I have taken Doorknob hostage in my cellar. He is "perfectly" trapped - solid walls, solid floor, solid roof, no windows, etc. The only way out is a steel door and the only way to unlock the door is by means of two-pan balance scale (accurate to the quacogram).
I have provided 100 weights. Each weight will have its own (non-zero positive integer) weight (in quacograms) accurately written on it. He must put a non-zero number of weights on either side of the scale pans such that they balance perfectly (no trickery on the weights allowed - just the weights). Doorknob has no computer or writing implements - also he is is always able to somehow manoeuvre the weights no matter their weights.
- How might I select the weights' weights such that he may NEVER escape?
- How might I minimise the total weight? (having achieved 1.)
- What is the lowest highest weight that a weight must have? (having achieved 1. and 2.)
- How much wood must the woodchuck chuck? Just kidding :D
As always explanations/"proofs" are maybe more important than the figures themselves.