What is the smallest positive integer $n$ such that $n=x^3+y^3$ for two different positive integer tuples $(x,y)$?
1. We need to find the smallest positive integer \( n \) such that \( n = x^3 + y^3 \) for two different positive integer tuples \((x, y)\).
2. We start by considering small values of \( x \) and \( y \) and compute \( x^3 + y^3 \) for each pair.
3. We need to find two different pairs \((x_1, y_1)\) and \((x_2, y_2)\) such that \( x_1^3 + y_1^3 = x_2^3 + y_2^3 \).

Let's compute some values:

\[
\begin{aligned}
&1^3 + 12^3 = 1 + 1728 = 1729, \\
&9^3 + 10^3 = 729 + 1000 = 1729.
\end{aligned}
\]

4. We observe that both pairs \((1, 12)\) and \((9, 10)\) yield the same sum \( 1729 \).

Thus, the smallest positive integer \( n \) such that \( n = x^3 + y^3 \) for two different positive integer tuples \((x, y)\) is \( 1729 \).

The final answer is \( \boxed{1729} \).