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Puzzle 1: How Many Palindromes? | by “Caveman Chuck” Coker
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Puzzle 1: How Many Palindromes?

How many palindromes can you find in this photo?

 

This photo was taken on the shoulder of the southbound Highway 30 in Highland or Redlands, California. (I don't know exactly where the city limits are.) There is another palindromic odometer here. I think this photo came out better than the first one I tried.

 

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Update: This image was used to accompany the article, Would a Mileage Tax Punish Green Drivers?, by Josh Loposer, on February 20, 2009.

 

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There is an article in the journal Nature about multiple palindromes: Abundant gene conversion between arms of palindromes in human and ape Y chromosomes. I have absolutely no idea what the authors are talking about.

 

Here is something interesting you can do by squaring numbers consisting only of the digits "1" (the products are all palindromes):

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321

1111 x 1111 = 1234321

11111 x 11111 = 123454321

111111 x 111111 = 12345654321

1111111 x 1111111 = 1234567654321

11111111 x 11111111 = 123456787654321

111111111 x 111111111 = 12345678987654321

 

The only palindromic year in the 21st century was 2002. If you type 2002 into a calculator and turn it upside-down, it will still read 2002.

 

A palindromic prime is a number that is simultaneously palindromic and prime. The first 17 (a base-10 prime number) base-10 palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, and 787.

 

10^150006 + 7426247 x 10^75000 + 1 is a 150007-digit palindromic prime.

 

In the April 1984 Scientific American "Computer Recreations" column, an article appeared about mathematical patterns (F. Gruenberger, Computer Recreations, "How to Handle Numbers with Thousands of Digits, and Why One Might Want To.", Scientific American, 250 [No. 4, April, 1984], 19-26.). Here's the algorithm:

 

1. Pick a number.

2. Reverse its digits and add this value to the original number.

3. If this is not a palindrome, go back to step 2 and repeat.

 

Do all numbers eventually become palindromes by this process? It was suggested that this is the case.

 

Most numbers become palindromes fairly quickly, in only a couple of steps:

 

13

1. 13 + 31 = 44

 

64

1. 64 + 46 = 110

2. 110 + 011 = 121

 

87

1. 87 + 78 = 165

2. 165 + 561 = 726

3. 726 + 627 = 1353

4. 1353 + 3531 = 4884

 

In fact, about 80% of all numbers under 10,000 solve in 4 or less steps. About 90% solve in 7 steps or less. A rare case, number 89, takes 24 iterations to become a palindrome. It takes the most steps of any number under 10,000 that has been resolved into a palindrome.

 

Does every number eventually become a palindrome? Nobody knows for sure, since it has never been proven. There are some numbers that do not appear to ever form a palindrome. The first one is 196. Such numbers are called Lychrels. The search to resolve this number has been referred to as the 196 Algorithm or the 196 Problem, but normally called the 196 Palindrome Quest.

 

The 20-digit number 10,200,000,000,065,287,900 solves after 257 iterations.

 

(The information starting with the paragraph about the April 1984 Scientific American issue and continuing to here is from a fascinating web page about the 196 Palindrome Quest.)

 

Not that it has a relation to the 196 Palidrome Quest, the 196th numerical palindrome is 9779.

 

I found a web site devoted to the 196 Palindrome Quest: http://www.p196.org/.

 

My favorite alphabetic palindrome is the sentence, "Do geese see God?" Most sentences like this don't make much sense.

 

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Taken on September 19, 2008