[ *July 14, 2002* ]

A very unique palindrome 549945

549945 is the only fixed-point palindrome

for the Kaprekar process for base-10 numbers

containing fewer than 16 digits.

A neat palindromic fact spotted by Terry Trotter

when he visited Walter Schneider's webpage at

MATHEWS : Kaprekar Process

( now alas a broken link! )

This Kaprekar process is illustrated as follows :

549945 995544 – 445599 549945

[ *July 19, 2002* ]

No person is better placed to explain the topic

than Terry Trotter (email) himself.

A Palindrome à la Kaprekar

Here is an interesting palindrome: 995544445599.

To see why, we must first introduce a popular idea that has

fascinated mathematicians and school students alike for sometime.

It's called Kaprekar's Ordered Subtraction Operation (OSO).

The OSO may be described thus:

Start with a four-digit number whose digits are not all equal,

arrange the digits in descending and ascending order, subtract

and repeat the process. Then the process __terminates__ on the

number 6174 after seven or fewer steps.

The amazing thing here is that 6174 always results, no matter

what the initial 4-digit number might be (excepting those consisting

of only one digit, like 3333, of course).

As one does the OSO process with numbers of more or fewer digits,

different outcomes can result. With 3-digit numbers, the constant

final outcome is always 495. I call that a __terminator__. With 2-digit

numbers, the following five-term __cycle__ is the only outcome

09 81 63 27 45 09

But things really become interesting when using numbers of 5 or

more digits. Here __cycles__ and __terminators__ can occur for a given case.

(See Walter Schneider's website for all the outcomes

up to 15-digit numbers in base 10.)

A quick examination of that table reveals the fact that there

is only one palindrome. It occurs for 6-digit numbers. 549945 is one

of two __terminators__ possible, accompanied by one __cycle__ of seven terms.

By now it should be obvious how the 12-digit palindrome above

was created. It is a number with the interesting property

that can be stated as:

A palindrome with an even number of digits such that if it is

separated into 'halves', the positive difference of the two parts

is itself a palindrome.

995544445599 995544 – 445599 549945.

Now, I can only but wonder if there are other palindromes

that share this special property.

ps1. regarding your doubt about fixed points...

I just call such things terminators or self-producers.

6174 is often called Kaprekar's Constant.

ps2. For every Kaprekar Palindrome, as we might call

those numbers with the split-half property, there is always

a companion. Just reverse the 'halves', like so:

445599995544

trivial, perhaps, but necessary for completeness.