Question

All digits in a dropdown number are different, and one of its digits is the average of all its digits. It has at least two digits. For example, 5021 is a dropdown number, but neither 4389 nor 6033 is a dropdown number.
Questions:
1. Find the smallest dropdown number.
2. What are the smallest and largest 4-digit dropdown numbers?
3. How many 3-digit dropdown numbers are there?
4. Is it possible for a pair of consecutive numbers to both be dropdown numbers? If not, explain why not. If it is possible, find the smallest such pair.

Answer 1

It is 9817 is the answer hope this help c3822

Answer 2

1. Start the search among 2-digit numbers. A dropdown number (DDN) with 2 digits is a number $ab$ such that

$\dfrac{a+b}2 = a \implies a + b = 2a \implies b = a$

or

$\dfrac{a+b}2 = b \implies a+b = 2b \implies a=b$

but both of these solutions violate the requirement that the digits are distinct, so there are no 2-digit DDNs.

A 3-digit DDN $abc$ is such that

$\dfrac{a+b+c}3 = a \implies a+b+c = 3a \implies b+c = 2a$

or $a+c=2b$ if the average is $b$, or $a+b=2c$ if the average is $c$. The smallest possible value for $a$ is 1 since we require 3 digits. Then $b+c=2$, and we can pick $b=0$ and $c=2$ to get the smallest DDN, 102.

2. In a 4-digit DDN $abcd$, we have

$\dfrac{a+b+c+d}4 = a \implies a + b + c + d = 4a \implies b+c+d=3a$

or $a+c+d=3b$ or $a+b+d=3c$ or $a+b+c=3d$.

We're free to fix $a=1$ and $b=0$ to try to get the smallest DDN. This leaves us with $c+d=3$ or $c+d=-1$ or $1+d=3c$ or $c=3d$.

The first two cases are impossible - the only choices for $c,d$ such that $c+d=3$ are 1 and 2, and the sum of two positive integers must be positive. The smallest possible value of $c$ is 2; this leaves us with $1+d=6$ or $2=3d$, but the latter case is impossible because 3 does not divide 2. So $d=5$, and the smallest 4-digit DDN is 1025.

To find the largest DDN, start with the largest possible values for $a$ and $b$. Let $a=9$ and $b=8$. Then $c+d=19$ or $c+d=15$ or $17+d=3c$ or $17+c=3d$. At most, we can have $c+d=13$ with 7 and 6, so the first two cases are impossible. If we maximize $c=7$, then either $17+d=21\implies d=4$ or $24=3d\implies d=8$ (which we don't want). So the largest 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

$\dfrac{1+2+4+9}4=4 \text{ and } \dfrac{1+2+5+0}4=2$

which happens to be the smallest pair. (Also found with brute force.)