Write down any two numbers a and b. Add b – a to the list as the third number, and continue this procedure: each number in the list is the previous number with the number before that subtracted from it. That is how we get b – a for the third number in the list: it's the second number minus the first number. Such a list might go on forever; however, how many different numbers will ever appear in it?
Solution
We try it in general. Here's the list:
a
b
b - a
-a (= b – a – b)
-b (= -a – (b – a) = -a – b + a)
a – b (= -b – (-a) )
a (= a – b – (-b) )
and now we're back at the beginning.
That makes six different numbers.