Jamshrivelson

Question: One morning, Phil was playing with his daughter, who loves to cut paper with her safety scissors. She especially likes cutting paper into “strips,” which are rectangular pieces of paper whose shorter sides are at most 1 inch long. Whenever Phil gives her a piece of standard printer paper (8.5 inches by 11 inches), she picks one of the four sides at random and then cuts a 1-inch wide strip parallel to that side. Next, she discards the strip and repeats the process, picking another side at random and cutting the strip. Eventually, she is left with nothing but strips. On average, how many cuts will she make before she is left only with strips? Extra credit: Instead of 8.5 by 11-inch paper, what if the paper measures $m$ by $n$ inches? (And for a special case of this, what if the paper is square?) [Solution]

Question: The Robot Weightlifting World Championship was such a huge success that the organizers have hired you to help design its sequel: a Robot Tug-of-War Competition! In each one-on-one matchup, two robots are tied together with a rope. The center of the rope has a marker that begins above position 0 on the ground. The robots then alternate pulling on the rope. The first robot pulls in the positive direction towards 1; the second robot pulls in the negative direction towards $-1.$ Each pull moves the marker a uniformly random draw from $\left[0,1\right]$ towards the pulling robot. If the marker first leaves the interval $\left[-\frac12,\frac12\right]$ past $\frac12,$ the first robot wins. If instead it first leaves the interval past $-\frac12,$ the second robot wins. However, the organizers quickly noticed that the robot going second is at a disadvantage. They want to handicap the first robot by changing the initial position of the marker on the rope to be at some negative real number. Your job is to compute the position of the marker that makes each matchup a $50:50$ competition between the robots. Find this position to seven significant digits—the integrity of the Robot Tug-of-War Competition hangs in the balance! [Solution]

Question: Hames Jarrison has just intercepted a pass at one end zone of a football field, and begins running — at a constant speed of $15$ miles per hour — to the other end zone, $100$ yards away. At the moment he catches the ball, you are on the very same goal line, but on the other end of the field, $50$ yards away from Jarrison. Caught up in the moment, you decide you will always run directly toward Jarrison’s current position, rather than plan ahead to meet him downfield along a more strategic course. Assuming you run at a constant speed (i.e., don’t worry about any transient acceleration), how fast must you be in order to catch Jarrison before he scores a touchdown? [Solution]