A triangular area is divided into 25 areas, each an equilateral triangle the same size and shape as the others. You need to set up a security system with one beam of laser light that shines through every area. Set up a projector, a receiver, and several mirrors to accomplish each of the five challenges set out below.

Mirrors: The mirrors are small, just big enough to reflect the beam. However, our diagrams show the mirrors being as long as one side of a small triangular area. This was done to help you understand how the mirrors are set. Consider the mirrors to be reflective on both sides if that helps your solution.

Beam: The beam is allowed to cross itself and to cross the same area more than once. To get credit for covering one of the areas you must have the beam pass all the way through the area from the middle of one side to the middle of another side. This means that the beam will travel through the area in a direction parallel to one of the three sides of the large triangle.

Consider this example to help understand how the system can be set up. The arrow represents the projector. The circle represents the receiver. The heavy black lines are mirrors. The beam passes through the middle of each area. Eight mirrors are used.

Challenge 1: Cover all 25 areas using only seven mirrors.

Challenge 2: Cover all 25 areas such that the beam enters each area only once and never crosses itself.

Challenge 3: With two mirrors already placed as shown in the diagram, cover all 25 areas by adding only 6 mirrors.

Challenge 4: The shaded areas in the diagram represent solid columns that the beam can’t pass through or bounce off of. Use only 6 mirrors to cover all 22 remaining areas.

Challenge 5: Accomplish Challenge 4 without having the beam travel outside of the large triangular area.