If $a, b, c,$ and $d$ are integers such that $a + b + c + d = 30$, then the minimum possible value of $(a - b)^2 + (a - c)^2 + (a - d)^2$ is

We need to find the minimum value of $(a-b)^2 + (a-c)^2 + (a-d)^2$ given that $a + b + c + d = 30$ and all variables are integers.

The expression $(a-b)^2 + (a-c)^2 + (a-d)^2$ represents the sum of squared differences between $a$ and each of the other three variables.

Key Insight: To minimize this expression, we want $a$ to be as close as possible to $b$, $c$, and $d$. This is because squaring any difference makes it positive, and smaller differences lead to smaller squared values.

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Since $a + b + c + d = 30$, the average value is $\tfrac{30}{4} = 7.5$.

Since all variables must be integers, we cannot make all four values exactly equal to 7.5.

We need to distribute the values as evenly as possible using integers. The closest we can get is:

Two variables equal to 8

Two variables equal to 7

Check: $8 + 8 + 7 + 7 = 30$

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To minimize $(a-b)^2 + (a-c)^2 + (a-d)^2$, we should:

Choose $a$ to be one of the "middle" values (either 7 or 8)

Make the other three variables as close to $a$ as possible

Let's set $a = 8$. Then to minimize the expression, we want $b$, $c$, and $d$ to be as close to 8 as possible.

The best choice is: $a = 8$, $b = 8$, $c = 7$, $d = 7$

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$(a-b)^2 + (a-c)^2 + (a-d)^2$

$= (8-8)^2 + (8-7)^2 + (8-7)^2$

$= 0^2 + 1^2 + 1^2$

$= 0 + 1 + 1 = 2$

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Therefore, the minimum possible value is 2.