A blind ant is walking on the coordinate plane. It is trying to reach an anthill, placed at all points where both the $x$-coordinate and $y$-coordinate are odd. The ant starts at the origin, and each minute it moves one unit either up, down, to the right, or to the left, each with probability $\frac{1}{4}$. The ant moves $3$ times and doesn't reach an anthill during this time. On average, how many additional moves will the ant need to reach an anthill? (Compute the expected number of additional moves needed.)
1. **Understanding the Problem:**
   - The ant starts at the origin \((0,0)\).
   - It moves one unit in one of the four directions (up, down, left, right) with equal probability \(\frac{1}{4}\).
   - The ant needs to reach a point where both coordinates are odd.
   - We need to find the expected number of additional moves required after the ant has already moved 3 times and hasn't reached an anthill.

2. **Possible Positions After 3 Moves:**
   - After 3 moves, the ant can be at any point \((x, y)\) where \(x\) and \(y\) are integers.
   - Since the ant hasn't reached an anthill, both \(x\) and \(y\) cannot be odd simultaneously.

3. **Expected Number of Moves to Reach an Anthill:**
   - We need to calculate the expected number of moves for the ant to reach a point where both coordinates are odd.
   - Let \(E(x, y)\) be the expected number of moves from point \((x, y)\) to reach an anthill.

4. **Setting Up the Equations:**
   - If the ant is at \((x, y)\) where both \(x\) and \(y\) are even or one is even and the other is odd, it needs to move to a point where both coordinates are odd.
   - The expected number of moves from \((x, y)\) can be expressed as:
     \[
     E(x, y) = 1 + \frac{1}{4} \left( E(x+1, y) + E(x-1, y) + E(x, y+1) + E(x, y-1) \right)
     \]
   - This equation accounts for the fact that the ant moves one step and then has an equal probability of moving to one of the four neighboring points.

5. **Simplifying the Problem:**
   - Consider the symmetry and periodicity of the problem. The ant's position modulo 2 is crucial.
   - Define \(E_{even, even}\), \(E_{even, odd}\), \(E_{odd, even}\), and \(E_{odd, odd}\) as the expected number of moves from points with respective parities.

6. **Solving the System of Equations:**
   - From the symmetry and periodicity, we can set up the following system:
     \[
     E_{even, even} = 1 + \frac{1}{4} \left( E_{odd, even} + E_{odd, even} + E_{even, odd} + E_{even, odd} \right)
     \]
     \[
     E_{even, odd} = 1 + \frac{1}{4} \left( E_{odd, odd} + E_{odd, odd} + E_{even, even} + E_{even, even} \right)
     \]
     \[
     E_{odd, even} = 1 + \frac{1}{4} \left( E_{odd, odd} + E_{odd, odd} + E_{even, even} + E_{even, even} \right)
     \]
     \[
     E_{odd, odd} = 0
     \]
   - Simplifying these equations:
     \[
     E_{even, even} = 1 + \frac{1}{2} E_{odd, even} + \frac{1}{2} E_{even, odd}
     \]
     \[
     E_{even, odd} = 1 + \frac{1}{2} E_{odd, odd} + \frac{1}{2} E_{even, even}
     \]
     \[
     E_{odd, even} = 1 + \frac{1}{2} E_{odd, odd} + \frac{1}{2} E_{even, even}
     \]
     \[
     E_{odd, odd} = 0
     \]

7. **Solving for \(E_{even, even}\):**
   - Substitute \(E_{odd, odd} = 0\) into the equations:
     \[
     E_{even, odd} = 1 + \frac{1}{2} E_{even, even}
     \]
     \[
     E_{odd, even} = 1 + \frac{1}{2} E_{even, even}
     \]
   - Substitute these into the equation for \(E_{even, even}\):
     \[
     E_{even, even} = 1 + \frac{1}{2} (1 + \frac{1}{2} E_{even, even}) + \frac{1}{2} (1 + \frac{1}{2} E_{even, even})
     \]
     \[
     E_{even, even} = 1 + \frac{1}{2} + \frac{1}{4} E_{even, even} + \frac{1}{2} + \frac{1}{4} E_{even, even}
     \]
     \[
     E_{even, even} = 2 + \frac{1}{2} E_{even, even}
     \]
     \[
     \frac{1}{2} E_{even, even} = 2
     \]
     \[
     E_{even, even} = 4
     \]

8. **Conclusion:**
   - The expected number of additional moves needed for the ant to reach an anthill is \(4\).

The final answer is \(\boxed{4}\)