If the product of n positive numbers is $n^n$, then what is the minimum value of their average for n = 6?

Let the 6 positive numbers be $x_1, x_2, x_3, x_4, x_5, x_6$.

Given: $x_1 \times x_2 \times x_3 \times x_4 \times x_5 \times x_6 = 6^6$

The average is: $\dfrac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6}$

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By the AM-GM inequality:

$\dfrac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \geq \sqrt[6]{x_1 \times x_2 \times x_3 \times x_4 \times x_5 \times x_6}$

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Substituting the product value:

$\dfrac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \geq \sqrt[6]{6^6}$

$\dfrac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \geq 6^{6/6}$

$\dfrac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \geq 6$

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The equality holds when all numbers are equal: $x_1 = x_2 = x_3 = x_4 = x_5 = x_6 = 6$

Therefore, the minimum value of the average is $6$.