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.