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Difference Dynamics

Age 14 to 18 Challenge Level:

 number cycle
Choose any three numbers. The differences between your numbers give you three new numbers. Repeat this operation to give a sequence.

In this example the sequence starts: $15, 39, 8 \to 24, 31, 7  \to  7, 24, 17  \to 17, 7, 10 ...$.

What happens to this sequence. Investigate for different starting points.

What do you notice? Can you explain what happens?
 
How about sequences starting with four numbers, or two numbers?
 
See the article Difference Dynamics Discussion.