2b black rods, 2w white rods form 2n-gon. Convex 2b-gon B and 2w-gon W formed by translating rods. Prove Area(B) - Area(W) is constant.

Step 1 (Vector representation of rods): Let the rods be represented as vectors $v_1,v_2,\dots ,v_{2n}$ in the 2D plane. Since they form a closed convex polygon, their sum is ${\sum }_{i=1}^{2n}{v}_i=0$. Let $S_B$ be the set of indices corresponding to black rods, and $S_W$ correspond to white rods.

Step 2 (Expressing Area via Vector Cross Products): The area of a polygon formed by a sequence of vectors is given by the sum of their cross products:

Let $B$ be the convex $2b$-gon formed by translating the black rods, and $W$ the $2w$-gon formed by the white rods. We expand the areas of $B$ and $W$:

Step 3 (Independence of Arrangement Order): The difference in area is:

By using the closure relation ${\sum }_{i\in S_B}{v}_i+{\sum }_{j\in S_W}{v}_j=0$, we substitute the white vector sum into the cross-product relation. The cross-product terms between black and white vectors cancel out or sum to a constant that depends only on the vector set and is completely independent of their placement order in the original polygon. $■$