# Containing a one

Imagine all the positive integers in the range 1-1,000,000,000 (inclusive).

How many of these numbers contain the digit ‘1’ in some location?

For example 43,252,003 does not, but 980,107 and 98,111,787 do. Take a guess before thinking about it.

## Solution

To find out how many contain, at-least, one occurrence of ‘1’, we can work out the complement; how many contain

__no__occurrences and subtract this from the total. If we consider that every positive number less than 1,000,000,000 is a nine digit number (prepending zeros, as necessary, to pad it out), then each of these digits can be one of the nine possible digits in the set [0,2,3,4,5,6,7,8,9] so there are 9^{9}combinations. However, there is a special case when every single digit is zero, and this is not valid, so we need to subtract for this case, so the answer is 9^{9}-1 = 387,420,488This means that the number that

__do__contain the digit ‘1’ is 1,000,000,000-387,420,488=612,579,512 which as a percentage is ≈ 61.258%This is a higher percentage than my initial gut feel!

*The percentage of the first billion numbers that contain the digit ‘1’ ≈ 61.258%*

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## Bigger Numbers

What happens if we use a different number? 1,000,000,000 = 10

^{9}, let’s look at how things change for 10^{n}n | 9^{n} - 1 | 10^{n} | Percentage | |
---|---|---|---|---|

1 | 8 | 10 | 20.000% | |

2 | 80 | 100 | 20.000% | |

3 | 728 | 1,000 | 27.200% | |

4 | 6,560 | 10,000 | 34.400% | |

5 | 59,048 | 100,000 | 40.952% | |

6 | 531,440 | 1,000,000 | 46.856% | |

7 | 4,782,968 | 10,000,000 | 52.170% | |

8 | 43,046,720 | 100,000,000 | 56.953% | |

9 | 387,420,488 | 1,000,000,000 | 61.258% | |

10 | 3,486,784,400 | 10,000,000,000 | 65.132% | |

11 | 31,381,059,608 | 100,000,000,000 | 68.619% | |

12 | 282,429,536,480 | 1,000,000,000,000 | 71.757% | |

25 | 717,897,987,691,852,588,770,248 | 10,000,000,000,000,000,000,000,000 | 92.821% | |

50 | 9^{50}-1 | 1E+50 | 99.485% | |

100 | 9^{100}-1 | 1E+100 | 99.997% |

As n gets larger:

We can see that, as the size of the number gets bigger, a higher, and higher concentration of the numbers contain the digit ‘1’ (or any other digit).

At primary school, we used to play a game called “Buzz”, where we would go around the classroom incrementing numbers

You can see that if you played this game for a

*1, 2, 3 …*but there would be a magic number of the day (say “5”), and instead of saying a number that contained that digit, you would say “Buzz”.*1, 2, 3, 4, Buzz, 6, 7, 8, 9, 10, 11, 12, 13, 14, Buzz, 16, 17 …*You can see that if you played this game for a

*loooong*time, it would become very boring.