I lifted a box that was too heavy this week and I screwed up my back so bad that every time I put weight on my right foot, searing pain shoots up my entire right side. Getting older is awesome!

The prize this week for the first correct response: FREE Beta Access to my book. A \$5 value! Read up on the deets, if this is the first you’re hearing of it.

How many positive integers less than 1,000 contain exactly one odd digit?

Put your answers in the comments, and I’ll post the solution and contact the winner (if there is one) on Monday. Good luck!

UPDATE: Congratulations to Chong Lee, who nailed it first. Welcome to the Beta, Chong Lee. I hope you enjoy the book.

Solution below the cut.

The best way to solve a question like this is to count possibilities for each digit. There are three places your single odd digit could appear: the hundreds digit, the tens digit, or the units digit. For each of those possibilities, you have to count all the possible positive outcomes.

If your odd digit is the hundreds digit, for example, then you count like this:

##### Hundreds digit is odd
• There are 5 odd digits (1, 3, 5, 7, 9) that could occur as the hundreds digit.
• There are 5 even digits (0, 2, 4, 6, 8) that could occur as the tens digit.
• There are 5 even digits that could occur as the units digit.

5 × 5 × 5 = 125 possible outcomes in which the single odd digit is in the hundreds place.

What adds an extra degree of difficulty to this question is that the number of possible odd digits and the number of possible even digits is always the same. Sorry about that. That’s why it’s a challenge question.

Moving on, we still have to account for the odd digit coming in the tens or ones places.

##### Tens digit is odd
• There are 5 even digits that could occur as the hundreds digit. (Remember, zero still counts. 010, or just plain old 10, has one odd digit.)
• There are 5 odd digits that could occur as the tens digit.
• There are 5 even digits that could occur as the units digit.

5 × 5 × 5 = 125 possible outcomes in which the single odd digit is in the tens place.

##### Ones digit is odd
• There are 5 even digits that could occur as the hundreds digit.
• There are 5 even digits that could occur as the tens digit.
• There are 5 odd digits that could occur as the units digit.

5 × 5 × 5 = 125 possible outcomes in which the single odd digit is in the ones place.

##### How many positive integers less than 1000 contain exactly one odd digit?

125 + 125 + 125 = 375.

Magic, right? I know. If you want to see it done in a way that it could never be done in a test environment, I’ve confirmed the results in a spreadsheet for all the doubters. Click here.