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___

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Solution

Given, a + b + c + d = 30 and a, b, c & d are integers.
Now for $(a - b)^{2} + (a - c)^{2} + (a - d)^{2}$ to have minimum value, a, b, c and d must be as close to one another as possible.
Thus, a, b, c & d can take 7, 7, 8, 8 in any order.
Let’s take a = 7, b = 7, c = 8, d = 8
$(a - b)^{2} + (a - c)^{2} + (a - d)^{2} = (7 - 7)^{2} + (7 - 8)^{2} + (7 - 8)^{2} = 2$

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