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Age 11 to 14

Challenge Level

- Problem
- 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 Ãƒ? 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.

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How many such students are there?

How many ways can you write the word EUROMATHS by starting at the top left hand corner and taking the next letter by stepping one step down or one step to the right in a 5x5 array?

A political commentator summed up an election result. Given that there were just four candidates and that the figures quoted were exact find the number of votes polled for each candidate.