[ July 16, 2001 ]
A palindrome story for 89
told by Terry Trotter (†) (email)

[ October 7, 2023 ]
and a Fibonacci story for its reciprocal \( {1 \over 89} \)
told by Ir. Jos Heynderickx & Dr. Luc Gheysens - goto second topic

Background

I first learned about palindromes in general, and the two famous
problems about the reverse-and-add procedure with the numbers
89 and 196, sometime in the late 60's. Stanley Bezuszka, of Boston
College, had come to Kansas to give a session at a math teachers'
conference, and he presented the problems to us. He commented
that math teachers often couldn't do the 89 problem correctly
on their first attempt. They would frequently make silly, little
errors, then think they'd found the palindrome. As their errors
were pointed out to them, they began to realize that what, at first,
seems so easy, becomes hard if they aren't careful.

The story

My first year working here in the Escuela Americana of San Salvador
(1981) I used this problem one Friday with a group of 20-25 seventh
graders. Supposedly it was an elite group of kids, and for the most
part it was. So every Friday was a 'fun day', in which I dug into my bag
of recreational math topics and presented "Trotter Math" to them.

After explaining the reverse-and-add procedure on the chalkboard
with about 3 examples, I then said, "Ok, anybody want to make some
money ?" Now I had their full attention.

I continued, "I'll give one colon to the first person who can bring
to me a palindrome, correctly computed, beginning with the
number...(pause)...89 ! Begin now."

Pencils were beginning to move rapidly on all their papers. Soon
someone shouted, "I've got it !" and rushed up to me with his paper.
"Sorry," I said, "look, there's an error here. Go back and work some
more." Another soon did the same, but with a different error. Then
another, and so on.

I decided there needed to be a new rule, to get them to slow down
and be more careful. I said, "Ok, now. If you bring me something with
an error on it, you must pay me 25 centavos as a fine. But if you're
correct, I'll pay you the colon, of course. Is that a deal ?" They agreed.

But they continued to come forward... with incorrect work. I was getting
a nice collection of 25-centavo coins. Some even had to borrow a coin
from a friend in order to show me wrong work.

Finally, a girl, who had been quietly working all the while, raised
her hand, and said softly, "Mr. Trotter, I think I have it." Well, Claudia
was not a good student in math. Never was, in all the 3 years she was in
my classes. So I was sure she had made a mistake, too. But I said, "Bring
your paper, please."

So she did. I'm sure you can guess the rest --- it was exactly right !
All 24 addition steps leading to the 13-digit palindrome were there. And
she was the first student I ever had, and one of the very few, to do it
on the first attempt.

Oh, yes, the money. I awarded her all the coins the fast students had
paid to me in fines, over 2 colones, as I recall. Everybody had a good time
and learned a good math lesson. And the whole affair didn't cost me a
'centavo' of my own money !

[ October 7, 2023 ]

The reciprocal of the spooky number 89 is the fraction \( {1 \over 89} \)
by Ir. Jos Heynderickx & Dr. Luc Gheysens

Apparently this fraction is the sum of an infinite row of numbers
whereby behind the decimal point the Fibonacci numbers (A000045) appear.

I was charmed by this topic in these gentlemen's vast and impressive collection
of various mathematical puzzles, riddles, curiosa, etc. (in Dutch).

Here are some links to those entries
Summary of all webpages
https://curiosa-deel7
https://101.jouwweb.be/
https://rekenraadsels.jouwweb.be/

But let us return to the core of this 89 topic
by visualizing the infinite sum.

The explanation of this remarkable sum phenomenon is given here
Chapter 6.12
It gives the generative function for the Fibonacci numbers and the fraction \( {1 \over 89} \)

Here is an attachment with introduction to the world of Fibonacci numbers.
Bijlage 7. Fibonaccigetallen