Consider an odd number <=999. Each of these gives a unique six-digit
Originally Posted by mathsmad
palindromic number as follows:
first add as many zeros to the front of the number as needed so that we
have exactly three digits. Reverse the digits and append reversed copy to
the front of the three digits.
It is self evident that any six-digit palindromic number can be generated in
Therefore there are exactly as many six-digit palindromic numbers as there
are odd numbers <=999. There are exactly 500 odd numbers <=999, so
there are exactly 500 six-digit palindromic numbers.
For seven-digit odd palindromic numbers the three most and the three least
significant digits are the digits of a six-digit palindromic number. Thus for
each possible middle digit there are 500 seven-digit palindromic numbers. The
possible middle digits are 0, 1, 2, .., 9. Hence there are 5000 seven-digit
odd palindromic numbers.