In playing around with numbers today, I came up with a nice little
brain teaser. After alpha-testing it on a friend I
thought I'd post it here for the world's perusal.

The number 71352 is a multiple of 3. Without the use of a pencil, paper,
calculator, etc. take that same set of digits, and rearrange them in such
a way that they form another multiple of 3.

The trick with this puzzle is that it's based on a simple rule for
determining whether or not a number is divisible by 3. If a number is
divisible by 3, then the sum of its digits is divisible by three as well.
As in this example, 7 + 1 + 3 + 5 + 2 = 18, which is indeed divisible
by three.

But of course, the order they're in doesn't matter. There is no way to
change the sum of the numbers by rearranging them, due to the
associative law.
This means that every possible combination you can make with those digits
is a multiple of three.

This puzzle could have another layer added to it by asking *how many*
ways the number can be rearranged to form a multiple of three. The answer
to that is the factorial of the number of digits in use, that being
how many possible combinations there are. In this case, there are 5 digits,
and 5! = 5 × 4 × 3 × 2 × 1 = 120, so there are 120
multiples of three expressed with that exact set of digits.

Click here to see the answer.