Let $m$ and $n$ be fixed distinct positive integers. A wren is on an infinite board indexed by $\mathbb Z^2$, and from a square $(x,y)$ may move to any of the eight squares $(x\pm m, y\pm n)$ or $(x\pm n, y \pm m)$. For each $\{m,n\}$, determine the smallest number $k$ of moves required to travel from $(0,0)$ to $(1,0)$, or prove that no such $k$ exists. 

[i]Proposed by Michael Ren
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   - If \(\gcd(m, n) > 1\), then no such \( k \) exists. This is because the wren's moves will always land on lattice points that are multiples of \(\gcd(m, n)\), making it impossible to reach \((1,0)\).
   - If both \( m \) and \( n \) are odd, then no such \( k \) exists. This can be shown using a chessboard coloring argument, where the wren will always land on squares of the same color, making it impossible to reach \((1,0)\) which is of a different color.

3. **Assuming \( \gcd(m, n) = 1 \) and \( m \) is even, \( n \) is odd:**
   - We classify the moves into two types:
     - Type 1: \((x, y) \to (x \pm m, y \pm n)\)
     - Type 2: \((x, y) \to (x \pm n, y \pm m)\)
   - Since the order of moves does not matter, we can assume all type 1 moves are done first, followed by type 2 moves.

4. **Using Type 1 Moves:**
   - We can move from \((x, y)\) to \((x \pm am, y \pm bn)\) if and only if \( a \equiv b \pmod{2} \).
   - The number of type 1 moves required is \(\max\{|a|, |b|\}\).

5. **Using Type 2 Moves:**
   - Similarly, we can move from \((x, y)\) to \((x \pm cn, y \pm dm)\) if and only if \( c \equiv d \pmod{2} \).
   - The number of type 2 moves required is \(\max\{|c|, |d|\}\).

6. **Combining Moves to Reach \((1,0)\):**
   - We want to achieve the following sequence of moves:
     \[
     (0,0) \xrightarrow[\text{type 1 only}]{(am, bn)} \xrightarrow[\text{type 2 only}]{(am+cn, bn+dm)} = (1,0)
     \]
   - Since \(\gcd(m, n) = 1\), \( m \mid b \) and \( n \mid d \), so \( b \) must be even. Thus, \( a \) is even too. Write \( a = 2s \).

7. **Finding the Minimum Number of Moves:**
   - We need \( 2sm + cn = 1 \). Any solution \((s, c)\) of this equation will lead to a series of \(\max\{2|s|, |b|\} + \max\{|c|, |d|\} \geq \max\{2|s|, m\} + \max\{|c|, n\}\) moves.
   - We aim to minimize this, which is basically minimizing \(|s|\).

8. **Conclusion:**
   - The smallest number of moves \( k \) required is given by:
     \[
     k = \max\{2p, m\} + \max\{q, n\}
     \]
     where \( p \) is the least \( t \geq 0 \) such that \( 2mt \equiv \pm 1 \pmod{n} \) and \( q \) is defined as either \(\frac{2mp - 1}{n}\) or \(\frac{2mp + 1}{n}\), whichever is the lower-valued integer.

The final answer is \( \boxed{ \max\{2p, m\} + \max\{q, n\} } \)