Today, Guest Andrew Boyd, chief scientist at the
PROS organization, shares an "age-old" story about
numbers. The University of Houston presents this
series about the machines that make our
civilization run, and the people whose ingenuity
created them.

A fourteen-year-old friend of
mine recently informed me that we were "reversed
ages." I wasn't sure what he meant until he pointed
out that I was 41, so that the digits in our
respective ages were reversed -- one-four,
four-one. My friend is autistic and often observes
these kinds of details, so it seemed a mere anomaly
until he pointed out that we would also be reversed
ages when he was 25 and I was 52, and then again
when he was 36 and I was 63.

This was his way of saying the process repeats
every 11 years, and a little thought shows why.
Adding eleven to a number increases each individual
digit by exactly one. One-four plus eleven equals
two-five; four-one plus eleven equals five-two.

Had the story ended here, I probably wouldn't have
thought much of it; but he continued with the
revelation that at his next birthday he would be
reversed ages with a favorite uncle.

Intrigued by my young friend's insight, I began to
wonder if some deeper mathematical structure lay
behind the problem. Once started, the process
repeats every eleven years, but what gets it
started? Is there some special condition that leads
to this two-digit two-step, or does it happen to
all of us at some time or another?

When pondering the idea driving to and from
work led nowhere, it occurred to me that I had
failed to ask the source of the insight to begin
with. When I again saw my friend, I asked him why;
what made people have reversed ages? His reply was
remarkably simple and to the point: "Multiples of
nine."

Now, neither 14 nor 41 is a multiple of nine, but
their difference, 27, is. An afternoon of working
out the details led to some fascinating
discoveries, not the least of which is that nine
and eleven being situated on either side of ten is
not coincidental. Lo and behold, when two people
share an age difference that is a multiple of nine,
they will find the digits in their ages
periodically reversed. When the age difference is
not a multiple of 9, they will never experience
reversed ages.

Autism is a perplexing condition. Brought into the
national spotlight by Dustin Hoffman and Tom Cruise
in the Oscar-winning film *Rain
Man*, autistics frequently demonstrate
normal or exaggerated abilities in some areas while
performing far below their peers in others. An
autistic child may spell complicated words without
difficulty, yet have trouble putting together basic
sentences. The rules of social interaction can be
learned, but the nuance of social interaction will
forever remain an enigma.

My friend exhibited uncommon creativity by
connecting reversed ages to multiples of nine. By
thinking differently, he observed something
delightful about numbers. But even more, he
reminded us that there are no rules governing
imagination and inventiveness; that by "thinking
differently," we are all capable of remarkable
feats of ingenuity.

I'm Andy Boyd, at the University of Houston, where
we're interested in the way inventive minds work.

(Theme music)

Those with specific interests in autism should check
the Autism Online Support Group: http://www.mdjunction.com/autism
Dr. Andrew Boyd is Chief Scientist and Senior Vice
President at PROS, a worldwide provider of pricing
and revenue-optimization solutions. Working with
leading academicians and practitioners, he directs an
international group of advanced-degree recipients in
Economics, Operations Research, Quantitative
Marketing, and Statistics. Dr. Boyd received his A.B.
with Honors at Oberlin College with majors in
Mathematics and Economics in 1981 and his Ph.D. in
Operations Research from MIT in 1987. Prior to
joining PROS, he enjoyed a successful ten-year career
as a university professor.

Items (1) and (2) below demonstrate the multiples
of 9 property -- that two people will periodically
have reversed ages if and only if their age
difference is a multiple of 9. The remaining items
serve as challenges for the mathematically
adventurous.

1. We start by showing that two people with an age
difference that is a multiple of 9 will at some
point have reversed ages. It turns out that if the
age difference is D, and D = 9 x n, then reversed
ages are first observed when the younger person is
n years old. For example, suppose the age
difference between two people is D = 27 years. Then
27 = 9 x 3, making n = 3. When the younger person
turns 03, the older person turns 30, and the digits
are reversed. Every 11 years thereafter until the
older person has a three-digit age (should she be
so lucky), the two people will have reversed ages.
A proof can be found in the following table for age
differences up to 81.

2. We next show that whenever two people have
reversed ages, their age difference must be a
multiple of 9. First, write the number representing
the age of the older person as ab, where a is the
first digit and b is the second digit. The age of
the older person is then (a x 10) + b, just as the
number represented by the digits '83' is (8 x 10) +
3. The age of the younger person is represented by
ba, and the age is (b x 10) + a. Subtracting the
ages yields [(a x 10) + b] - [(b x 10) + a] = (9 x
a) - (9 x b) = 9 x (a - b). Not only is the number
a multiple of 9, it is 9 times the difference of
the two digits. 83 - 38 = 45 = 9 x (8 - 3).

3. Consider any number with increasing digits; for
example, 1256, 367, or 245,689. Any such number,
when multiplied by 9, has the property that the sum
of the digits in the result must be exactly 9. 9 x
1256 = 11,304, and 1 + 1 + 3 + 0 + 4 = 9. This
general property is easy to prove. Write any number
with, say, 5 digits as abcde, where a < b < c
< d < e. Rather than multiplying by 9
directly, multiply by (10 - 1); that is, multiply
the number by 10, which simply adds a 0 to the end
of abcde, and then subtract abcde. Add up the
digits of the result.

4. The reversed ages property holds for number
systems that use a base other than 10. Let B be the
base of a number system. Prove that two people will
share reversed ages if and only if their age
difference is a multiple of B - 1, and that the
period for reversed ages once they get started is B
+ 1. Hint: Replicate the arguments in items (1) and
(2).

5. It turns out that for any two numbers that have
reversed digits, not just two-digit numbers, the
difference is always a multiple of 9. In fact, the
result is true of any two numbers made up of the
same set of digits! For example, consider the
number 736,253. The number 236,735 consists of the
same set of digits, but rearranged. The difference
between these numbers is 736,253 - 236,735 =
499,518 = 9 x 55,502. Show that the difference
between any two numbers made up of the same set of
digits is always a multiple of 9. Hint: Pick a
number with, say, 5 digits and express it as abcde.
Write the number as (a x 10,000) + (b x 1000) + (c
x 100) + (d x 10) + e. Do the same for any
rearrangement of the digits, and take the
difference. Along the way, show that the difference
of any two powers of 10 is a multiple of 9. Expand
the argument to an arbitrary number of digits.n of
reversed ages, the reversed ages

6. Item (5) tells us that when two numbers have
reversed digits, the difference will always be a
multiple of 9. However, for any difference that is
a multiple of 9, is it always possible to find two
numbers with this difference that have reversed
digits? If so, then we have shown the reversed ages
property for all numbers, not just for the two
digit ages of items (1) and (2). The table in item
(1) hints at the problem. Up to a difference of 81
= 9 x 9, it is easy to find an age at which digits
first become reversed. However, the process breaks
down at 90 = 9 x 10. Following the same logic
embodied in the table, it should be that when the
younger person is 10 and the older is 100 the ages
will be reversed. But this is not the case. 010 is
not 100 reversed. Are there reversed numbers that
have a difference of 90? Yes, quite a few,
including 1101 and its reversed counterpart. When
the older person reaches an age of 1101, the
younger person will be 1011. What if the age
difference is 189 = 9 x 21? In this case reversed
ages are achieved when the older person reaches
1090 and the younger person is 0901. Question: for
any difference that is a multiple of 9, is it
always possible to find two numbers with this
difference that have reversed digits? The author
(A. Boyd) has not worked out the answer.

The Engines of Our Ingenuity is
Copyright © 1988-2002 by John H.
Lienhard.