Martin Gardner and Recreational Mathematics
Most people don’t do mathematics for recreation. In part, this is probably due to our schooling, which positions “school mathematics” as the only type of mathematics that most of us experience. You can go your whole life thinking of “mathematics” as calculations, a bit of algebra, and maybe the quadratic formula. In most Kindergarten to grade 12 curricula, calculus and algebra are the peaks of mathematics learning, with the early grades a steady march through mastering calculation skills, and on to more abstract “things”, which then serve as gatekeepers for who gets to do college or university mathematics. This is the approved knowledge base that is used to prepare us for life and work, debatable though it is (what topics are to be included, and why?)
Once our formal schooling is over, we probably have little pieces of mathematics that we use in our jobs and lives: measurements, taxes, and so on. Doing mathematics purely for recreation is not really a consideration to most people. Relaxation time in the evenings is usually given over to less taxing pursuits.
But what if we did mathematics purely for recreation, rather than for research, or for practical reasons?
If you had bad experiences with school mathematics, this idea may make you cringe, or inspire fear. Why do that, you might ask? My answer to this is also my answer to many things: because it’s interesting. At risk of tautology, I might state this as an axiom of life:
It’s interesting to think about interesting things.
The category “recreational mathematics” has and should include things like puzzles, chess problems, sudoku, logic puzzles, and games. There should be something for everyone in this category.
Martin Gardner’s column in Scientific American, which ran from 1956 to 1981, was called “Mathematical Games”, and spawned many of the most well-known recreational math problems. It is worth noting that Gardner himself wasn’t a world-class mathematician.
In 2004, he had this to say:
I go up to calculus, and beyond that I don’t understand any of the papers that are being written. I consider that that was an advantage for the type of column I was doing because I had to understand what I was writing about, and that enabled me to write in such a way that an average reader could understand what I was saying. If you are writing popularly about math, I think it’s good not to know too much math. (AMS interview, quoted on Wikipedia)
His last statement resonates and should be a lasting statement. From Eugenia Cheng, to Stephen Strogatz, to Jordan Ellenberg, we are in a bit of a golden age of mathematical explanation for the layperson. And indeed, untrained mathematicians such as myself, benefit immensely from an explanation of mathematics without extensive and dense notation, of the kind that professional mathematicians work with.
His problems are designed to get you thinking, and this is a thinking-averse world. Humans are terrific at thinking, however, when we choose to do so- it’s often just easier not to.
Gardner made his problems so you don’t need extensive expertise to come up with a “line of attack” on them. You won’t be deploying set theory, or higher dimensional constructs with them, just logic, and common sense.
These problems puzzle and surprise though, and they are often delightful in their simplicity, once you see through them.
Consider this famous problem from Mr. Gardner, often called the “Mutilated Chessboard”. Alex Bellos’ version of the same problem is here.
The naive answer is, “yes, of course, 31 x 2 is 62”, and that works. A few more minutes of playing around might afford you greater insight. The reason for the answer can be explained quite simply when you *see* it. (Visualization is key to this sort of problem).
Another favourite, cited by Alex Bellos (who is carrying the torch for recreational mathematics with his books and newspaper columns), is this one:
Ten red socks and ten blue socks are all mixed up in a dresser drawer. The 20 socks are exactly alike except for their colour. The room is in pitch darkness and you want two matching socks. What is the smallest number of socks you must take out of the drawer in order to be certain that you have a pair that match?
Again, once you see the answer, you will be surprised by its simplicity. There is a very simple mathematical principle at play in this problem. I suggest staging this problem with actual socks!
Or, consider this one, from my deck of “Mathematical Games Challenge Cards”:
Joe throws an ordinary die, and then Moe throws the same die. What is the probability that Joe will throw a higher number than Moe?
Can you explain why the answer is 5/12?
“Perhaps in playing with these puzzles you will discover that mathematics is more delightful than you expected.”
Martin Gardner. (Entertaining Mathematical Puzzles, 1961).
In the same book, Gardner speaks of the satisfaction of problem-solving:
How about this proportional reasoning problem, from the same book?
Here is a problem about the power doubling, from the same book (often called by teachers “the penny problem”.
Does this result surprise you?
This one is about earning money:
Which would you choose? Why?
Optimal pie or cake cutting is often a big deal at parties. Here is a problem about that:
Is this entertaining math, or serious math.
In Gardner’s words, from his last column in 1998:
The line between entertaining math and serious math is a blurry one. Many professional mathematicians regard their work as a form of play, in the same way professional golfers or basketball stars might.
You might say that math *is* play (or at least it *can* be).
Gardner, in the same column, lamented the lack of progress in getting recreational mathematics into curricula around the world:
For 40 years I have done my best to convince educators that recreational math should be incorporated into the standard curriculum. It should be regularly introduced as a way to interest young students in the wonders of mathematics. So far, though, movement in this direction has been glacial.
This is still the case. Mathematics as a school subject is still too often seen as medicine that must be taken, rather than as an activity that sparks joy, fun, and play. This must change. Tackling one of these puzzles with your students (if you are a teacher), or for yourself, might be a good start.
Matthew Oldridge believes that “math is play, or at least it can be more play-ful”.
