An arithmetical prodigy from 1899

An arithmetical prodigy from 1899

GRIFFITH, Arthur F., a new arithmetical prodigy, attracted the attention of the scientific world in December, 1899, through the efforts of Professors E. H. Lindley and W. L. Bryan, of the University of Indiana.

The present case is more than a mere curiosity, as he has been subjected to psychological investigation. This has been done in the case of only two rapid calculators before the present. M. Binet, of Paris, made a scientific investigation into the mental development and methods of Inaudi and Diamandi.

The new American calculator is a youth nineteen years old by the name of Arthur Frederick Griffith, born in Milford, Kosciusko County, Ind. His father is a stonemason in poor circumstances. He is the eldest of six children, and began at an early age to exhibit signs of his present extraordinary facility with numbers. His mother could keep him still when a very young child by getting him to count the various objects in the room, and this habit of enumeration gradually extended to counting for three succeeding summers the number of grains of corn fed to his father's chickens.

There has been no very great aptitude for figures displayed by any of his ancestors or relatives as far as is known, nor has there been any mental or physical abnormality (with one slight exception) in himself or in his family. He was sent to the public school at the age of ten years, and continued there for seven years, making a good record in all studies, but astonishing his teachers by his wonderful command of numbers. His mathematical education went no higher than arithmetic. His progress in that branch will be appreciated when it is said that his teachers used to employ him to write on the blackboard perfect squares and cubes of numbers for the other pupils of his class to find the roots of. He began when twelve years old to invent short cuts in which largely he owes his wonderful results.

He was brought to the psychological laboratory of Indiana University in November, 1899, and Professors Lindley and Bryan made a careful examination of his physical and mental state, finding his nerve signs, his sensory and motor abilities normal. His memory for figures is extraordinary. He recalls 21 digits after a single hearing. His memory is not much above the normal in ordinary psychological retentiveness, but is systematic and highly trained in certain lines. He knows the multiplication table up to 130 by 130, and gradually increases this by remembering for months the problems given him. He has memorized the squares of all numbers up to 130, the cubes of all up to 100, the fourth powers up to 20, the fifth powers of many numbers, and all the powers of the numbers 2 and 5 up to and including the thirty-third power. He is one of the most rapid calculators on record, multiplying two-place figures in a second and a half, and a four by a three-place number in 3 seconds. He adds three columns of figures as quickly as an expert accountant, and extracts the square root of six-place numbers in about five seconds, and the cube root of nine-place numbers in a slightly longer time.

His methods, which have been studied by Professors Lindley and Bryan, show that he has an extraordinary originality, anticipating many of the methods of higher mathematics and making discoveries of his own about the relations of numbers. He has fifty short methods of multiplying, some of which, though he never studied algebra, approximate the binomial theorem; he has six abbreviated methods for adding and the same number for dividing. One of his short cuts is illustrated by the fact that he raises 991 to the fifth power in 13 operations.

A further study of Griffith and his methods is in progress, and it is thought that it will be of great value, not only to psychology, but to mathematics as well. Griffith was exhibited before the annual meeting of the American Psychological Association at New Haven, Conn., December 27 and 28, 1899.

Introduction.

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There are many myths related to arithmetic prodigies. The most common is that great arithmetic feats are due to some exceptional inborn abilities, and extraordinary memory. For those who want to see a more realistic evaluation of exceptional arithmetic skills, we recommend a book written by Arthur Benjamin:

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Arthur Benjamin, Mathemagics: How to Look like a Genius Without Really Trying (with M.B. Shermer) (Book-of-the-Month-Club Selection), McGraw Hill, New York, 218 pages, 1993, (revised edition, 1998).

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Arthur Benjamin is a mathematician and a professional magician. His magic repertoire includes arithmetic feats. Magicians usually guard their professional secrets, but Arthur Benjamin made an exception for his math magic. So his book provides a detailed "first hand" description of the skills involved in his amazing performances.

Basics.

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There are two resources needed to execute arithmetic algorithms: know-how and memory.

Know-how includes some memorization, the multiplication facts to 12 times 12, squares of numbers up to fifty, prime numbers up to 100, and some other bits and pieces of arithmetic lore that sometimes come in handy. But the main part is knowledge of a variety of algorithms which make apparently difficult computations easy. For example, you can compute 37 times 43 instantly if you know that

37*43 = (40-3)*(40+3) = 40² - 3².

You simply subtract 9 from 1600, giving 1591 as the answer!

The memory that is needed is "working term memory", which keeps the partial results of a computation for a few seconds until they are used again and can be forgotten. Most people, without any training, can keep around seven decimal digits in their working memory. But when numbers make sense to a person who is very familiar with their properties, the ability to remember numbers is greatly increased. Think about remembering 36144324. For a person who sees this as 36 = 6², 144 = 12², and 324 = 18², this number is not a "random" bunch of digits; it can be remembered easily for quite a while.

Doing tasks.

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To see what is essential in performing arithmetic feats without straining your skills and memory, try to do some task dealing with multi-digit numbers using a simple calculator. A calculator will greatly extend your knowledge of memorized facts. Also, use paper and pencil to write partial results, which will extend your short-term memory. In this way you will be testing only your know-how.

Compute 991⁵.

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(According to the article above, Arthur Griffith can do this in 13 operations!)

An example of an efficient computation.

Know-how (Pascal's triangle):

					1
				1		1
			1		2		1
		1		3		3		1
	1		4		6		4		1
1		5		10		10		5	1

The coefficients in the binomial expansion of (a + b)⁵ are 1, 5, 10, 10, 5, 1.

The coefficients in the binomial expansion of (a - b)⁵ are 1, -5, 10, -10, 5, -1

We rewrite 991 as 1000 - 9.

991⁵ = (1000 - 9)⁵ = 1000⁵ - 5*9*1000⁴ +10*9²*1000³ - 10*9³*1000² + 5*9⁴*1000 - 9⁵

Calculate:		Write:
(mental) 1000 - 45 = 955		955 000 000 000 000
(calculator) 81000 - 9³ = 80271		802 710 000 000
(calculator) (5000 - 9)*9⁴ = 32745951		32 745 951
(written) Add the numbers.		955 802 742 745 951

So 991⁵ = 955,802,742,745,951.

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