## Friday, September 2, 2011

### A Water and Oil Puzzle

I think we may have had this puzzle in the past. However as this is one of the most popular puzzles all around the world, I decided to discuss it again. With an extra twist.

A wise man told me this week that he loves to observe how in math simple problems and their solutions can be extended to provide answers to a far more complex problems. His example was a famous water and oil puzzle.

Take a jar of water and a jar of oil. Both jar are identical and have the same amounts of liquid. Pour 1 cup of water from the water jar to the oil jar. Now, take 1 cup from the jar containing the mix of water and oil and pour it into the water jar. Do you have more water in the oil jar or oil in the water jar now? Please explain your solution.

Now, imagine you go to the kitchen to cook dinner and your kids discover the oil and water jars. They keep playing with them, taking a bit from one jar and pouring into another, repeating it again and again with various amounts of liquid (using toy cars and Strawberry Shortcake cups as water containers). Miraculously they manage not to spill anything.

When dinner is ready you see that both jars have the same total amount of oily substance. Now, do you have more water in the oil jar or oil in the water jar?

Answer ideas accepted any time until midnight on Saturday September 3rd (EST), on our Family Puzzle Marathon. They will be hidden till then and everyone who submitted a valid solution will get a puzzle point.

Tom said...

Too many unknowns for me to be comfortable at first. Starting quantities matter, time matters, size of the jars can matter if they are small, and specific gravity matters. Or they seem to matter.

“Do you have more water in the oil jar or oil in the water jar now? Please explain your solution.” More than before? Yes, surely. More than the other? Let’s see.

If we consider the issue of specific gravity (density), the oil will rise above the water level, and water will sink beneath the oil; they don’t mix very well, especially if given some time. After pouring the first cup of water into the oil, the water will rather quickly sink to the bottom. (If the oil jar was smaller than 2 cups, some oil will now spill away and be lost, and that would very much affect what comes later. Let us assume the puzzle doesn’t want spillage, and both jars => 2 cups capacity.)

If there was ONLY one cup of water to begin with, that water jar could now be empty. If there was 1.5 cups or water at the start, half a cup remains. That starting quantity matters.

So there is one cup of water in the oil jar, but we don’t know how much oil is there, except given it is more than one cup.

Taking the next cup from the “mix” of oil and water, that liquid could be almost entirely oil if we are very careful to skim from the top. But if we just reach in at this time, it’s quite impossible to know what this second cup will contain, depending on how deep we dig and how fast we move; it could contain almost no oil at all, could be around 50/50 or anything else.

Now pouring this second cup (of unknown mix) back into the water jar which contains an unknown quantity of remaining water….really quite impossible to know the percentages now. There COULD be, now, one cup of water in the oil jar, but maybe less; and there COULD be one cup of oil in the water jar, or maybe less.

But we are not asked for the percentages in the mixture! Come to think of it, if we catch some water on the second transfer (say one ounce), it will also displace oil, even though they don’t MIX. That will leave this much water in the oil jar (one cup less one ounce), and this much oil in the water jar (one cup less one ounce). They’d be equal transfers after this first pair or transfers! Very Cool.

Feels like we’d get less equality and more random results after a series of transfers, since the fluids don’t mix well. However the same displacement “rule” will hold while the kids play. If the jars are level after each pair of transfers (no spills and no loss), then the amount of water in the oil jar must equal the reverse.

And IF finally, after the kids have done their play, there IS an equal amount of oil in each jar, seems both jars must be 50% oil and 50% water if they were equal at the start. Wouldn’t HAVE to end up at 50%, of course.

Great fun with this, thank you.

anne-marie said...

Oil and water do not mix.
If I take one cup of water from the water jar and put it in the oil jar, the oil will stay at the bottom because it is less dense than water. So, if I take one cup of substance from the second one, there is a big chance that I will take a cup of oil that I will add to the first jar of water.
After the kids play with it, I'll say that there isa good chace that both contain the same amount of oil and water.

Ilya said...

Let's say both jars have equal volume (V) of liquid after a number of exchanges between them. One of them has X volume of oil. That means the water volume is V-X. In the other jar, we have V-X of oil (since we originally started with V of both oil and water and nothing has been spilled), and X of water for the same reason. Therefore the amount of water in one jar is always the same as the amount of oil in the other.

Maria said...

You are all right - some intuitively and some with the precise math explanation. The amount of water in the oil jar will be equal to the amount of oil in the water jar.

Let's try some specific example. Say, we ended up with 1/3 of water in the oil jar. As jar is full, we have 2/3 of oil in it. So the ratio of water-to-oil in this jar is 1/2.

What about the other jar, the one that was originally filled with the water? If 1/3 of water was taken out of it, then 2/3 are left. And as the jar is full, then the remaining 1/3 is oil. So, the ratio of oil-to-water in this jar is 1/2. Same as water-to-oil ratio in the other jar.

And it doesn't matter how many times we pour the stuff.

For a general case Ilya provided a great brief explanation. Puzzle point for everyone.

Have a great Labor Day weekend!