I found out of it, took me a while.
2^2010 is a 606 digit number beginning with 1, which means 2^2009 is a 605 digit number.
this means there are 605 numbers beginning with 1
each time you multiple a number starting with 1, it will become a number starting with 2 or 3
which means there are 605 numbers beginning with 2 or 3.
each time you multiply a number starting with 2 ot 3, it will become a number starting with 4,5,6, or 7
which mean there are 605 numbers beginning with 4,5,6, or 7
The number 8 or 9 can only origin from 4
we can conclude that 2010-(3*605) = 195
which mean 195 numbers begin with 4
Quote (smashT @ 25 Nov 2013 09:14)
I was going to say... probably close to a third of 605.
Since the 606 digit number (largest one) starts with a 1, there are 605 possibilities for numbers starting with a 4. 4x is missed, as is 4xx...
Tough one.
and yea, that made me think
Quote (mungoflago @ 25 Nov 2013 22:12)
the answer is approximately 2009/10
the beauty of power of 2's is that is virtually repeats...therefore you can conclude:
2^1=2
2^2 = 4
2^3 =8
2^4 = 16
2^5 = 32
2^6 = 64
2^7= 128
2^8=256
2^9=512
2^10=1024
2^11=2048
2^12=4096
2^13=8192
this will forever repeat....so 2^2, 2^12. 2^22, 2^32, 2^42, 2^52, etc etc etc -- eventually it will turn to something like 2^155, but it will only happen once ever 10 digit increases...this is a quick and dirty way of estimating that approximately 200 numbers will have a 4-handle.
Yeah, I thought about that.