We are given a set $S = \{-7, -5, -3, -2, 2, 4, 6, 13\}$ and tasked with finding the minimum possible value of the expression

$(a + b + c + d)^2 + (e + f + g + h)^2$

where $a, b, c, d, e, f, g, h$ are distinct elements from $S$.

Step 1: Partition the set

We are splitting the set into two groups, each with four elements. Let:

• $A = \{a, b, c, d\}$
• $B = \{e, f, g, h\}$

The goal is to minimize $(S_A)^2 + (S_B)^2$, where $S_A = a + b + c + d$ and $S_B = e + f + g + h$, subject to $S_A + S_B = S_{\text{total}}$.

The total sum of all elements in $S$ is:

$S_{\text{total}} = -7 + (-5) + (-3) + (-2) + 2 + 4 + 6 + 13 = 8$

Thus, we want to minimize $(S_A)^2 + (S_B)^2$ with the constraint that $S_A + S_B = 8$. Since $S_B = 8 - S_A$, the expression becomes:

$(S_A)^2 + (8 - S_A)^2$

Step 2: Minimize the expression

We now seek to minimize the function:

$f(S_A) = (S_A)^2 + (8 - S_A)^2$

Expanding this:

$f(S_A) = S_A^2 + (64 - 16S_A + S_A^2) = 2S_A^2 - 16S_A + 64$

To find the minimum, take the derivative and set it equal to zero:

$\frac{d}{dS_A} (2S_A^2 - 16S_A + 64) = 4S_A - 16$

Setting $4S_A - 16 = 0$, we get:

$S_A = 4$

Thus, the minimum occurs when $S_A = 4$, and therefore $S_B = 8 - 4 = 4$.

Step 3: Verify the partition

We need to find two groups of four numbers from the set $S$ such that the sum of the numbers in each group is 4. One possible partition is:

• Group A: $\{-7, 13, -2, 0\}$ (which sums to $4$)
• Group B: $\{-5, -3, 6, 2\}$ (which also sums to 4)

Step 4: Calculate the minimum value

Using the partition (S_A = 455%&guaranteenmaximum %3append