## Wednesday, December 3, 2008

### Wednesday Math, Vol. 50: Little Karl shows he's clever

Math, like music, is famous for its prodigies, people whose talent shines through at a very young age. This is certainly the case with Karl Friedrich Gauss, whose master work Disquisitiones Arithmeticae was completed by the time he was 21, and published two years later in 1801. Gauss' name is always included on any list of the greatest mathematicians of all time, and besides producing great work when he was just a teenager, there are many stories of his displays of mathematical talent when he was just a wee lad.

A story that has been passed down through the ages is when little Karl is in primary school. I tell the story in class complete with voices and a bad German accent.

So the class of little kids is being rambunctious, and the teacher wants some order.

"You vill be QUIET!" The teacher shouts. (When I tell this story, I slap my hand on the desk. It helps to wake up any drowsy students.) "Everyone will take their slates and they will add all the numbers from 1 to 100, and I WILL HEAR NO NOISE UNTIL YOU ARE DONE!"

Now it's quiet. All you can hear is chalk on little chalkboards. Skritch, skritch, skritch.

Less than a minute later, little Karl raises his hand. "Herr Doktor Profethor?" (I use a high squeaky voice for Little Karl. The lisp makes him more appealing.)

"Jah?"

"fünftauthend fünfthig?"

Pause. "Jah."

1+2+3+4+...+98+99+100 = 5,050.

How did little Karl know so fast?

What he did was put the numbers together from the front of the list and the back of the list.

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
...
49 + 52 = 101
50 + 51 = 101

That means we have 50 pairs of numbers all of which add up to 101. 50 x 101 = 5,050.

He wasn't the first person in history to know about this, but he did figure it out independently at a very early age. The sum from 1 to n is known as the nth triangular number, and the formula for finding it without all the adding is 1/2 x n x (n+1). This is because a triangle can be thought of as half a rectangle as shown below.

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This picture has two triangles of the size 1+2+3+4+5+6+7, which makes a rectangle of 7 by 8. That means we can divide by two to get 1/2 x 7 x 8 = 28.