There are many facts which can be proved just by looking at a picture! For instance:x^{2} – y^{2} = (x+y) (x-y)

is illustrated in Figure 1. At left is a large square (of side length x) with a small square (of side length y) cut out of the corner. (Thus its total area is x^{2} – y^{2}).

This figure can be cut into blue and green rectangles, and by rearranging them, we can form a rectangle of side lengths (x-y) and (x+y), on the right side of Figure 1!

**Presentation Suggestions:**

Write down the identity and then draw the picture, asking students to see if they can figure out how the picture explains the identity. Or, draw the picture and ask if they can figure out what it should prove!

You might challenge students to take any algebraic identity that they know and develop a proof without words.

**The Math Behind the Fact:**

Almost any “Proof without Words” from Mathematics Magazine or Nelsen’s recent book (see reference) will serve as a short, snappy Fun Fact!

**How to Cite this Page:**

Su, Francis E., et al. “Proofs without Words.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**References:**

R. Nelsen, *Proofs without Words*.

**Fun Fact suggested by: **

Lesley Ward