If a+b+c+d=30, where a,b,c and d are integers, then the minimum possib

Question

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

A. 3
B. 2
C. 0
D. 1
E. 5

sach24x7 · Answer

Superb question. Simple but tricky.

To minimize the overall value individual values of a-b , a-c, a-d should be minimum and that is possible only when they are same digit or minimum difference. 
Values would be 7 8 8 7 which gives answer 2

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pandajee · Answer

IMO - B

(a - b)^2  +  (a - c)^2  +  (a - d)^2  - Since in the question the difference of digits is squared, so the min value can be zero. It cant be negative.
Lets assume the min value is zero. It means a=b=c=d
=> 4a = 30
=> a= 30/4 = 15/2 = 7.5 But 'a' cant be a decimal number as the question states that the digits are integers. So, lets assume that a=7 or a=8
In case, a= 7, sum of all digits is 7 x 4 = 28 ( Assuming all digits to be of same value so as to get zero as the difference).
It falls short of 30 by 02. So, we need to increase two 07 into two eights. Hence the numbers are 7,7,8,8

In case, a=8, sum of all digits is 8 x 4 = 32 ( Assuming all digits to be of same value so as to get zero as the difference).
It exceeds 30 by 02. So, we need to decrease two 08 into two seven. Hence the numbers are 8 , 8 , 7 , 7

Now, (8-8)^2 + (8-7)^2 + (8-7)^2 = 0 + 1 + 1 = 2 Answer

ScottTargetTestPrep · Answer

The only way (a - b)^2 + (a - c)^2 + (a - d)^2 could equal 0 is if a = b = c = d. However, in this case, we have a + b + c + d = 4a = 30, and there are no integer solutions to this equation. Thus, the minimum possible value of (a - b)^2 + (a - c)^2 + (a - d)^2 is greater than 0.

If (a - b)^2 + (a - c)^2 + (a - d)^2 = 1, then two of the terms in the summation are equal to 0, and one of the terms is equal to 1. This means that a is one more than one of the variables b, c, or d, and a is equal to both of the remaining variables. In this case, we get a + a + a + a + 1 = 4a + 1 = 30. Once again, there are no integer solutions to this equation. We see that the minimum possible value of (a - b)^2 + (a - c)^2 + (a - d)^2 is greater than 1.

If (a - b)^2 + (a - c)^2 + (a - d)^2 = 2, then two of the terms in this summation are equal to 1, and one of the terms is equal to 0. This means that a is equal to one of the variables b, c, or d, and a is one more than the remaining variable. In this case, we get a + a + a + 1 + a + 1 = 4a + 2 = 30. We see that a = 7 salsifies this equation.

Since (a - b)^2 + (a - c)^2 + (a - d)^2 cannot equal 0 or 1, but it can be equal to 2, it follows that the smallest value of this expression is 2.

Answer: B

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If a+b+c+d=30, where a,b,c and d are integers, then the minimum possib

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