I am a laboratory man dealing with random medicine.
Day 1
I have 10 bottles of pills. 9 of them contain good pills that weigh 1.000g each, and the other one contains bad pills that weigh 0.999g each. There are uncountably many pills in each bottle.
I have a digital scale that is as precise as exactly 1mg, and can measure at a rather high value (even more than several kilograms) while maintaining the precision. How many times do I have to operate the scale to identify the bottle with bad pills?
Day 2
The factory has gone crazy. They send me another 10 bottles of pill, some of them (maybe zero, maybe all) contains bad pills. Weight specifications remain the same as above. I need to find out all bottles of bad pills with the same tool above, operating it for as fewer times as possible.
Day 3
Finally I'm crazy. I have once more 10 bottles of pills. There are two kinds of bad pills today - one of them weighs 1.001g each and the other weighs 0.999g each. How can I identify all bottles into "good", "overweighed" and "underweighed", with minimal use of my only scale (yet precise and powerful)?