mathschallenge.net

Problem

A chain of numbers is made by using the following rule:

For example, starting with the number 7,

It can be seen that the chain has returned to 7.

Which positive integer below 10 will never come back to itself?

Solution

Listing the chain for each starting number:

1 5 1
2 6 10 2
3 7 11 15 3
4 8 12 16 20 4

Having worked through the starting numbers 1-4, we can see what happens to 5-8. The only number that has not appeared is 9:

9 13 17 21 25 5 1 5

9 is unusual as it works its way into the 1,5 chain, never returning to itself.

Which numbers do not come back to themselves in general?
What happens if you change the rule from +4/÷5?