Another Proof Without Words

Here's another pretty famous proof without words - it shows that the sum of the first n odd integers is n². Starting with the upper left hand corner, we have 1 red bead. Then adding 3 clear beads (3 being the 2nd odd integer), we get 2² which is 4 beads in total. Adding the 3rd odd integer 5, we get 3² = 9 beads. 

Said another way, 

1=1=1²

1+3=4=2²
1+3+5=9=3²
1+3+5+7=16=4²
1+3+5+7+9=25=5²

You could realize this via algebra as well. Consider the two consecutive square numbers n² and (n+1)². Expanding (n+1)², you get  n²+2n+1. Then the difference between one square number n² and the next square number (n+1)² is simply (n²+2n+1)-n² = 2n+1. So to get from the nth square number to the next square number, you just add the next odd number, 2n+1.