

Nir Dahan Inner circle Munich, Germany 1390 Posts 
Let's look at the fibonacci series mod 10, here are the first few elements:
1,1,2,3,5,8,3,1,4,5,9 .... the question is whether the pattern ever repeats itself, if not prove that it can't, if yes what is the pattern's length? N. 
Jonathan Townsend Eternal Order Ossining, NY 27087 Posts 
Are we looking to see if a digit string, minus the commmas and any belonging to an number in the sequence, can repeat? ie, somewere down the road we might get 314 in the sequence?
...to all the coins I've dropped here

stanalger Special user St. Louis, MO 996 Posts 
Yes! Half of 5!

TomasB Inner circle Sweden 1143 Posts 
We know that it must repeat itself due to the pigeon hole principle. The series is infinite but there are only 100 possible different pairs of adjecent digits, so as soon as two identical pairs has appeared we are back on track and the series will repeat itself. So the maximum possible period length is 100. So I just wrote the sequence down until I got a pair repeated which threw the sequence into its black hole.
/Tomas 
Nir Dahan Inner circle Munich, Germany 1390 Posts 
Very nice tomas, that was quite fast.
I took this problem from "problem solving strategies" by Arthur Engel I strongly recommend it for HEAVILY math oriented problems. it is basically a collection of principles with tons of solved examples to prepare you for any mathematical olympiad... http://www.amazon.com/gp/product/0387982......n=283155 enjoy, N. 
TomasB Inner circle Sweden 1143 Posts 
I sense that Stan solved it first.
/Tomas 
stanalger Special user St. Louis, MO 996 Posts 
Quote:
On 20060124 16:32, stanalger wrote: Translation: Yes, the pattern repeats. It has period 60. (60 is half of 5!=120.) 
Nir Dahan Inner circle Munich, Germany 1390 Posts 
Stan,
do you have a proof or was it by experimenting. cause I was thinking on the pigeon hole principle mentioned by tomas... nir 
Lawrence O Inner circle Greenwich (CT) 6799 Posts 
Magic is the art of proving impossible things in parallel dimensions that can't be reached

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