If a + b + c + d = 13, find the minimum possible value of a3 + b3 + c3 + d3, given that a, b, c and d are all positive integers.

1. Break Down the Problem

We need to minimize the expression $a^3 + b^3 + c^3 + d^3$ given the constraint that $a + b + c + d = 13$, with $a, b, c, d$ being positive integers.

2. Relevant Concepts

To minimize the sum of cubes $a^3 + b^3 + c^3 + d^3$, we can apply the concept of power mean inequality which suggests that for a fixed sum, the sum of cubes is minimized when the values are as equal as possible.

3. Analysis and Detail

1. Given $a + b + c + d = 13$, we want to distribute 13 among $a, b, c,$ and $d$ to keep values as close to each other as possible while still being positive integers.

2. Since we have four variables, we can start by dividing 13 by 4: $\frac{13}{4} = 3.25$ This suggests we can use values around 3.

3. Assigning the values that are close to 3:

   • Let's start with $a = 3, b = 3, c = 3, d = 4$.
   • Now check the sum: $3 + 3 + 3 + 4 = 13$ This satisfies the constraint.

4. Calculate the sum of cubes for these values: $a^3 + b^3 + c^3 + d^3 = 3^3 + 3^3 + 3^3 + 4^3 = 27 + 27 + 27 + 64 = 145$

4. Verify and Summarize

To ensure this is minimized, let's consider other nearby combinations:

• Trying $a = 2, b = 3, c = 4, d = 4$:

  • Check sum: $2 + 3 + 4 + 4 = 13$
  • Calculate cubes: $2^3 + 3^3 + 4^3 + 4^3 = 8 + 27 + 64 + 64 = 163$

• Trying $a = 2, b = 3, c = 3, d = 5$:

  • Check sum: $2 + 3 + 3 + 5 = 13$
  • Calculate cubes: $2^3 + 3^3 + 3^3 + 5^3 = 8 + 27 + 27 + 125 = 187$

After testing these combinations, we find:

• The values $a = 3, b = 3, c = 3, d = 4$ give the minimum sum of cubes of 145.

Final Answer

The minimum possible value of $a^3 + b^3 + c^3 + d^3$ is $\boxed{145}$.