Inspired by Project Euler

• For $n=2k$ , there are $20\times 30^{k-1}$ n-digit Reversible numbers

• For $n=4k+1$ , there are $\text{no}$ n-digit Reversible numbers

• For $n=4k+3$ , there are $100\times 500^{k}$ Reversible numbers

Analysis

• There are no 1-digit reversible numbers

• In case of a two digit number $\overline{\text{ab}}$ , $a+b$ must be odd and less than 10. There are $20$ such numbers.

• For $\overline{\text{abc}}$ to be a three-digit solution, $a+c$ must be odd and greater than 10, $2b$ must be less than 10. Five choices for b, 20 choices for ac, 100 solutions.

• For $\overline{\text{abcd}}$ to be a four-digit solution, $a+d$ must be odd and less than 10, neither $a$ nor $d$ can be zero, $b+c$ must be odd and less than 10. Twenty choices for $ad$ , thirty for $bc$ , 600 solutions.

• There are no five-digit solutions.

• For $\overline{\text{abcdef}}$ to be a six-digit solution, $a+f$ , $b+e$ , $c+d$ must all be odd and less than 10. $a$ and $f$ cannot be zero. 20 choices for $af$ , 30 for $be$ , 30 for $cd$ , 18000 solutions.

• For $\overline{\text{abcdefg}}$ to be a seven digit solution, $a+g$ odd and greater than 10, $b+f$ even and greater than 10, $c+e$ odd and greater than 10, $2d$ less than 10. 5 choices for $d$ , 20 for $ce$ , 25 for $bf$ , 20 for $ag$ , 50000 solutions.

• Eight digit numbers will be same as 2 digit, 4 digit and 6 digit.

In general, the result given above, holds good.