Proof without words

Today I came across William Dunham's awesome book titled The Mathematicial Universe with the following fascinating picture. It is a wordless proof.


This is silently proclaiming that 1+2+3+...n = (n)(n+1)/2! (Ok, that's not a factorial, but excitement!)

So the story is about the young Gauss who had a teacher that made him add 1+2+3+...100 so to keep the students busy all day. Gauss figured out that you can pair up the first and last number, 1 and 100, add them up so you have 101. The next pair, 2 and 99 also add up to 101. There are 50 such pairs so the answer is instantly 5050.

Let's decipher the picture.

Each white column is indicating 1,2,3... When you add them all up, it covers exactly half the area in the n by n+1 grid!

The author calls it immensely elegant. And I agree! Oh my I want to frame this!

Math is not always dry and boring.

Here is another one. It is saying adding odd numbers 1+3+5+7+9... you can always get a perfect square.



I noted that when I was a kid! One day I got so bored that I start counting tiles on the floor and noted the pattern.

So how do you prove it, in words?

Odd numbers can be represented by 2k+1, where k is an integer.
So 3 = 2(1)+1, 5=2(2)+1, 7 = 2(3)+1, and so on.

Let's add 1+3+5, we have 1+2(1)+1+2(2)+1, this is 2(1+2)+2+1
Let's add 1+3+5+7, we have 1+2(1)+1+2(2)+1+2(3)+1 = 2(1+2+3)+3+1.

See the pattern? Add n odd numbers, we get 2(1+2+3...n)+n+1.
Now apply the formula: 2(n)(n+1)/2 + n + 1, we have n(n+1)+n+1 = n²+2n+1. That is a perfect square.