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Problem of the Day #132: Albert’s Lasers July 29, 2011

Posted by Billy in : potd , trackback

Albert has just finished meticulously setting up a row of $100$ lasers for an advanced optics project, all of which are initially pointed upwards. While Albert is taking a break from his many hours of work, Sreenath stumbles upon the lasers and, not knowing why they are there, decides to play with them. For each laser, Sreenath randomly chooses to turn it $60^\circ$ to the right, $30^\circ$ to the right, $30^\circ$ to the left, $60^\circ$ to the left, or not to turn it at all. This causes the beams of some of the lasers to intersect. For example, if Sreenath turns one laser to the right $30^\circ$, it will intersect all other lasers to the right except those already pointing $30^\circ$ or $60^\circ$ to the right. Albert’s Rage Factor when he returns from the break is equal to the number of pairs of lasers that intersect. What is the expected value of Albert’s Rage Factor?

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