palindromes
when i was playing with my calculator, i discovered an interesting fact. i have explained it below.
first study the table below:
integer remarks
405 not palindrome
3223 palindrome
8120 not palindrome
1823281 palindrome
3245 not palindrome
2700 not palindrome
explanation
- first we take an integer ‘a’, and invert it to make ‘a*’.
- then we add together ‘a’ and ‘a*’ to obtain a new integer b which could be either a palindrome or not.
- again reversion of ‘b’ is done to obtain ‘b*’. then ‘b’ and ‘b*’ are added together to obtain another new integer ‘c’. again, ‘c’ could be anything
- the process is continued in the same fashion infinitely as shown below:
a+a*=b
b+b*=c
c+c*=d
d+d*=e…………….
if we keep on adding sum and its reverse, i found out that we can obtain many, most probably infinite symmetrical numbers frequently in the process. these numbers are same even if the digits are written backwards. my calculator was able of computing only 10 digit integers and therefore my findings are confined to integers of few digits. but i hope that we can receive interesting results from further research. i attached a small “fortran 77″ program that i made to simulate my theory. the program is also limited to 10 digit integers and most probably won’t run on vista (in which case download the coding here), but i hope it will clarify my theory.
magic no. 9
pick any whole number. call it ‘a’. rearrange all the digits in the number ‘a’ to form another number ‘a*’. now take the absolute value of the difference between ‘a’ and ‘a*’. call this new number ‘b’, which is our magic number. this number is always exactly divisible by 9. see for yourself. below is a example:
│98765-75896│= 22869; 22869/9= 2541
│524484-444258│= 80226; 80226/9= 8914
│4721-7241│= 2520; 2520/9= 280
