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From A Golden March from the futility closet.

Circle from Fulity Closet - A Golden March

Draw a circle whose circumference is the golden mean. Choose a point and label it 1, then move clockwise around the circle in steps of arc length 1, labeling the points 2, 3, and so on. At each step, the difference between each pair of adjacent numbers on the circle is a Fibonacci number.

When would the first collision occur? I think this problem could be solved by some modular arithmetic; however, I only know how to use integers, not Fibonacci numbers when doing modular arithmetic.

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yiyi
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  • the question requires some clarification, to make sense. what is meant by "the distance is a Fibonacci number"? and what is meant by collision. as Ross has just pointed out if it means coincidence of points, this will never happen. – David Holden Dec 09 '13 at 14:14
  • @DavidHolden: the difference between the numbers of neighboring points is a Fibonacci number. When the first $10$ points are placed, as in the diagram, the differences are $3,5,8$. I have not proven the claim that they are always Fibonacci numbers. – Ross Millikan Dec 09 '13 at 14:27
  • @DavidHolden I was thinking as Ross Millikan answer. Didn't realize that it could be thought of in different methods. – yiyi Dec 09 '13 at 14:32

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There will never be a collision because the golden ratio is irrational. If there were a collision, the first would be some point $m$ landing on $1$. The distance traveled would be $m-1$. That would have to be a multiple of $\phi$, which is impossible.

Ross Millikan
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