Quick sums

How long would it take you to add all the whole numbers from 1 to 100? A famous story, told many times, says that a teacher in Germany once gave this sum to his class to keep them busy. One young boy, Carl Friedrich Gauss, wrote the answer, 5050, almost at once. The details may have grown in the telling, but Gauss was real, and he grew up to be one of the greatest mathematicians who ever lived. His trick is simple, and once you have seen it you can add long lists of numbers in seconds. This page also shows a second surprise: when you add the odd numbers, the answer is always a square number.

Pair the ends

Start with a smaller sum: 1 + 2 + 3 + ... + 10. Instead of adding from left to right, pair the first number with the last, the second with the second-to-last, and so on.

Every pair adds up to 11. There are 10 numbers, so there are 5 pairs. The total is 5 × 11 = 55. You can check it the slow way: it really is 55.

1. Add the first and the last number. That is the sum of every pair.
2. Count the numbers, and halve that count to get the number of pairs.
3. Multiply the number of pairs by the sum of a pair.

Why the rule always works

Write the sum twice, once forwards and once backwards, one above the other. Then add the two rows column by column.

There are 100 columns, and each one adds to 101. So the two rows together make 100 × 101 = 10100. But that is the sum written twice, so the sum itself is half of it: 5050. This way also works when the count is odd. For 1 + 2 + ... + 9: two rows make 9 × 10 = 90, and half of that is 45.

Sums that do not start at 1

The pairing trick works for any adding pattern, where the numbers go up by the same step each time. Every pair still adds to first + last.

The odd numbers make squares

Now add up the odd numbers, starting from 1, and watch the answers.

Every total is a square number. The picture shows why. Start with 1 dot. Add 3 dots around it in an L shape, and you have a square of 2 by 2. Add 5 more in a bigger L, and you have a square of 3 by 3. Each new odd number is exactly the L that turns one square into the next.