Suppose a standard 8x8 chessboard has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2x1 so as to cover all of these squares?

A group of miners in Colorado in the late 19th century are working for a ruthless mining company. The mining company dictates that each day the miners should have exactly 30 minutes for lunch. On top of this, the miners are not provided watches or time keeping devices since the company owners believe they will just get destroyed when the miners inevitably blow their arms off with dynamite or die in some other horrific mining accident. Instead, the miners are given two pieces of rope, but it is no ordinary rope, this rope has the special property that when burned it acts as a timer. Of the two pieces they are given, one piece of rope burns for 15 minutes and the other for 45 minutes. Although the rope acts as a timer, the rope is unpredictable in so much as it does not burn linearly with respect to length and time. Another way to think about this is that if you cut the 15 minute rope into thirds, each third would not burn for 5 minutes, it would instead burn for an indeterminate amount of time. How can the miners use a lighter and these ropes to make sure their lunch is only 30 minutes long?

Suppose you're in a hallway lined with 100 closed lockers. You begin by opening every locker. Then you close every second locker. Then you go to every third locker and open it (if it's closed) or close it (if it's open). Let's call this action toggling a locker. Continue toggling every nth locker on pass number n. After 100 passes, where you toggle only locker #100, how many lockers are open?