When the square of the sum of two numbers are added to the square of their difference, we get 416. The difference between the square of the sum and square of the difference is 384. What are the numbers ?

Let the two numbers be x and y. Based on the problem, we can set up two algebraic equations:

1. (x + y)² + (x - y)² = 416
2. (x + y)² - (x - y)² = 384

By adding the two equations:
2(x + y)² = 800 → (x + y)² = 400 → x + y = 20

By subtracting the second equation from the first:
2(x - y)² = 32 → (x - y)² = 16 → x - y = 4

Now we have a simple system of linear equations:
x + y = 20
x - y = 4

Adding these gives 2x = 24, so x = 12. Substituting x back gives 12 + y = 20, so y = 8. The numbers are 8 and 12.