## Not Much of a Puzzle

The otherwise excellent puzzle book Super Puzzles, by Jean-Claude Baillif, contains this strange puzzle:

Can you find a number of ten digits, each of them different, that can be multiplied by two to produce another number of ten different digits?

You might want to try it. Monsieur Baillif gives his solution as follows:

2 x 4,938,271,605=9,876,543,210.
In the first number, the digits 4, 3, 2, 1, and 0, in that order, alternate with the digits 9, 8, 7, 6, and 5.

The above implies that this might be the only solution. And his explanation sounds as if he is giving us a clue to his strategy for solving the puzzle. Look at the final number, and you will notice his real strategy. This is the largest ten-digit number with all of the digits different. Divide by two and you get his other number, which luckily turns out to be composed of ten different digits. So, it would seem that he stumbled upon a solution with his first try.

Let's start at the other end, with the smallest possible ten-digit number with all of the digits different (1,023,456,789) and multiply by two (2,046,913,578). We too have stumbled upon a solution with our first try. Just how many solutions are there? A little computer work shows that there are 184,320 solutions (out of 1,451,520 starting numbers in the right range). I actually printed them all out, which was quite a waste of paper. This is not much of a puzzle.