South Carolina High Energy Mathematics Teachers' Circle

Problems to Chew on at Meals (stolen kindly from Paul Zeitz)

Problem 1
Consider the following diagram. Can you connect each small circle on the top with its same-letter mate on the bottom with paths that do not cross one another, nor leave the boundaries of the large circle?

Problem 2
Factor x4 + x2 +1.

Problem 3
You are in the downstairs lobby of a house. There are 3 switches, all in the "off" position. Upstairs, there is a room with a lightbulb that is turned off. One and only one of the three switches controls the bulb. You want to discover which switch controls the bulb, but you are only allowed to go upstairs once. How do you do it? (No fancy strings, telescopes, etc. allowed. You cannot see the upstairs room from downstairs. The lightbulb is a standard incandescent 100-watt bulb.)

Problem 4
For 10 days, you must take one A pill and one B pill at noon. Otherwise, you die. If you take too much or too little medicine, you will die. The pills are indistinguishable! All goes well until day 3. On this day, you shake one A and TWO B pills into your palm.

Can you survive? If so, HOW?

Problem 5
You are locked in a 50 × 50 × 50-foot room which sits on 100-foot stilts. There is an open window at the corner of the room, near the floor, with a strong hook cemented into the floor by the window. So if you had a 100-foot rope, you could tie one end to the hook, and climb down the rope to freedom. (The stilts are not accessible from the window.) There are two 50-foot lengths of rope, each cemented into the ceiling, about 1 foot apart, near the center of the ceiling. You are a strong, agile rope climber, good at tying knots, and you have a sharp knife. You have no other tools (not even clothes). The rope is strong enough to hold your weight, but not if it is cut lengthwise. You can survive a fall of no more than 10 feet. How do you get out alive?