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© Copyright 1999, Jim Loy
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.
Addendum #1:
So, is there a better puzzle waiting to be discovered here. How about a ten-digit number (all of which are different), which when doubled produces a ten-digit number (all of which are different), and when tripled (or quadrupled) also produces a ten-digit number (all of which are different)? Unfortunately, there are hundreds of solutions under these criteria. How about doubled and tripled and quadrupled? There are more than 20 solutions for that. How about a further stipulation that multiplying by five (or six) also produces a ten-digit number (all of which are different)? There seem to be no solutions for that further stipulation. So far, I have found no decent puzzle here.
Addendum #2:
Fred. Schuh, in The Master Book of Mathematical Recreations, has this to say about such puzzles (with thousands of solutions):
The person who poses the puzzle knows that success is possible, but knows nothing of other solutions, when there may be many, perhaps millions. Such a thing is wrong. There is no proper idea behind the puzzle.
In the example that he gave (perhaps hypothetical), the composer began with a solution and "composed" a puzzle to fit it. And so, the composer knew that there was at least one solution.