes?
- Can a 6 by 6 by 6 cube be made with 27 bricks that are each 1 by 2 by 4 units?
- Smith gave a hotel clerk $15 for his room for the night. When the clerk discovered that he had overcharged by $5, he sent a bellboy to Smith's room with five $1 bills. The dishonest bellboy gave only three to Smith, keeping the other two for himself. Smith has now paid $12 for his room. The bellboy has acquired $2. This accounts for $14. Where is the missing dollar?
 The CoM is an open-source event — if there isn't one already going near you, consider starting one!
 Instead I had the good fortune of crossing paths with a group of teachers in high school who managed to convey the excitement, joy, and beauty to be found in learning interesting things. My career in math is in large part due to the insufficiently rewarded efforts of Carlton Urdahl and Al Heine at Buffalo High School in Buffalo, MN.
 Sprouts animation borrowed from Byrdseed here.
 “Place four coins on the bottom row of circles (G, D, E, and R), so that the letters MARTIN are exposed. Your challenge is to slide these coins along the graph edges, covering the top row of circles, to expose the letters GARDNER. Easy, right? There's a catch — at no time are two coins allowed to be next to each other along an edge. You'll find this makes the task much more interesting. (And yes, you have to slide the coins one at a time, all the way from one circle to another!) For example, the G coin cannot move, initially — it would wind up adjacent to the D coin. But D can move to T. (or D-T). Be careful to pay attention to all edges as you slide the coins — the I-R edge is particularly easy to miss. Good Luck!” Puzzle from here.