Mad Mix

Imagine we have two large reservoirs of fluids. One contains the liquid ‘supertox’ and one contains purified water. We want to get certain proportional mixtures of the two.

There is also a drain for waste. Supertox is very dangerous, however, so we’d like to pour as little down the drain as possible, even in dilute form. Supertox is polar, so supertox and water mix perfectly and instantly whenever one is poured into the other.

In addition, we have two opaque measuring vessels, one of 10 liters and one of 7. We call these the 10-vessel and the 7-vessel for short. There are no measurement lines on the vessels. When we fill a vessel from a reservoir, we fill it from a faucet, as our client is nervous about contamination from handling.

So we can fill a vessel to completion from a reservoir or we can pour liquid from one vessel to the other, stopping either when the receiving vessel is full or the source vessel is empty. We are allowed to pour pure water back in the water reservoir and to pour pure supertox back into the supertox reservoir.

Warm-Up

We want to create a 10-liter solution having 40% of supertox without pouring anything down the drain. How do we do it?

Solution to Warm-Up

Fill the 7-vessel with supertox. Pour into the 10-vessel. Fill the 7-vessel with supertox again. Pour into the 10-vessel until the 10-vessel is full. So we have 4 left in the 7-vessel. Pour the contents of the 10-vessel back into the supertox container. Pour the contents of the 7-vessel into the 10-vessel, and then fill the rest of the 10-vessel with water.

1. We want a mixture having 2.7 liters of supertox in a 3-liter mixture. We don’t want to pour more than 10 liters of any liquid laced with supertox down the drain. How can we achieve this?

2. Can we get a mixture that is 2/9 supertox and the rest water without pouring any supertox down the drain?

Now, chemists require extremely precise mixtures, so let me explain our terminology with an example. A 25% concentration of supertox in the 10-liter container would correspond to 2.5 liters of supertox and 7.5 liters of water.

Also, we have a big empty bucket at our disposal that can hold over 100 liters, though we don’t know exactly how big it is.

3. Can we get a 26% concentration of supertox in the 7-liter container?

4. Suppose that we want to get mixtures which are all supertox, 1/2 supertox, 1/3 supertox, 1/4 supertox, … 1/50 supertox. You are allowed to specify the size of the two vessels. You may even include fractional sizes as long as the smallest vessel contains at least one liter and the largest contains at most 30 liters. In addition, you have the big bucket again with capacity over 100 liters. Which two vessel sizes would you choose in order to obtain each of these mixtures? Given those vessel sizes and given a target mixture, how would you obtain it?