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De-Trix: Magical Matrix


This trick was performed on television by the famous magician Blackstone. But, it was invented by Mel Stover, who started doing magic tricks at the age of 10 but later worked for 49 years as a technician for the Faculty of Medicine at the University of Manitoba.

You are the Mathemagician performing before an audience. With no one able to see what you are doing, write the number "34" on a piece of paper. Fold the paper up and give it to someone to hold (without opening it up).

Ask for a volunteer. Show them this matrix of numbers on a large card (that audience can see as well):

 1	 2	 3	 4

 5	 6	 7	 8

 9	10	11	12

13	14	15	16
Give the volunteer a marking pen and then these instructions, one by one:
  • Circle any number in Row 1, then cross out all remaining numbers in its column/row.
  • Circle any "uncrossed-out" number in Row 2, then cross out all remaining numbers in its column/row.
  • Circle any "uncrossed-out" number in Row 3, then cross out all remaining numbers in its column/row.
  • Circle any "uncrossed-out" number in Row 4...there should be only one choice!
  • Add the four circled numbers.
Now, with great presence and magical gestures, ask the person holding the paper to unfold it and reveal your written number. The audience will be amazed to see that with your great mathemagic skills, you have successfully predicted the volunteer's sum!

Your Task 1: Determine the mathematics underlying this magical trick to reveal why it works.

Your Task 2: Generalize the mathemagic trick to a n x n matrix involving the numbers 1 - n2. What will be the magic number sum for this generalized case?

 


Hint: Peform the trick multiple times. You should always get the magical answer of 34 everytime. Why? Look for patterns "hidden" in how the matrix was constructed.

Also, this hint suggests why you should not perform this trick twice before the same audience, as it destroys any sense of magic on your part.

 


Solution Commentary: Amongst magicians, this matrix is known as being a "force matrix" in that it forces a resultant sum number of 34 everytime.

Rename each of the numbers in the matrix as a multiple of 4 less some value. For example, the first row would be: 1=(4x1)-3; 2=(4x1)-2; 3=(4x1)-1; 4=(4x1)-0...and the second row would begin: 5=(4x2)-3; 6=(4x2)-2... etc.

Now, perform the "trick" yourself using these new cell numbers. For example, suppose you originally had picked 3, 6, 9, and 16. Now, as a sum, you would write this as: [(4x1)-1] + [(4x2)-2] + [(4x3)-3] + [(4x4)-0].

But, this becomes: [4(12+3+4)-(1+2+3)] = 34.

For the generalized case of a n x n matrix, did you get a magic sum of [n(1+2+...+n)-(1+2+...+(n-1)]?

Can you generalize the trick even further to where the n x n matrix starts at a random whole number m?