golf balls arranged on my basement floor

I realized I could find out how many golf balls there are in a triangle by (1) adding 1 to the number of rows, (2) multiplying this by the number of balls in the base, and (3) dividing the result by 2.

What I’m actually doing here is just adding all of the numbers from 1 to 16.

Visual proof

from Bill the Lizard’s blog:

1 + 2 + 3 + ... + n = n * (n+1) / 2

When I asked Bill for an explanation, here’s what he said:

Hi Josh, thanks for commenting. There’s an old story about Gauss that might shed some light on the fourth proof. (I’m quoting from my own post about Gauss, but this story has been retold hundreds of times.)

“When Gauss was in school at age 10, his teacher, perhaps needing a quiet half hour, gave his class the problem of adding all the integers from 1 to 100. Gauss immediately wrote down the answer on his slate (this would have been in the year 1787). He had quickly noticed that the sum (1 + 2 + 3 + … + 100) could be rearranged to form the pairs (1 + 100) + (2 + 99) + (3 + 98) + … + (50 + 51), and that there were 50 pairs each equaling 101. This reduced the problem to the simple product 50 x 101 = 5050.”

The fourth proof here is illustrating the same idea. It’s just arranging the numbers in a different way than Gauss did over 200 years ago.

I hope that helps.

Pretty cool, huh?