The Locker Problem (Driscoll, 1999)

1. The Locker Problem (Driscoll, 1999) There are 20 lockers in one hallway of the King School. In preparation for the beginning of school, the janitor closed all of the lockers and put a new coat of paint on the doors, which are numbered from 1 to 20. When the 20 students from Mrs. Mahoney’s class returned from summer vacations, they decided to celebrate by working off some energy. They came up with a plan: the first student ran down the row of lockers and opened every door. The second student started with locker #2 and closed every second door. The third student started with locker #3 and changed the state of every third locker door. The fourth student started with locker #4 and changed the state of every fourth locker door, the fifth student started with locker #5 and changed the state of every fifth locker door, and so on, until all 20 students had passed by the lockers. Which lockers are still open after the twentieth student is finished? Which locker or lockers changed the most? Suppose there are 200 lockers. Which lockers are open after the 200th student is finished? Which locker or lockers changed the most?

2. Something Nu(An Extension to the Locker Problem; Driscoll, 1999) Consider the operation of counting the factors of a whole number. This function is usually called “v” (the lowercase Greek letter for “nu”). For example, the number 6 has factors 1, 2, 3, and 6, so v(6)=4. Here’s some practice: • If the input to v is 5, what is the output? What if the input is 12? • What is v(24)? v(288)? v(23 x 32 x 54)? • Find some numbers that v takes to 6. • Classify all numbers n so that v(n)=3. Classify all numbers n so that v(n)=2. • What can you say about a number m if v(m)=12? • Find two number n and m so that v(nm) = v(n) v(m). Find two more. Compare with what other people have found.