## Ask Professor Puzzler

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Vipin sends us the following question:

_+_+_+80+90=100 , no negetive number in the question.

There are multiple ways to solve this equation, but I have no idea if the ones I came up with are the ones the problem creator was looking for. That's okay. If you come up with a solution that works, it doesn't matter if it's "the right one."

For people reading this question, I'm about to propose a solution, so if you want to try to figure out a solution for yourself, do it now, before you read any further!

--- SOLUTIONS BELOW ---

The first thing I did to this was, I subtracted 80 and 90 from both sides of the equation, which turned it into:

_ + _ + _ = 100 - 80 - 90

_ + _ + _ = -70

If you don't mind, I'm going to replace those blanks with variables, because then I can talk about each individual blank by name:

a + b + c = -70, where a, b, and c are not negative.

I came up with a few possibilities; the first two use complex numbers.

**Solution One**

a = 0, b = -30 + i, c = -40 - i

I know, you'll be tempted to say, "Wait! -30 and -40 are negative numbers!" To which I reply, "No, -30 is part of the complex number -30 + i. Complex numbers are *neither negative nor positive*, so I have filled the blanks with numbers which are not negative."

Maybe you'd like to argue that the point is to not use the negative symbol at all. In which case, I give you solution #2:

**Solution Two**

a = 0, b = 30i^{2}, c = 40i^{2}

I've used no negative symbols, but b and c evaluate to -30 and -40 respectively, giving the desired result.

Maybe you'd like to argue that you can't use negative symbols *and *you can't use expressions that *evaluate* to negative numbers. In which case, I would say, "Stop adding to the rules and find your own solution!" ;)

If you're willing to accept solution two, it opens up a boatload of possibilities, like using trig, or other functions:

**Trig Solution**

a = 0, b = 0, c = 70cos(180º)

If our readers aren't willing to accept any of these answers, please use the "Ask Professor Puzzler" link at the bottom of the page to suggest your own solution. Be sure to mention what blog post you're writing about! If we get any solutions we like, we'll add them to the bottom of the post.