Arithmagic

Watch him pull a rabbit out of his hat? Not exactly.

Arthur T. Benjamin eschews the usual trappings of the magician's trade. Calling himself a mathemagician, he astonishes audiences with amazing feats of mental arithmetic. Behind the scenes, he reveals how you, too, can look like a genius without really trying.

A math professor at Harvey Mudd College in Claremont, Calif., Benjamin has brought his particular brand of prestidigitation to a wide variety of appreciative audiences. He has appeared on television programs and performed in night clubs, classrooms, and banquet halls. Last July, he delighted a crowd of mathematicians gathered for the Toronto Mathfest.

Benjamin's act is a striking demonstration of how miraculous fairly ordinary, simple mathematical machinery can appear when you don't really know what's going on under the hood.

One of my first encounters--many, many years ago--with arithmetic shortcuts was in a book called Cheaper by the Dozen, the story of efficiency expert Frank B. Gilbreth and his family of 12 children. In a chapter about dinner conversation, Frank B. Gilbreth Jr., and Ernestine Gilbreth Carey write, "Also of exceptional general interest was a series of tricks whereby Dad could multiply large numbers in his head, without using pencil and paper."

For example, to multiply 46 times 46, you figure out how much greater 46 is than 25. The answer is 21. Then you calculate how much less 46 is than 50. The answer is 4. You square 4 and get 16. Putting 16 and 21 together gives the answer 2116.

That's the sort of venerable procedure that Benjamin uses to do fast multiplies and to square four-digit numbers faster than someone using a calculator. Turning a repertoire of such calculating tricks into a real show, however, requires developing a memory for numbers and learning how to calculate from left to right, performed at the speed of rapid-fire chatter.

Suppose you want to multiply 378 by 7. Starting from the left, you would get 2100 (300 x 7) plus something more. In the next step, 70 x 7 equals 490, which is added to 2100 to give 2590, plus something more. Finally, 7 x 8 equals 56, which is added to 2590 to give 2646. One advantage of using this method is that you can start saying the answer while you're still calculating it, Benjamin remarks.

Here's a neat way to square two-digit numbers. Suppose the number to be squared is 37. That number is 3 less than 40, and 34 is 3 less than 37. Multiply 40 by 34 to get 1360, then add the square of the difference, 3² or 9, to get 1369. The trick is to choose the difference so that the multiplication is easy. For example, to square 59, choose a difference of 1. Go up to 60 and down to 58. Multiply 60 times 58 to get 3480, then add 1² , to obtain 3481.

The proof is in the algebra: a²= (a + d)(a - d) + d^{2 .} The same idea can be extended to squaring 3-digit and 4-digit numbers.

Interestingly, when Benjamin performs calculations involving long sequences of digits, he relies on a phonetic code to remember the numbers. "There's no mental blackboard," he says. "It's much more an auditory process."

Benjamin turns sequences of digits into words that add up to some sort of crazy scenario. For example, he can mentally convert the sequence 9 6 4 8 3 7 5 4 8 3 1 2 7 5 9 6 into the words "pitcher fume color fume tinkle beach" as a handy mnemonic for (9 6 4) (8 3) (7 5 4) (8 3) (1 2 7 5) (9 6).

The underlying scheme assigns different consonants to different numbers, and the memorizer supplies the vowels: 1 (t, d), 2 (n), 3 (m), 4 (r), 5 (l), 6 (j, ch, sh), 7 (c, k, g), 8 (f, v, ph), 9 (p, b), 10 (z, s). This is the modern version of a number alphabet originally proposed by Pierre H&eacuterigone and published in Paris in 1634. Three consonants--w, h, and y, spelling "why"--do not appear in the list.

Benjamin can also handle magic squares (see More than Magic Squares, 10/14/96), natural logarithms, cube roots, and much more. "As a kid, I liked to show off," he says. "Now I get paid to do this!" Besides, the techniques are so easy that even an elementary-school student can master them.

References:

Gardner, M. 1989. Tricks of lightning calculators. In Mathematical Carnival. Washington, D.C.: Mathematical Association of America.

______. 1988. Memorizing numbers. In Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Mathematical Puzzles and Games. Chicago: University of Chicago Press.

Gilbreth, F.B., Jr., and E.G. Carey. 1948. Cheaper by the Dozen. New York: HarperCollins.

Arthur Benjamin's Web page is at http://www.math.hmc.edu/~benjamin/.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.