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Number Chain

Problem

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

Divide by 5 if the number is divisible by 5 otherwise add 4.

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 maps 5 maps 1
2 maps 6 maps 10 maps 2
3 maps 7 maps 11 maps 15 maps 3
4 maps 8 maps 12 maps 16 maps 20 maps 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 maps 13 maps 17 maps 21 maps 25 maps 5 maps 1 maps 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?

Hint: Try +3/÷4 and +2/÷3 to start with.

Problem ID: 80 (Oct 2002)     Difficulty: 1 Star

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