Plus Advent Calendar Door #18: Clever sums

How would you go about adding up all the integers from 1 to 100? Tap them into a calculator? Write a little computer code? Or look up the general formula for summing integers?

Carl Friedrich Gauss as depicted on the (now defunct) German 10 Marks note.

Legend has it that the task of summing those numbers was given to the young Carl Friedrich Gauss by his teacher at primary school, as a punishment for misbehaving. Gauss didn't have a calculator or computer, no one did at that time, but he came up with the correct answer within seconds. Here's how he did it.

Notice that you can sum the numbers in pairs, starting at either end. First you add 1 and 100 to get 101. Next it's 2 and 99, giving 101 again. The same for 3 and 98. Continuing like this, the last pair you get is 50 and 51 and they give 101 again. Altogether there are 50 pairs all adding to 101, so the answer is 50 x 101 = 5050. Easy — if you're Gauss.

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