UNO High School Problem of the Week Competition
Problem 1
The natural numbers are printed one immediately following the other: 12345678910111213...Note that the numbers greater than 9 take up two digits (spaces) to print and the numbers greater than 99 take up three or more digits to print. What is the 1000th digit printed?
Solution
There are a couple of ways to get the solution for problem 1.
Method One:
Write out all the numbers and count! However, this method is considered time-consuming. If you have a couple of hours to waste, then you are welcome to use this method. :-)
Method Two:
There are 9 digits between 1 and 9.
There are 90 two-digit numbers from 10 to 99. Thus, 90x2 is 180 digits.
There are 900 three-digit numbers from 100 to 999. So, 900x3 is 2700 digits. Since 2700 > 1000, we know that the 1000th digit is a part of a three-digit number.
So, we have 1000 (9 + 180) = 811 digits. Knowing that a three-digit number contains the 1000th digit, we take 811 / 3 = 270.33333. Next, we have 270x3 = 810. This indicates that the 1000th digit is the first digit of the 271st three-digit number. The 271st number is 371. So, the 1000th digit is 3.
From the given information the ratio of the distances covered by First, Second, and Third, in that order, is 10 : 8 : 6. Since the speeds remain constant the ratio of the distance remains constant, so that, when Second completes an additional 2 miles, Third completes an additional 1 1/2 miles since 2/x = 8/6 where x is the additional distance completed by Third. Therefore, when Second completes 10 miles, Third completes 7 1/2, that is, Second beats Third by 2 1/2 miles.