Wednesday, February 01, 2006

Mathematical Mind Reading

The old "Sum and Difference" game is a simple trick that even very young children can play to amaze and frustrate playmates or (better yet) adults (even teachers) whose algebra skills have atrophied.

It's a great way to introduce the idea of "unknown" numbers to beginning algebra students.



Pick any two numbers. Add them (first number plus second number), and report the total. Then subtract them IN THE SAME ORDER (i.e. first number minus second number), and report the total.

You can determine the two numbers using simple algebra:

1st number = x (perhaps: 7).

2nd number = y (perhaps: 12).

Let: a = x + y, and let: b = x – y. The player will report to you the values of "a" and "b," the sum and difference of the two numbers (in our example, these numbers are 19 and -5). Add these results in your head (in this case, the sum is 14), and what you have will always be twice one of these numbers [since a + b = (x + y) + (x – y) = 2x.] Divide by two (14 ÷ 2 = 7), and you now have one of the numbers. Since you already know the sum of the two numbers, subtract to find the second one (19 – 7 = 12).


An entertaining variation is the "Birth Date Trick, below.”



Add your birth date plus birth month (example: for June 17, 1990, the birth date would be 17, the birth month would be 6, and the total would be 23), and report this number. Then, subtract these same two numbers IN THE SAME ORDER (i.e. birth date minus birth month). Again, be careful – this second result could be negative (in our example, the difference would be 6 – 17 = -11). Since this is really just another “sum and difference” exercise like the one above, all you have to do is add these two results and divide by two to get the birth month, then subtract the birth month from the total to get the birth date!


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