1. **Problem statement:** The sum of two numbers is 50. We need to find the least value of the sum of their squares. 2. **Let the two numbers be** $x$ and $y$. Given: $$x + y = 50$$ 3. **We want to minimize:** $$S = x^2 + y^2$$ 4. **Using the constraint, express** $y$ **in terms of** $x$: $$y = 50 - x$$ 5. **Substitute into** $S$: $$S = x^2 + (50 - x)^2 = x^2 + (2500 - 100x + x^2) = 2x^2 - 100x + 2500$$ 6. **To find the minimum, take the derivative of** $S$ **with respect to** $x$ **and set it to zero:** $$\frac{dS}{dx} = 4x - 100 = 0$$ 7. **Solve for** $x$: $$4x = 100 \implies x = 25$$ 8. **Find** $y$: $$y = 50 - 25 = 25$$ 9. **Calculate the minimum sum of squares:** $$S = 25^2 + 25^2 = 625 + 625 = 1250$$ **Final answer:** The least value of the sum of their squares is **1250**.