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By day I'm a propeller-head geek. I design software for electronic components for a major automotive supplier. When I'm not earning a paycheck, I enjoy playing music -- primarily jazz and classical but I dabble in other genres as well. I also compose, arrange, and play with electronic gadgets and toys. My other hobbies include photography, colored pencil drawing, genealogy, model railroading, and crosswords.

Tuesday, November 22, 2005

Palindromes

You're about to get a glimpse of the way the Mind of Kevin Bowman works. This is also an example of why people who have ADD tendencies should stay away from the internet.

For reasons I can't remember, I was talking with a work mate about strange audio processesing algorithms when I recalled seeing a farcical processing device on the internet that named all of the controls and such with palindromes. A quick search found the device in question to be the palindrometer (found here). Then I remembered hearing a satirical skit on NPR (my buddy Dave calls it National Proletariat Radio) that was an interview with a guy named Bob, who spoke entirely in palindromes. Of couse, this instigated a thourough search of the 'net for a transcript of that dialog. I did not find what I was looking for but did find several sites dedicated to palindromes and one site led me to another on which I discovered an intersting (to me) mathematical problem involving numerical palindromes (numbers like 12321).

So now I'm completely intrigued by this mathematical problem, which you can read about here. I started thinking about a related subject: what does it mean to "reverse a number"? And I discovered a couple of interesting things:
1) The absolute value of the difference of a number and it's reverse is a number evenly divisible by nine. This was really no surprise once I remembered the old accounting trick related to transposed digits (if the error is a multiple of 9, you've probably made a transposition error).
2) The result of dividing the difference mentioned above by 9 is often a palindrome! This was true for all of the numbers I first experimented with. My first experiments were with sequencial digits, like 123, 456, 987654, etc. I even tried 196 (196 - 691 / 9 = 55). However, I was able to find numbers that, when subtracted from thier reverse and divided by 9, did not result in a palindrome (72157 - 75127 / 9 = 330).
3) At least for some numbers, the process of "reverse and subtract" (and taking the absolute value) results in a palindrome. The number 196 "solves" in this case:
196 - 691 = 495
495 - 594 = 99

Subtraction is nothing more than adding with negative numbers. Perhaps this will open the door to generalizing the "reverse and add" process and help gain some understanding with the 196 problem. Or perhaps I've just found another way to waste a bunch of time.

3 comments:

Anonymous said...

As the author of the p196.org page linked above, I'm glad you stopped in and found something of interest. Be careful though...

For some people, thinking about this problem is like a virus there is no cure for! I've been sick with it for over 8 years!! :-)

Loved the link to the Palindrometer! I'd never seen that one in my wanderings. I'll have to add a link to it as well to my Blackboard.

ninth freak said...

boss, this exercise which involves reversing nos. and then taking their difference & further dividing by 9 will always,mind u always give u a palindrome irrespective of the no. of digits involved.

Anonymous said...

For any five digit number such as 96321 when you reverse it and subtract the two numbers and divide by 99 you get a palindrome. 96321-12369=83952

83952/99=848

What I find interesting is that when you look at the digits in the palindrome you find some interesting patterns. If you look at the original number 96321 you will notice that the palindromes digits folow that the outside digits are equal to the difference of the outside digits of the original number and the inside palindrome digit is equal to the difference of the second and fourth digit of the original number

96321

9-1=8
6-2=4

848

Therefore take any five digit number such as

85731

8-1=7
5-3=2

727*99=71973

85731-13758=71973