Think of a number

For two years you have used algebra to describe situations and to find unknown letter-numbers. This chapter uses it in a new way: to explain why tricks and puzzles work, and to invent new ones.

Here is a trick from Grade 7:

• Think of a number.
• Double it.
• Add four.
• Divide by two.
• Subtract the number you first thought of.

I predict the answer is 2.

Try it.

Always 2.

But testing more numbers can never prove it, because there are endless numbers to try. Algebra can, because a letter-number stands for all numbers at once.

Think of a number: x. Double it: 2x. Add four: 2x + 4. Divide by 2: x + 2. Subtract the starting number: x + 2 − x = 2.

The x cancels at the last step. What is left is the 2 that came from 'add four, then halve'. That is the whole secret.

So you can choose the answer you want. Add six instead of four and the answer is always 3. Add ten and it is always 5. The final answer is always half of whatever you added.

You can also build longer tricks. Any chain of steps that gets back to x and leaves a fixed number behind will work. More steps make it look more magical, but the algebra stays just as short.