Topic: Fibonacci mod 10
 Message: Posted by: Nir Dahan (Jan 24, 2006 01:33PM)
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.
 Message: Posted by: Jonathan Townsend (Jan 24, 2006 02:57PM)
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?
 Message: Posted by: stanalger (Jan 24, 2006 03:32PM)
Yes! Half of 5!
 Message: Posted by: TomasB (Jan 24, 2006 04:10PM)
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
 Message: Posted by: Nir Dahan (Jan 24, 2006 04:52PM)
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/0387982191/102-6628468-1751367?v=glance&n=283155

enjoy,

N.
 Message: Posted by: TomasB (Jan 25, 2006 12:02AM)
I sense that Stan solved it first. ;)

/Tomas
 Message: Posted by: stanalger (Jan 25, 2006 09:17AM)
[quote]
On 2006-01-24 16:32, stanalger wrote:
Yes! Half of 5!
[/quote]

Translation:

Yes, the pattern repeats.
It has period 60. (60 is half of 5!=120.)
 Message: Posted by: Nir Dahan (Jan 25, 2006 04:29PM)
Stan,

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

nir
 Message: Posted by: Lawrence O (Dec 20, 2010 01:50PM)
http://www.maa.org/columns/colm/cardcolm200706.html