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Mathematical Magic That Usually Works

Mathematical magic tricks are fun, but often they can be done without knowing why they work. That is, they work without the user knowing or understanding the underlying magical mathematics.

For example, consider this trick...Can you "prove" why it works?

With your back turned, ask someone to write down a n-digit number, preferably where n>5 to have greater magical effect.

Suppose the person wrote 89426354
Now, ask the person to add the digits...without telling you the sum...
8+9+4+2+6+3+5+4 = 41
Now, ask the person to subtract this total from the original number...
89426354 - 41 = 89426313
Now, ask the person to cross out one digit in this difference...
Suppose the 6 is crossed out to leave 8942 313
Now, ask the person to add up the remaining digits and tell you the total...
8+9+4+2+3+1+3 = 30
From this total, with your back turned, you can magically tell the person what digit they crossed out...the 6!

Question 1: How is the trick done...i.e. what is the mathematical magic? (see the hint if you give up)

Question 2: Will the mathematical magic always work for any n-digit number?

Question 3: Can you prove why this mathematical magic works?


Hint: The Mathematical magic secret: Take the final sum, add its digits (and re-add) until a single digit is produced. Subtract this remaining digit from 9, and PRESTO, that is the crossed out digit....usually.

In the example, 3+0 = 3 and 9-3 = 6.


Solution Commentary: First,trouble occurs when the announced final sum produces a digit-sum of 9 (or 0 if the prson's original number was a string of zeros). This means that the crossed out digit was either 0 or 9, so you have a 50-50 chance of being correct.

As to the proof, I will leave that for you to ponder...It involves some knowledge of number theory.