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Fido the Magician

This week's resource is narrow-in-focus but is offered just for fun...and surprise...and it can lead into a solid mathematics exploration.

Without a further intro, go to the Fido Puzzle. Do as it says, clicking on the lower-right-hand charater to move from frame-to-frame.

Be sure to test the trick using the following:

• 2-, 3-, 4-, 5-, 6-, ..., n-digit numbers. Does the trick always work?
• Try a case where you circle a zero obtained in the difference...what happens then? That is, does the person predict the right digit? And if not, how is the predicted "wrong digit" a big clue...
• What happens if your initial number was 2222?
Now, try to determine how the trick is done by the magician, but not necessarily the underlying mathematics. My suggestion is to record your initial set of digits, the obtained difference, and the circled digit that was correctly predicted. A pattern is lurking here...!

Finally, why does this work mathematically? Note: This question is not trivial and will probably require some background in number theory.