Solution:

To solve this problem, let's consider the motion of the particles from an observer's frame of reference. We can assume that the square is located in the x-y plane, with one particle at each corner.

Step 1: Analyzing the Motion

Let's denote the particles as A, B, C, and D, with A being the particle at the bottom left corner and moving clockwise. Since each particle maintains a direction towards the next particle, we can conclude that their velocities are always directed towards the center of the square.

Step 2: Describing the Motion Equations

Let's consider the motion of particle A. Its velocity can be written as a vector equation:

v_A = -v_i + v_j

Here, v_i and v_j represent the unit vectors along the x and y directions, respectively. The negative sign indicates that the velocity is directed towards the center of the square.

Similarly, we can write the velocity vectors for particles B, C, and D:

v_B = -v_i - v_j
v_C = v_i - v_j
v_D = v_i + v_j

Step 3: Finding the Time of Meeting

To find the time at which the particles meet, we can calculate the distance traveled by each particle and equate them. Let's consider the meeting point as the origin (0, 0) and define the position vectors as:

r_A = (-a/2, -a/2)
r_B = (a/2, -a/2)
r_C = (a/2, a/2)
r_D = (-a/2, a/2)

Using the velocity vectors and position vectors, we can write:

r_A = (-v_i + v_j)t
r_B = (-v_i - v_j)t
r_C = (v_i - v_j)t
r_D = (v_i + v_j)t

Now, let's calculate the distances traveled by each particle using the distance formula:

d_A = √[(-a/2 + vt)^2 + (-a/2 + vt)^2]
d_B = √[(a/2 + vt)^2 + (-a/2 - vt)^2]
d_C = √[(a/2 + vt)^2 + (a/2 - vt)^2]
d_D = √[(-a/2 + vt)^2 + (a/2 + vt)^2]

Equating the distances traveled by each particle, we can solve for t, the time at which they meet.

Step 4: Calculating the Time of Meeting

By equating the above distances, we can simplify the equation and solve for t. After solving the equation, we find:

t = a/(2v)

Therefore, the particles will meet each other after a time of a/(2v).

Conclusion:

The time taken by the particles to meet each other is given by the equation t = a/(2v), where 'a' is the side length of the square and 'v' is the constant speed at which the particles move. This result holds true as long as the particles maintain a constant speed and direction towards the next particle in succession.