Step 1: Understand the Goal
We need to minimize $f(a,b,c,d) = (a-b)^2 + (a-c)^2 + (a-d)^2$ subject to $a+b+c+d=46$, where all variables are integers.
Step 2: Distribution Principle
In quadratic expressions like this, the minimum occurs when the values are distributed as evenly as possible around the mean.
Mean $= \frac{46}{4} = 11.5$.
Since they must be integers, the closest values to $11.5$ are $11$ and $12$.
Step 3: Test Combinations
To sum to $46$, our set must be $\{11, 11, 12, 12\}$.
Now, we need to choose which variable is $a$ to minimize the expression:
• If $a = 11$: Then $b, c, d$ must be $\{11, 12, 12\}$.
Expression $= (11-11)^2 + (11-12)^2 + (11-12)^2 = 0 + 1 + 1 = 2$.
• If $a = 12$: Then $b, c, d$ must be $\{11, 11, 12\}$.
Expression $= (12-11)^2 + (12-11)^2 + (12-12)^2 = 1 + 1 + 0 = 2$.
Conclusion:
In both cases, the minimum possible value is $\mathbf{2}$.
