Let the two-digit numbers be $xy \Rightarrow 10x+y.$
When the digits are reversed $yx$ the number increased by $18.$
$10y+x = 10x+y+18$
$\Rightarrow 10y-y+x-10x=18$
$\Rightarrow 9y-9x=18$
$\Rightarrow y-x=2$
$\Rightarrow \boxed{y=x+2}$
All the positive two-digit numbers possible $= 10x+y = 10x+x+2 = 11x+2$
Now, we get all such numbers.
- $x=1 \Rightarrow 13 \longrightarrow 31$
- $x=2 \Rightarrow 24 \longrightarrow 42$
- $x=3 \Rightarrow 35 \longrightarrow 53$
- $x=4 \Rightarrow 46 \longrightarrow 64$
- $x=5 \Rightarrow 57 \longrightarrow 75$
- $x=6 \Rightarrow 68 \longrightarrow 86$
- $x=7 \Rightarrow 79 \longrightarrow 97$
- $x=8 \Rightarrow 90 \longrightarrow 09 (90+18=108)$ (Not possible)
$\therefore$ The number of other two-digit numbers is $6.$