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The Magic and Math of Mental Calculation

“If I were performing for a different sort of audience, like, say, at the Magic Castle in Hollywood, I might follow this up with other feats of magic and mind,” mathemagician (and Harvey Mudd College professor) Art Benjamin told the 300-plus people gathered at the Carnegie Institution for Science on October 23 to hear him speak.

Martin Gardner Celebration of Mind 2013

Benjamin had just wowed the crowd by squaring two-, three-, and four-digit numbers in his head; constructing a magic square based on a volunteer’s birth date; and correctly pairing days of the week with dates decades past.

“But for an audience such as this one,” he continued, “here for MAA’s Celebration of Mind, I actually prefer to break the magician’s rule and explain to you most of what you’ve seen me do up here—and more.”

And so began the lengthy question-and-answer portion of the MAA’s fourth annual Martin Gardner Celebration of Mind. Celebrations of Mind are held worldwide on or around the October 21 birthday of the late Martin Gardner, bringing puzzlers, math enthusiasts, and Gardner fans together to share in the sorts of mental enrichment Gardner himself enjoyed.

This year’s MAA event—a presentation by Arthur Benjamin called “The Magic and Math of Mental Calculation” held two days after what would have been Gardner’s 99th birthday—was cosponsored by Math for America–DC and ThinkFun. ThinkFun donated enough copies of its popular games Chocolate Fix and Rush Hour for each attendee to leave with one.

They left, too, with insights into Benjamin’s methods.

Identifying days of the week, for instance, comes down to memorizing and combining numeric codes for days, months, and years. How to remember that the code for October is six? “You might think of ‘tricks’ or ‘treats’”—each six letters long—or note that “‘tricks’ rhymes with ‘six.’”

And the “missing-digit trick”? When the mathemagician asked volunteers to multiply 5,778 by any three-digit number and then read off in any order six of the seven digits of the resulting product, how could he identify the digit they had omitted (four out of four times!)? Benjamin suspected that some of his math-savvy listeners had figured this feat out for themselves. “Finish my sentence,” he prompted them. “That was just based on the magic property of the number...”

“Nine,” thundered the audience in response. (If you’re unfamiliar with the property in question, watch the video of Benjamin’s presentation to learn more.)

Benjamin even disclosed how he made the “magic number” he had derived from volunteer Skyler’s birth date appear mind-bogglingly often in a four-by-four magic square. He did so, though, only after he mourned the tendency of audiences to react to the revelation of a magician’s secrets with disappointment.

“I’ll explain how I do the magic square,” he said, “but just remember how impressed you were before you heard the explanation.”

Perhaps most instructive for the many youngsters in the Carnegie auditorium was Benjamin’s response to a question about how he became the marvel he is today.

“The short answer is, I did a lot of homework,” he said, addressing by name the 7-year-old who had posed the question (and who had introduced herself to Benjamin before the talk). “The only way to get good at math—or anything—is to put a lot of time and practice into it. I was not born doing these kinds of calculations.”

Benjamin continued: “Did I waste a lot of my youth thinking about ways to multiply numbers? Yes I did, and I’m proud of that.”

Those seated in the front row may at this point have caught Benjamin’s eye sparkling even a bit more than usual.

“You might call it The Products of a Misspent Youth,” he mused aloud. A title for an autobiography, perhaps? “I’ve never used that line—I’m gonna keep that one.”

Until that autobiography hits the shelves years hence, interested parties will have to content themselves with Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks or the Arthur Benjamin courses available from the Teaching Company or one of the two books the mathemagician/math professor has coauthored for the MAA. —Katharine Merow

Watch the lecture on YouTube.

You can also watch a short version or hear Benjamin tell the story of how he became a mathematician.