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Let ABCD be a square with side length 1. A laser is located at vertex A, which fires a laser beam at point X on side BC, such that BX = 2/3. The beam reflects off the sides of the square, until it ends up at another vertex; at this point, the beam will stop. Find the length of the total path of the laser beam.

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To solve the problem, we first place the square \( ABCD \) in the coordinate plane with the following coordinates:

  • \( A(0, 1) \)
  • \( B(0, 0) \)
  • \( C(1, 0) \)
  • \( D(1, 1) \)

The point \( X \) on side \( BC \) is located at coordinates \( (0, 2/3) \).

Initially, the laser beam is fired from point \( A(0, 1) \) towards point \( X(0, 2/3) \). The trajectory of the laser can be modeled as a straight line that reflects off the walls of the square.

Step 1: Finding the path

  1. Initial Line Segment:

    • The line from \( A(0, 1) \) to \( X(0, 2/3) \) has a vertical orientation, going downward along the line \( x = 0 \).

  2. Reflection at \( BC \):

    • When the laser beam reaches \( X(0, 2/3) \), it reflects off side \( BC \) (the vertical side). The reflection will follow the law of reflection which states that the angle of incidence is equal to the angle of reflection. Therefore, it will proceed to the right along the line \( y = 2/3 \) towards the side \( CD \), which is horizontal.

  3. Continuing the path to \( CD \):

    • We calculate where this line intersects line \( CD \) (which is located at \( y = 0 \)).
    • The point moving horizontally to the right will always have \( y = 2/3 \) until it reaches \( D(1, 1) \).

Moving to the right, the laser will eventually reach the right corner:

  • The distance from \( X \) to \( D \) along the horizontal component (along line \( y = 2/3 \) up to the endpoint):
    • The coordinates would reach \( (1, 2/3) \).

Step 2: Reflection and Continuing to \( D \)

Next, we adjust the aim for the next horizontal direction reflecting downward hitting \( CD \):

  • It will reflect down to reach the point \( (1, 0) \), which is point \( C \).

Step 3: Straight path to vertex \( C \)

Moving from \( (1, 0) \) to \( D(1, 1) \):

  • It is straightforward from \( C(1, 0) \) to \( D(1, 1) \) this can be calculated as: \[ \text{Horizontal distance from } B \rightarrow C = 1 \]

Finally, we calculate the total distance traveled by the laser beam:

  • Distances covered:

  1. From \( A \) to \( X \):

    • \( 1 - \frac{2}{3} = \frac{1}{3} \)

  2. From \( X \) to the reflection at \( (1, \frac{2}{3})\): \[ \sqrt{ (1 - 0)^2 + \left(\frac{2}{3} - \frac{2}{3}\right)^2 } = 1 \]

  3. From \( (1, 2/3) \rightarrow (1, 0) \):

    • Just skipping down fully, which is another \( \frac{2}{3} \)

  4. From \( C \) to \( D \) = 1 final square unit path.

Total Length Calculation

Sum all path lengths:

\[ \frac{1}{3} + 1 + \frac{2}{3} + 1 \]

Adding these gives:

\[ 1 + 1 + 1 = 3 \]

Final Result

The total length of the laser beam's path is \[ \boxed{3} \]