Question 1209945
Let's compute the values of f(4,1), f(4,2), f(4,3), and f(4,4) using the given properties.

1.  **f(a,1) = a**
    * f(4,1) = 4

2.  **f(a + 1, b) = b[f(a,b) - f(a,b - 1)]**

3.  **f(a,b) = 1 if b > a**

Now, let's calculate the required values:

* **f(4,1) = 4** (given)

* **f(4,2):**
    * f(4,2) = 2[f(3,2) - f(3,1)]
    * f(3,1) = 3
    * f(3,2) = 2[f(2,2) - f(2,1)]
    * f(2,1) = 2
    * f(2,2) = 2[f(1,2) - f(1,1)]
    * f(1,1) = 1
    * f(1,2) = 1 (since 2 > 1)
    * f(2,2) = 2[1 - 1] = 0
    * f(3,2) = 2[0 - 2] = -4
    * f(4,2) = 2[-4 - 3] = 2[-7] = -14

* **f(4,3):**
    * f(4,3) = 3[f(3,3) - f(3,2)]
    * f(3,2) = -4
    * f(3,3) = 3[f(2,3) - f(2,2)]
    * f(2,2) = 0
    * f(2,3) = 1 (since 3 > 2)
    * f(3,3) = 3[1 - 0] = 3
    * f(4,3) = 3[3 - (-4)] = 3[7] = 21

* **f(4,4):**
    * f(4,4) = 4[f(3,4) - f(3,3)]
    * f(3,3) = 3
    * f(3,4) = 1 (since 4 > 3)
    * f(4,4) = 4[1 - 3] = 4[-2] = -8

Now, sum the values:

f(4,1) + f(4,2) + f(4,3) + f(4,4) = 4 + (-14) + 21 + (-8) = 4 - 14 + 21 - 8 = 3

Therefore, f(4,1) + f(4,2) + f(4,3) + f(4,4) = 3.

Final Answer: The final answer is $\boxed{3}$