BackAble was I ere I saw Lychrel
Are there any numbers that will not take you to a palindrome? The answer: Indeed there are, but not too many.
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Numbers lend themselves to endless interesting manipulations. Many people use them to wow audiences at parties. (I’ve tried, though wow may not quite apply). Done well, it can seem like magic; and indeed, there’s a significant overlap between mathematics and magic—go find the compelling “mathemagicians" on YouTube, for example.
So this column is going to tell you a little bit about one of the many intriguing patterns that numbers can throw at you, and the host of intriguing details that emerge from fiddling with them.
To start, pick a number that has two or more digits. Reverse the digits and add the two numbers. Repeat with the new number. Keep doing this and, somewhat surprisingly, you will pretty quickly get a number that is a palindrome. (Well, not always, but I’ll come to that in a moment).
For example, start with 469. Reverse to get 964. Add those two, you get 1,433. Reverse that to get 3,341. Add to 1,433, you get 4,774: there’s your palindrome. Try another? 5,194 takes you to 112,211 in three steps. One more? 382 goes to 2,552, also in three steps.
Who would have thought that a number you choose at random might generate this satisfying pattern—a number that reads the same backwards and forwards? There’s a taste of what you can do with numbers.
As you can imagine, mathematicians have thoroughly investigated this curious little phenomenon. They found that while most numbers produce a palindrome pretty quickly—in two or three steps—some get there rather more slowly. When you try 9,518, for example, you meander through 15 iterations before arriving at 133,697,796,331. The record here even has a charming name, the Most Delayed Palindrome: 261 iterations for this 119-digit palindromic behemoth: 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544.
But pretty soon, some mathematician asked the obvious question that I alluded to in passing above, and plenty of other mathematicians must have nodded their heads: Are there any numbers that will not take you to a palindrome?
The answer: Indeed there are, but not too many. The smallest such is 196, and the next few are 295, 394, 493, 592, 689 and 691.
Start with 196 and iterate away as long as you care: I guarantee you will not come to a palindrome.
Now I can’t actually prove that you won’t get a palindrome—nobody has proved it—so how can I guarantee it? Because far more passionate and dedicated men than me have started at 196 and gone to great lengths searching for palindromes. Like John Walker: in 1987, he programmed his computer to do this reverse-and-add routine starting with 196. Three (!) diligent years later, it stopped after 2,415,836 iterations, having produced a number that was—are you ready?—a million digits long. No palindrome in those two-million-plus steps, though. And if you think that’s a lot of work for Walker’s machine, in 2011 Romain Dalbeau reached a billion iterations, producing a number with a mind-boggling 413,930,770 digits. Still no palindrome.
These are seriously big numbers. I mean no disrespect, but you’re not going to chug through a billion iterations. Because even if you can manage one every second, 24 hours a day, year in year out, the effort will keep you till about December 2046. So I can safely guarantee you won’t find a palindrome when you start with 196.
But in passing, note something about Dalbeau’s and Walker’s efforts. If you divide the number of iterations by the number of digits in the resultant non-palindrome, in both cases you get about 2.416. In fact, that ratio holds all the way through, regardless of how many iterations you have done. Another puzzle.
Incidentally, these numbers that don’t produce a palindrome are known as Lychrel numbers.
But oops! Above, I said that there are “not too many" of them. In one sense, that’s true: below 700, for example, there are just the ones I listed, a total of seven. Seven isn’t too many, compared to 700, yes. But in another sense, we can actually claim there are as many Lychrels as non-Lychrels. Why? Because the first Lychrel is 196; the second, 295; the third, 394; the fourth, 493; and we can go on and on like this. There is a 234th Lychrel, a 19,781st Lychrel, a 589,444,067th Lychrel. Pick any number, no matter how large—and as far as we now know, there’s a Lychrel corresponding to that number.
I’ll leave you to savour what’s really a small flavour of infinity. But what I really should have said was not that there are “not too many" Lychrels, but that they are rarer than non-Lychrels. There’s a difference.
We owe the name, incidentally, to a Wade VanLandingham, who became obsessed with these numbers and with 196 in particular. He mangled his girlfriend’s name, Cheryl, to come up with “Lychrel". She’s now his wife. As he writes on his website, p196.org, “she was so happy to have Lychrel Numbers named after her, she said yes".
I told you there’s magic in numbers.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences.
Comments are welcome at dilip@livemint.com. To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza
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