1. Start the search among 2-digit numbers. A dropdown number (DDN) with 2 digits is a number 
such that
or
but both of these solutions violate the requirement that the digits are distinct, so there are no 2-digit DDNs.
A 3-digit DDN 
is such that
or 
if the average is 
, or 
if the average is 
. The smallest possible value for 
is 1 since we require 3 digits. Then 
, and we can pick 
and 
to get the smallest DDN, 102.
2. In a 4-digit DDN 
, we have
or 
or 
or 
.
We're free to fix 
and to try to get the smallest DDN. This leaves us with 
or 
or 
or 
.
The first two cases are impossible - the only choices for 
such that are 1 and 2, and the sum of two positive integers must be positive. The smallest possible value of is 2; this leaves us with 
or 
, but the latter case is impossible because 3 does not divide 2. So 
, and the <em>smallest</em> 4-digit DDN is 1025.
To find the largest DDN, start with the largest possible values for and . Let 
and 
. Then 
or 
or 
or 
. At most, we can have 
with 7 and 6, so the first two cases are impossible. If we maximize 
, then either 
or 
(which we don't want). So the <em>largest</em> 3-digit DDN is 9874.
3. I don't have an analytical solution to this, but using brute force (program) the total count is 112.
4. It is possible; consider 1249 and 1250, with digital averages
which happens to be the smallest pair. (Also found with brute force.)
