February 2012 Brain Teaser Solution

Q: Find three-digit numbers that the number itself, its double and its triple contain each digit from 1 to 9 exactly once. For example, 192 works because 192, 384, 576 contain 1 to 9 each once. 273 also works because 273, 546, 819 contain 1 to 9 each once.

What are the other two numbers that also have this property?

A: 219 and 327.
We need to keep the numbers low because if we are going to double and triple and maintain only 3 digits, then that means that the number must be less than 334 (334 tripled makes 1002 -- 4 digits).
We can rule out any numbers where the digits repeats as we have to use all 9 digits (1 through 9).
I used a guess and check to solve this problem.  I kept the first digit 1, 2 or 3 and kept out any repeated digits.  This led me to 219 and 327 but those are also the same digits as those in the given (192 and 273)!!! 


It was then that I realized that if you swap 19 and 2, and swap 27 and 3, you'll get them.