You have two two-liter bottles, one (marked W) containing one liter of pure water and one (marked A) containing one liter of pure alcohol. The two things you can do are to pour liquid from either bottle down the drain or pour liquid from one bottle into the other, but you must always keep at least one liter of liquid in bottle A.

What is the minimum alcohol concentration that can be achieved in bottle A? Assume that there is no volume change on mixing water and alcohol.

Clearly you want to continue to pour out as highly concentrated alcohol mixture as you can while still (a given) maintaining no less than 1 L in the A bottle.  If you put in a bit, mix and dump out that same amount, you might just optimize this.  You would do this until you ran out of water to dilute with.  In the limit, you would be adding and dumping infinitesimally small amounts an infinite number of times.  in this case I believe the answer is (100/e)% or approximately 36.79% alcohol.  Other schemes,  such as adding and dumping 1/2^n amounts, give an inferior result. I suspect we can do no better.

Posted by Kenny M on 2025-02-03 17:03:57