Writ Large
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.
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