The infinity and the mystery about 196

I’m writing this one in english because it should be a guest post on Leah Libresco’s blog Unequally Yoked. Due to many logistical reasons this was never published.

It was an extremely hot Saturday in 1987. Leah (see note above) was far from being due in this World, and I was in the sixth grade at São Bento School in Rio de Janeiro, Brazil.

One of the Benedictines Monks together with a math teacher, Mrs. Sandra Carelli, brought to our school, in the past ten years, a new way to teach math for kids based on the educational theory of a Belgian mathematician named Georges Papy.

Papy advocated the use of sets as a main framework to teach math. The method, although very expensive in terms of teacher preparation, was producing a remarkable success in Brazil.

This digression is necessary to introduce the topic of this post, from the Naming Infinity Bookclubbers, about the fascination some people feel with numbers and the infinity when they are a child. When I read the book I got those memories back.

In that particular hot saturday, we just saw the proof of the cardinality of R being distinct of N, and for some 11-year-old boys it was too much for one day. We are dreaming about going home and playing soccer, but we still had 50 minutes more of math class to survive. Mrs. Carelli was having trouble keeping the order, and to keep us quiet she told us about a magic set: The Lychrel Numbers Set.

What are those Lychrel numbers? Well, it’s quite straightforward to define what are not a Lychrel number: if you take an integer, 305 for instance, and reverse its digits (503) and then add both numbers, you’ll get a palindrome: 808. In some cases, you’ll have to repeat the process more than once to get a palindrome, but you’ll eventually get one: 307 + 703 = 1010; 1010 + 0101 = 1111.

There’s, however, one magic number: 196.

This number can never be “palindromed”, and it creates a series of numbers that can never be “palindromed”: 691, 788, 887, 1675,  5761, 6347, 7436, and so forth.

The set of those numbers is known as Lychrel number, and it’s cardinality is supposed to be similar to N.

She didn’t tell us that, but challenged us to find out that 196 was not “magical”. The full classroom with 30 boys started summing and inverting digits and filling many sheets of papers with calculation for the next 50 minutes. At the end of the class, she told us that until that time no one had found a palindrome derived from 196, and it was believed that 196 would never be “palindromed”.

I went home and felt sorry for 196. Poor number! It’s not an ugly number, it was not prime, it wasn’t even an odd number. Why was it doomed to be forever inverted without reaching the comfort of being “palindromed”?

The years passed, and Lychrel numbers slipped to most obscure areas of memory but never vanished. Upon entering the University in 1993 when SPARCstations 2 were the most powerful desktop computers we had access, one of the first programs I developed was to spend idle computer time to calculate thousands of iterations for 196. It still seemed unfair that 196 could not rest in piece with its palindrome.

Still today, 196 reversed more than 10ˆ13 times, has no palindrome.

Sometimes, for fun, I thought about a somewhat mystical meaning for 196, related with the mystery of the Holy Trinity. As if God wanted to play with us and gave a number that has 1 (one God) and two multiples of 3 (the Trinity) with this “magical” property.

Some people ask when I talk about this subject: what’s the purpose of the Lychrel numbers in mathematics? None, as far as I know, but so was the prime numbers, the quaternions and other theories that later found a prominent place in the engineering, computer science or physics. Maybe the Lychrel numbers never find its use but for at least two thing it was very useful: to help a teacher calm down a classroom in 1987 and to make me love mathematics and get intrigued with the infinity.

naming infinity