Directions: Use the integers -9 to 9, at most one time each, to fill in the boxes and make a real number product with the greatest value.

### Hint

What has to happen to the imaginary terms to make a complex number a real number?

How can we accomplish that?

How can we accomplish that?

### Answer

Lauren Gaffney found the product with the greatest value is (9 + -6i)(6 + 4i) which has a product of 78.

Source: Robert Kaplinsky

We got 78 for an answer with (9-6i)(6+4i). Did we do anything wrong?

You can’t repeat values maybe.

HEre is part of the beauty of open-middle:

No values were repeated. Negative six is different from positive six.

I agree 78 is the highest possible product.

I forgot to mention the beauty of open middle: It creates discussion and answers you wouldn’t think about.

And I’d add that the beauty of Open Middle is that the person writing the problem also gets it wrong pretty often because someone finds a better answer and puts in in the comments! Thanks team. I’ll update the answer.

We found a couple ways to get 78. (-4+6i)(-6+-9i) is another.

Is 78 the greatest product?