Add 1 to 100 in Seconds: Gauss's Trick (With a Picture)

Step by step

1. $1, $2, $3 … $100 over 100 days. Total?
   Save one dollar on day one, two dollars on day two, and so on, up to a hundred dollars on day one hundred. How much do you save?
2. Start small: 10 days → a staircase of blocks
   Let's start smaller: ten days. Draw each day's savings as a stack of blocks. It makes a staircase.
3. Way 1: copy the staircase, turn it upside down
   Way one: make a copy of the staircase, and turn it upside down.
4. They fit → a 10 × 11 rectangle
   Slide it onto the first one. They fit together into a rectangle, ten blocks wide and eleven tall.
5. 10 × 11 = 110 → half = 55
   That's ten times eleven, a hundred ten blocks. But that's two staircases, so one staircase is half: fifty-five!
6. 100 × 101 = 10100 → half = 5050
   For a hundred days, the rectangle is a hundred by a hundred and one. That's ten thousand one hundred, and half of it is five thousand and fifty dollars!
7. Way 2: the pairing trick (the Gauss story)
   Way two is the famous pairing trick. The story goes that a young Carl Friedrich Gauss used it in school.
8. 1 + 100 = 101 · 2 + 99 = 101 · every pair = 101
   Pair the first day with the last: one plus a hundred is a hundred and one. Two plus ninety-nine is a hundred and one too. Every pair makes a hundred and one.
9. 50 pairs × 101 = 5050
   A hundred days make fifty pairs. Fifty times a hundred and one is five thousand and fifty. Same answer!
10. last number × next number ÷ 2
    So the total is the last number, times the next number, divided by two.
11. On a test: 2S = 100 × 101 = 10100 → S = 5050
    On a test, write it like this. Call the total S. Write it forward, then backward. Add the two lines: every column makes a hundred and one. A hundred columns make ten thousand one hundred. Half of that is five thousand fifty dollars.