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.

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?

Yes! Half of 5!

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

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.

I sense that Stan solved it first.

/Tomas

Translation:

Yes, the pattern repeats.
It has period 60. (60 is half of 5!=120.)

Stan,

do you have a proof or was it by experimenting.
cause I was thinking on the pigeon hole principle mentioned by tomas...

nir

http://www.maa.org/columns/colm/cardcolm200706.html