**Problem Analysis:**

We are given that there are six particles situated at the corners of a regular hexagon. Each particle moves at a constant speed v and maintains a direction towards the particle at the next corner. We need to calculate the time it will take for the particles to meet.

**Solution:**

To solve this problem, let's consider the motion of two adjacent particles. We can assume that the particles start moving at time t = 0.

- Let's consider two adjacent particles A and B. Particle A is at the corner and particle B is at the next corner.
- At time t = 0, both particles start moving towards each other with a constant speed v.
- Let's assume that the angle between the direction of particle A and the line joining the centers of the two particles is θ.
- As the particles move towards each other, the angle θ will decrease at a constant rate.
- The time it takes for the particles to meet can be calculated by finding the time it takes for the angle θ to become 0.

**Calculating the time taken for the particles to meet:**

- Let's consider the initial position of the particles at time t = 0. The angle θ between the direction of particle A and the line joining the centers of the two particles is 60 degrees (since we have a regular hexagon).
- As the particles move towards each other, the angle θ decreases at a constant rate. The rate at which θ decreases can be calculated by considering the relative velocity of the particles.
- Since the particles are moving towards each other, the relative velocity of particle A with respect to particle B is 2v (since the distance between the particles is a).
- The rate at which θ decreases is given by dθ/dt = (2v) / a.
- We can integrate this equation to find the time it takes for θ to become 0.
- Integrating both sides of the equation gives us ∫dθ = ∫(2v) / a dt.
- Integrating, we get θ = (2v / a)t + C, where C is the constant of integration.
- At time t = 0, θ = 60 degrees. Substituting these values into the equation gives us 60 = (2v / a)(0) + C.
- Solving for C, we get C = 60.
- Substituting the values of θ and C into the equation gives us θ = (2v / a)t + 60.
- We want to find the time it takes for θ to become 0, so we set θ = 0 and solve for t.
- Setting (2v / a)t + 60 = 0 gives us t = -60a / (2v).
- Since time cannot be negative, we take the absolute value of t, which gives us t = 60a / (2v).
- Simplifying, we get t = 30a / v.
- Therefore, the time it takes for the particles to meet is 30a / v.

**Final Answer:**

The time it takes for the particles to meet is 30a / v.