Writ large
Problem
Suppose you had to begin the never ending task of writing out the natural numbers: 1, 2, 3, 4, 5.... and so on.
What would be the 1000th digit you would write down. In what number would it occur?
What number would contain the millionth digit?
When would you have written six for the 6000th time?
Student Solutions
Suzanne and Nisha, The Mount School Yorkfound that the 1000 th digit is the 3 of 370 and their method for the 6000 th occurrence of the digit 6 was almost correct. Joeof Madras College, St Andrews found the millionth digit is the initial 1 in 185185. Well done.
Suzanne and Nisha's solution for the thousandth
digit:
argued as follows:
From 1 to 99 there are 189 digits.
From 100 to 200 there are 303 digits.
From 200 to 300 there are 300 digits.
From 1 to 300 there are 792 digits.
So the 1000 th digit lies somewhere within the numbers 300 and 400 and is the 208th digit counting from the 3 in 301.
208/3 = 69.333
So the 1000 th digit is the 3 in 370.
Joe's solution for the millionth
digit:
I worked out how many digits there are in each group of 1 digit, 2
digit, 3 digit, 4 digit and 5 digit numbers.
| 1 digit | 1 to 9 | 9 x 1 | 9 |
| 2 | 10 to 99 | 90 x 2 | 180 |
| 3 | 100 to 999 | 900 x 3 | 2700 |
| 4 | 1000 to 9999 | 9 000 x 4 | 36 000 |
| 5 | 10 000 to 99999 | 90 000 x 5 | 450 000 |
| total | 488 889 |
|---|
I then took 488 889 from 1 million to leave 511 111. Therefore I needed as many 6 digit numbers as would give a total of 511 111 digits.
511 111/6 = 85 185.1666
488 889 + 6(85 185) = 488 889 + 511 110 = 999 999
Starting from 100 000 the 85185 th number is 185184 so the millionth digit is the first digit of 185185.
The six thousandth six
| From 1 to 100 you write '6': | 10 times as the units digit 10 times at the tens digit |
20 times |
| From 1 to 1000 you write '6': | 100 times as the units digit 100 times as the tens digit 100 times as the hundreds digit |
300 times |
| From 1 to 10000 you write '6' | 1000 times as the units digit 1000 times as the tens digit 1000 times as the hundreds digit 1000 times as the thousands digit |
4000 times |
| From 10000 to 15999 you write '6': | 6 Ã? 300 = 1800 times | |
| From 16000 to 16099 you write '6': | 120 times | |
| From 16100 to 16159 you write '6': | 66 times | |
| From 16160 to 16165 you write '6': | 12 times | |
From 1 to 16165 you write the digit '6' altogether 5998 times.
So you write '6' for the 6000th time as the second '6' in the number 16166.