Puzzle of the Hundred Steps

There is a famous story that is said to have happened in a German primary school. A young student named Gauss had been misbehaving so his teacher gave him a task: add all the numbers from 1 to 100.

Seven year old Gauss is said to have given the correct answer within seconds, to the astonishment of his teacher. Gauss had spotted a pattern with the numbers that made the problem much easier to solve, but what was the pattern?

Can you solve these problems?

• Calculate the total of the whole numbers from 1 to 10
• Calculate the total of the whole numbers from 1 to 20
• Calculate the total of the whole numbers from 1 to 100
• Calculate the total of the whole numbers from 1 to 1000
• Gauss used a method of pairing numbers to yield identical sums. Can you explain his method?

If you want to challenge your student' in the same way Gauss' teacher did here's my take on the puzzle:

Alternatively you can click here to download it!

DON'T FORGET THE ANSWER SHEET, DOWNLOAD IT HERE.