Given:
- Integers a, b, c, and d such that a + b + c + d = 30

To find:
- The minimum possible value of (a - b)^2 + (a - c)^2 + (a - d)^2

Solution:

Step 1: Analyzing the equation
- We are given the equation a + b + c + d = 30, which implies that a = 30 - (b + c + d).
- Substituting this value of a in the expression to be minimized, we get:
(a - b)^2 + (a - c)^2 + (a - d)^2
= (30 - (b + c + d) - b)^2 + (30 - (b + c + d) - c)^2 + (30 - (b + c + d) - d)^2
= (30 - 2b - c - d)^2 + (30 - b - 2c - d)^2 + (30 - b - c - 2d)^2

Step 2: Expanding the equation
- Expanding the equation by squaring each term inside the brackets, we get:
(900 - 120b - 60c - 60d + 4b^2 + 4bc + 4bd + c^2 + 2cd + d^2) +
(900 - 60b - 120c - 60d + 4bc + 4c^2 + 4cd + b^2 + 2bd + d^2) +
(900 - 60b - 60c - 120d + 4bd + 4cd + 4d^2 + b^2 + c^2 + 2bc)

Step 3: Simplifying the equation
- Simplifying the equation by combining like terms, we get:
2700 - 240b - 240c - 240d + 8b^2 + 8c^2 + 8d^2 + 12bc + 12bd + 12cd

Step 4: Rearranging the equation
- Rearranging the equation to group the similar terms, we get:
(8b^2 - 240b) + (8c^2 - 240c) + (8d^2 - 240d) + 12bc + 12bd + 12cd + 2700

Step 5: Completing the square
- To find the minimum value of the expression, we need to complete the square for each quadratic term.
- For a quadratic term of the form ax^2 + bx, the minimum value is given by -b^2 / (4a).
- Applying this concept to each quadratic term in the equation, we get:
(8b^2 - 240b) = 8(b^2 - 30b) = 8(b^2 - 30b + 225) - 8(225) = 8(b - 15)^2 - 1800
(8c^2 - 240c) =

To be minimum two no would be 7 and other two would be 8 and " a " will sure be 8