how do you turn any four digit number to a single digit number using one unchanging rule?
for instance
1000 goes to 3
2000 goes to 3
9000 goes to 4
8000 goes to 5
1234 goes to 0
4321 goes to 0
You should be able to do it in your head
I can't see a direct pattern, but consider this, a number of string length >2 can be reduced to a single number using modular arithmetic.
In your case consider $\displaystyle a \equiv b \mod n$ when $\displaystyle n\in [2,9] \cap \mathbb{Z}$
For example $\displaystyle 9 \equiv 4 \mod 5$ as 4 is the remainder of 9 once divided by 5
For longer strings
$\displaystyle 39 \equiv 9 \mod 10$
$\displaystyle 217 \equiv 1 \mod 6$
$\displaystyle 1001200304 \equiv 2 \mod 9$
@Plato, the rule's actually pretty simple (it's not like, take the sum of the digits then multiply by 11 and divide by the product of the digits and take the floor value).
I know the solution as well so it's not that obscure. For example:
8244 --> 4
4567 --> 2
8189 --> 5 (Chook943 can verify these to be correct)