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See comments above for many different solutions.
Here are some solutions to a similar puzzle from a book published in 1908:
Teacher, do your students have
access to computers?
Here a concise URL for a version of this page without the comments.
Here is the link which will take them to a student version of this activity with different levels for which they can earn Transum Trophies!
Use the digits zero to nine to make the first ten digits (one at a time!)
Here are the solutions to the extension challenge as published in 'A Scrap-Book of Elementary Mathematics' in 1908: