How To Be A Little Gauss

There is a story about Carl Friedrich Gauss. Supposedly, when he was a little boy, his teacher asked the class to add up the numbers one through a hundred (1+2+3 etc., all the way up to 100). The teacher wanted to get some work done, or get some sleep, or whatever. Anyway, to the teacher's annoyance, little Gauss [Here the lecturer holds his hand out to show that little Gauss was about 2 feet tall, to the amusement of the audience]... To the teacher's annoyance, little Gauss came up to the teacher with the answer, right away. The teacher probably had to spend the rest of the class time verifying little Gauss's [2 feet tall] result.

Some people find that story hard to believe, even impossible. I think that the story has the ring of truth to it. I believe that the story is true, or close to it. There are versions of the story, in which the numbers are one to a thousand [murmur in the audience].

I think that you people can duplicate little Gauss's [2 feet tall] trick [doubt in the audience]. I'm going to give you two very small hints. But, that's all you will need, to be just like little Gauss [2 feet tall].

Nobody use your calculators, or even paper and pencil for a while. You are going to be slower than little Gauss [Lecturer hesitates, then shows "2 feet tall"]. But, you're going to be just as smart.

Here's the problem:

1 + 2 + 3 + ... + 99 + 100 = X
We want to find X. Well, it's going to take 99 additions to solve this. Let's see, 1+2=3, +3=6, +4=10, +. It's going to take a while. There's got to be an easier way.

What if we start at the other end:

100 + 99 + 98 + ... + 2 + 1 =
Do we get the same answer? [Various forms of "yes" from the audience] How many think we get the same number, raise your hands? [Most raise their hands] Yeah. It was algebra, right? Associative Law? It doesn't matter what order you add things up, you get the same answer. So "yes" we get the same answer [Lecturer writes "X" to the right of the equal sign].

That was your first hint, "Associative Law."

Let's see, 100+99=199, +98=297, +. That's going to take just as long, isn't it? There are 99 additions there, too. That didn't help. Any other way?

What if we add up the even numbers (that's 49 additions), then add up the odd numbers (that's 49 additions), and then add up the two totals? That's, uh, 99 additions. Darn, that's no better. How about this?

1 + 2 + 3 + 4 + ... + 48 + 49 + 50
51 + 52 + 53 + 54 + ... + 98 + 99 + 100
When we finally total them up, we get the same answer, right? Associative Law? Does that look helpful? Not [hesitate] really.

How about:

1 + 2 + 3 + 4 + ... + 48 + 49 + 50
100 + 99 + 98 + 97 + ... + 53 + 52 + 51
Does that help? This is the second hint, by the way [points at those numbers]. Do you see something magical about that? Anybody see it?

[A couple people in the audience say that the columns all add up to 101].

[There is general agreement].

Right. Do you all see it? [The lecturer points at each column in return] 101, 101, 101, ..., 101, 101.

How many 101s do we have? [Pause. Some say "50." There is agreement] Right. 50 times 101:

50 x 101
Can we do this in our heads, or do we have to use our calculators for that?

[Some people say "5050"].

[The lecturer writes a big 5050 on the board, and begins applauding the audience].

5050
[The audience applauds].