## Magic Problems

The basic "magic" problem is this. I am thinking of a number. When I multiply my number by 4, then add 7, the result is 31. What is my number?

### Guessers

The most primitive way to solve this problem is to guess at an answer, then check to see if it is right, then guess again if it is wrong.

This is not the best way to solve the problem, so eventually you will probably want to teach the guessers a better method. But first, let's appreciate guessing. You would like your students, no matter how they solve the problem, to check their answer by doing exactly what the guesser is going. Second, the guesser is going through the steps of the problem in a way to learn the structure of the problem. You would want everyone to understand that structure, even if they used a different method of first finding the answer.

You might also want to give the guesser problems that the nonguessers would not find interesting. Examples:

I am thinking of a number. When I add 173 to my number, then I subtract 173, the result is 295. What is my number?

You could also give a page of problems of the form:
I am thinking of a number. When I add 245 to my number, then subject 124, the result is 5. What is my number?
I am thinking of a number. When I add 245 to my number, then subject 124, the result is 27. What is my number?
I am thinking of a number. When I add 245 to my number, then subject 124, the result is 250. What is my number?
One could hope that the guesser would save time by eventually adding 121 rather than adding 245 and then subtracting 124.

Other problems involve adding zero and multiplying by 1, hopefully to show that these don't change the number. For example, I am thinking of a number. If you add 5 to my number, then multiply by 1, the answer is 9. What is my number?

### Letting Them Be the Magician

They can think of a number and give the problem to you. This is presumably excellent practice for understanding the problem. My magic trick: Someone thinks of a number, but does not tell me their number. They multiply this number by 4. Then they add 7. They tell me the result. Then I tell them their original number.

This of course is a simple algrebra problem. If there final result is 31, I am trying to solve the problem 4x + 7 = 31. (I vary what they multiply by and what they add.)

I am the one doing the algebra. In my experience, everyone wants to be the magician once they are comfortable with the problem. So my students will get their chance.

But to start, let's use the fact that they are doing the

But to start, they are on exactly the right side of the equation. They are taking a number, multiplying I gave my high school student the problem of solving 3x + 2 = 11. He subtracted 2 from both sides, just like I had taught him. That gave him 3x = 9. Then he subtracted 3 from both sides to get x = 6. That's a typical result from trying to remember rules -- close in memory, but essentially useless for solving the equation.

What should I do? I could have just repeated my rules again, in the hope that he would eventually understand them. But I decided to try for some understanding. First I did some loox problems, and discovered that he could solve 3x + 2 = 11 as a loox problem.

What next? A clue is that the answer 6 is in some sense ridiculous -- 3 times 6 plus 2 does not equal 11. That suggests an inability to check his answer, which means he couldn't solve the problem in a backwards direction. So I had do the backwards problem -- he thought of a number, muliplied it by something, added something to it, and told me the answer. Then I had to guess what his number is.

I appeal to your intuition. Doesn't it seem like someone should be able to do the problem in this direction? Can you imagine trying to solve the problem in the correct direction if you couldn't do it in the backwards direction? So I think this was a good exercise for him. It seemed to be both fun and challenging for him, which is a very good sign for learning (although it could have been fun or challenging for the wrong reasons).

I then tried this same perspective on my third-graders. They choose a number from 1 to 10, multiplied it by a number, then added something to it and told me the answer. I told them what number they started with. They weren't impressed. So I thought of a number, multiplied it by something, added something to it, and gave them the result. I wrote it as an algebra equation 3 x k + 7 = 19. I am not sure that one of them got it, but the other two subtracted 7 from 19 to get twelve, then asked themselves what number times 3 equals 12, getting the answer 4.

So, to my surprise, they were solving algebra equations. Subtracting instread of adding didn't stop them. I choose a larger number and suggested dividing to find out what number times 8 equals 120. We were having a contest, and I was losing.

### A More Complex Equation

Just to make things a little more challenging, I changed the equation to something like 4k + 5 = k + 17. They declared this to be impossible to determine. I should note that to understand the equation, they rewrote it substituting "some number" for k. In a sense this shows an incomplete mastery of variables, because they shouldn't need to do that. But in another sense, it shows that they know exactly what a variable means and they are treating it meaningfully, not just mindlessly manipulating symbols.

So, following my logic, I let them make up an equation like that for me to solve. In the time remaining, they couldn't do that. (To be more precise, they had to know the solution to the equation, which means deciding on their answer in advance and then constructing an equation around that solution.) They wanted to continue this next time. I predict that when they can make up a problem like that for me to solve, then they will be able to solve my problem, or at least come close to solving it (or understand the method if I tell it to them).

### The "Know-Your-Number" Trick

There is a "magic trick" that goes like this. You select a number, you add 3, you multiply by 2, you subtract 10, you divide by 2, then tell the magician the final result. The magician then tells you your original number. These problems are just a variation on that trick.