Along a road, there exists marks for every passed kilometer after every kilometer, at the start where the mark shows 0. A hiker who starts walking at the roads start devote himself to count the numbers on the marks. How many kilometers has the hiker walked/passed at the mark showing the number 2013?
I have the solutions aswell but I don't understand the solution. You can see it below.

Solution: The single digit numbers add up to 10 numbers and corresponds to 9 passed kilometers. The two digit numbers together holds a total of 180 numbers(I don't get this part). When you pass the sign with the number 99, which is the biggest two digit number, you have walked a total of 9+90=99km. All the three digit numbers together contain 3 · 900 > 2013 numbers. So, when one have passed the sign with the 2013 number/digit one have walked atleast a three digit number of kilometers. We have the following: 2013 = 10 + 180 + 6 · (3 · 100) + 3 · 7 + 2; which means that at the sign with the 2013th digit, one passed 9 + 90 + 6 · 100 + 7 + 1 = 707 kilometers.

 

If somebody could explain this I would be greatful.

 

Regards!

Edited September 16, 201411 yr by dawoodr

Seems a very silly problem - I think you are counting the cumulative number of printed characters on the kilometre posts.

 

single digits 0-9 inclusive - 10 kilometre posts - 10 characters - 9 kilometres (0-9)

 

double digit 10-99 incl - 90 kilometre posts - 180 characters - 90 kilometres (9-99)

 

triple digits 100-999 - 900 km posts - 2700 characters - 900 km (99-999)

 

so as the solution says you will pass the 2013 character during the three digits km posts

 

I would do it this way:

 

total number of characters passed at 99 km is 190. 2013-190 = 1823. So you have 1823 characters in the three character range. 1823 / 3 = 607.66. So you have to pass the 608th km post

 

so

9 km for single character km posts

90 km for double character km posts

608 km for a portion of the triple character km posts

9+90+608=707km