So what is a Palindromic Numbers ? What is so fascinating about them ?
Palindromic numbers are numbers that read the same from left to right and from right to left. In other words, they are "symmetrical" numbers.
Here are some examples:
The number of palindromic numbers with two digits is 9:
11, 22, 33, 44, 55, 66, 77, 88, 99.
There are 90 palindromic numbers with three digits
101, 111, 121, 131, 141, 151, 161, 171, 181, 191, …, 909, 919, 929, 939, 949, 959, 969, 979, 989, 999
There are also 90 palindromic numbers with four digits
1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, …, 9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999
There are many fascinating palindromes involving consecutive digit sequences.
For example in base 10 we can construct palindromes with consecutive digits starting
with 1.
For example 121 is such a palindrome. It is divisible by 11 with the result = 11 which of course is also a palindrome.
(Note that in this example the highest digit 2 is even.)
With the requirement that the highest digit is even the next such palindrome is 1234321. Again it is divisible by 11 with the result = 112211.
Continuing on in this fashion the next such palindrome is 12345654321. Again it is divisible by 11 with the result = 1122332211.
This pattern will repeat itself with the final single even digit in base 10 as highest digit.
Thus 123456787654321 divided by 11 = 11223344332211.
Here are some interesting facts on Palindromic numbers:
- The year 2002 is a palindromic year! So was the year 1991. Did you realize the previous occurrence of two palindromic years in one person’s lifetime was the years 999 and 1001! The next such pair will be 2992 and 3003. So this happens about once every 1000 years. The next normal palindromic year will be 2112.
- Palprimes - these are prime numbers that are also palindromic.
- Universal Day Of Symmetry: For example, at 10.01am on 10 January 1001, the time can be written as 100110011001. Another one would be at 20.02pm on 20 February 2002, which can be written as 200220022002. Both of these are palindromic numbers, and these are the only 2 cases, since it is impossible for us to have 300330033003 based on the 24 hour clock system.