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Making a Magic Trick More Magical...O' Ye Beast of a Number

Try this trick...

  • Wrte down a 3-digit number (e.g. 276)
  • List all the 2-digit numbers that can be made from your 3-digit number (e.g. 27, 26, 72,...)
  • Sum all of these 2-digit numbers (e.g. 27+26+72+... =S)
  • Divide this sum S by the sum of the 3 digits in your original number [e.g. S/(2+7+6)]
  • With my great magical powers, I predict that your final quotient is 22!
Task 1: Prove that this always works...that regardless of the 3-digit number, the final quotient magically is always 22.

Task 2: Generalize the trick to using 4-digit numbers, but still using it to make 2-digit numbers. What is the magic quotient now?

Task 3: Generalize the trick to using 4-digit numbers, but now using it to make 3-digit numbers. What is the magic quotient now?

Task 4: Generalize the trick to using 4-digit numbers, but using it to make both 2-digit numbers and 3-digit numbers. What is the magic quotient now?

Task 5: Generalize the trick to using n-digit numbers, explore different options of just 2-digit numbers, just 3-digit numbers....just (n-1)-digit numbers (or even their combinations). Does the trick still work with a predictable magic quotient?

 

Source: Adapted from B. Sones & R. Sones' "Math Acrobatics," Bellingham Herald, 1/24/2013


Hint: Fisrt try the trick with several different 30-digit numbers.

Second, try the trick with the three digit number, which can be written like xyz, or more properly as 100x+10y+z.

With a little algebra, the "magic" underlying the predictable quotient of 22 should become clear...

 


Solution Commentary: Some interesting numbers you should discover along the way in generalizing this magical trick: 22, 33, 666, etc.