Question

Suppose you are given the task to labelthreeunlabeled pill bottles. One of the bottles contains a placebo instead of pills with the active ingredient. Each bottle contains between 18 and 20 pills. Pills with the active ingredient weigh 1 gram each while the placebo pills weigh 0.9 grams each. You have at your disposal a digital scale that is accurate to tenths of grams. (The scale has been calibrated to automatically remove the weight of the bottle from the weight displayed. Thus, you can safely concentrate on the weight of the contents alone.)

1) Outline the method you would follow to discover which bottle contains the placebo. Your method should resort to physically counting pills in a bottle only in the worst case considered.

2) Use explicit enumeration of the possible outcomes, combinatorial arguments or simulation to find / estimate the number of times you must resort to physically counting pills. How often must the pills in more than one bottle be counted?

Answer 1

First, arrange the bottles on shelf and now take, 1 pill from the first bottle, 2 pills from the second bottle, 3 pills from the third bottle, and so on.

Ideally the weight should be 1*1gram + 2*1gram + 3*1gram = 6grams (if all the bottles had the active pill)
but if the weight turns out to be 5.9, that would mean that the first bottle from which we took 1 pill contains the placebo. Similarly, if the weight is 5.8, the placebo is in second bottle and if the weight is 5.7, the placebo is in the third bottle.