# December 2017 Brain Puzzler Solution

Q: For the positive integers up to and including 2017, how many of those integers have at least one zero?

A:  There are $2017$ positive integers in total to consider.
No one digit numbers have a 0 (1-9).
Below is the list of the qualifying numbers:

 There are 9 two digit numbers: 10,20, 30…90 9 There are the numbers 100 – 110 and 200-210 up to 900 – 910 99 Also 120 130 140…190 and then 220 230 etc 72 Then 1000-1099 100 Then same # of #s 1000 more than b 99 Then same # of #s 1000 more than c 72 2000-2017 18 Total 469

Another solution is to exclude the numbers that do not have a zero and subtract from 2017:
9 one digit numbers
81 two digit numbers (9 x 9 : 9 choices for each digit 1-9)
729 three digit numbers (9 x 9 x 9: 9 choices for each digit 1-9)
For the numbers between 1000- 1999, there are the same amount as for the three digit numbers as they are the same numbers with a “1” in the thousands place value.
9 + 81 + 729 + 729 = 1548
2017 – 1548 = 469