1. **Problem statement:** We have a chessboard divided into four pieces: two right triangles and two trapezoids. Each small square has area 1 unit. We need to find the area of each piece and check if their sum matches the total area of the chessboard. 2. **Chessboard total area:** The chessboard is 8 by 8 squares, so total area is $$8 \times 8 = 64$$ units. 3. **Areas of the four pieces:** - Each small square has area 1. - The chessboard is divided into 4 pieces: two right triangles and two trapezoids. 4. **Calculate areas of triangles:** - Suppose the two right triangles are formed by cutting along diagonals or lines that create right triangles with legs along the squares. - For example, if one triangle covers 1 square by 8 squares, its area is $$\frac{1}{2} \times \text{base} \times \text{height}$$. 5. **Calculate areas of trapezoids:** - The trapezoids are formed between these triangles. - Area of trapezoid is $$\frac{1}{2} (b_1 + b_2) h$$ where $b_1$ and $b_2$ are the lengths of the parallel sides and $h$ is the height. 6. **Sum of areas:** - Add the areas of the two triangles and two trapezoids. - The sum should be 64 units if no area is lost or gained. 7. **Explanation:** - The puzzle is known for an apparent paradox where rearranging the pieces seems to create or lose area. - This happens because the pieces do not fit perfectly; there is a very thin gap or overlap that is not obvious. **Final answer:** - The areas of the four pieces add up to 64 units, the total area of the chessboard. - The apparent paradox arises from the subtle gaps or overlaps when rearranged, so the Area Postulate is not violated.