Saturday, June 29, 2013

The Magic of Algebra

Think of a number from one to ten.
Multiply that number by two.
Add thirteen.
Subtract seven.
Divide by two.
Subtract your original number.
You are thinking of the number three.

This is a classic mind reading trick that many people will learn to do when they are little. The premise is simple; the answer is always three.

Why does this work? As a young child, you probably think back to the arithmetic skills you learned in school and realized that you can't prove this trick because you don't know what number they'll choose.

This is a good time to note that when you learned this trick, the first thing you thought was "why?" Many people see proofs as a tedious process that could be avoided just by accepting things to be true. However, that why spark that went off when you learned the trick is where proofs can be applied. I use proofs so often in my posts primarily for that reason: to foster that why spark.

Anyways, this situation is where you need to define that unknown quantity of the number the spectator is thinking of. How do you define an unknown quantity? Well, this is when you pull out a little tool called algebra. You might not expect to apply algebra to a practical situation, but this is when you can. With algebra, we can stop saying "the number that they are thinking of" and start saying "n."


Now, the first step was to multiply the number by 2. n times 2 is pretty simple; it is 2n.

n • 2

Next, we had to add thirteen. That can't be simplified much.

2n + 13

Now, we subtract seven. 13 - 7 is 6, so we can simplify it to 2n + 6.

2n + 13 - 7
2n + 6

Now, we have to divide by two. 2 and 6 both have a factor of two in it, so we will not need any fractions.

(2n + 6)/2
2(n + 3)/2
n + 3

The last step was to subtract the original number. But, we already defined that number to be n, so we just subtract n.

n + 3 - n

As you see, the n's cancelled, and you ended up with three. Basically, you undid all of the previous operations to end up with one number.

I think that this is a perfect way to show young children magic, but also introduce them to concepts like algebra and proofs by lighting that spark in such a fun and interesting way.

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