I am a laboratory man dealing with random medicine.
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?
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.
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)?