Jon,

The student was asked to add up the first 100 integers

1 + 2 + 4 + ... + 97 + 98 + 99 + 100

See what happens when you write the same sequence directly

underneath:-

1 + 2 + 3 + 4 + ... + 97 + 98 + 99 + 100

100 + 99 + 98 + 97 + ... + 4 + 3 + 2 + 1

Instead of adding up the rows, add up each individual column. In

other words, 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101 and so on all

the way up to 99 + 2 = 101, and finally 100 + 1 = 101

Clearly, every "column" equals 101. The cunning bit I guess is that we

know how many columns there are. There must be 100 (since we've

written out 1, 2, 3, etc up to 100).

100 lots of 101 equals 10,100. So far so good. But why is the answer

5050? Remember that we added up the columns. Each column

contained 2 rows, that is we added up the first 100 integers *twice*.

Therefore we need to divide the answer by 2.

We had 100 lots of 101 and needed to divide by 2.

(100x101)/2 is the same (written slightly differently) as what Gilbert

and Hotpawn wrote.

Gauss realised all this in his head in a few seconds. The other children

would have got as far as 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10 and so on,

adding each number in turn to the total of the previous numbers

added together. This is cumbersome and heavy handed.

As is so often in Mathematics, it wasn't that Gauss was brilliant at

adding up each number in turn, but rather he found an alternative,

simplified, more beautiful way of solving the problem.

Godfrey Hardy (might have been someone else) said "there is no

place for ugly mathematics".

Mark

The Squirrel Lover