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Magical 45

The problem as originally stated in "Uncle Art's Funland," a regular feature many Sunday newspapers:

Tell a friend that you can subtract 9 numbers that total 45 from 9 numbers that also total 45 and still have a remainder totalling 45.

 

Source: "Uncle Arts Funland," Bellingham Herald, August 28, 2005.


Hint: Clarity is not a feature in the presentation of this problem.

The "9 numbers" in both instances imply that you are to find two nine-digit numbers, each made up from all of the nine digits 1, 2,...8,9 which total (i.e. add-up to) 45.

 


Solution Commentary: The "amazing" solution given by "Uncle Art" is:

987654321-123456789=864197532

Now, why even give students such a problem? Because of the new questions it raises! Some examples:

  • One solution is given...is it the only solution? How do you prove/disprove the existence of other solutions?
  • Does anything interesting happen with other sequences of n-digits: abcde...rst - tsr...edcba =?, where n = 1, 2,3, 4, etc. For example, I found n=2 to be interesting in that multiple soutions can be found, all exhibiting an interesting "side" pattern.
  • Are there solutions for the original problem using other interpretations of its wording? For example, suppose the original set of 9 numbers was {10,11,14,5,1,1,1,1,1}add to 45. Can you find another set of 9 numbers {a,b,c,...,i} that add to 45 and such that (10+11+14+...+1+1) - (a+b+c+...+h+i) is a number whose digits total 45? Note: This question actually provides a hint to understanding of the wording of the original problem and its suggested solution.
  • What does this problem have to do with the process of casting out 9's?
  • Or....