## Friday, October 21, 2011

### Magical fragrance - how to share

Fairy god mother granted two of her favorite fairies a bottle of magical fragrance pictured below.  She commanded them to split the fragrance equally.

As fairies were in a state of war with each other they were not keen on sharing the fragrance daily.  So they decided that one of them will use half of the fragrance and pass over the bottle with the remaining half to the second fairy.  The bottle is semitransparent but not symmetrical in either horizontal or vertical direction.  How can the second fairy verify whether the amount she gets is indeed half of the original amount?  The bottle was completely full to start with. No measuring marks are present on the bottle and fairy doesn't want to pour any drops of the magical liquid out.

Answer ideas accepted any time until midnight on Saturday October 22nd (EST), on our Family Puzzle Marathon. They will be hidden till then and everyone who submitted something reasonable will get a puzzle point.

Tom said...

And in fact, how could the First Fairy know when she'd used half, and only half? I don't know but of course fairies are magic so they could know. Or else the fairy god mother could employ her own magic and just tell them when to swap. Or in fact, since the liquid itself is magical, the liquid could just inform the fairies.

The puzzle states that the Second Fairy doesn't want to pour our any drops, but the puzzle does not forbid the First Fairy from doing so; he/she is permitted (by this puzzle) to do some direct measuring -- siphon it all out, split it, and so forth. (Are all fairies female?)

Or, they could take turns, which is not expressly prohibited by the puzzle. Inequality could still result, however.

Eyeballing it seems not to work. Weighing it seems out, or a bouyancy scheme. Got me. Must be the magic!

Tom said...

More on fairies and fairness. More answers occur to me.

My younger brother and I would often split the last remaining wedge of pie or cake by agreeing, "you cut and I'll choose," or vice versa. While it was not a perfect solution to the millimeter, we were usually very close and it was fair enough. So the fairies could override their desire not to pour any out, split it as we did, and then they simply argue over who gets to keep the pretty bottle.

Another solution, not expressly prohibited by the puzzle, is to share the fragrance over time. #1 gets it today, #2 gets it tomorrow, back and forth; or week by week. Not perfect but perhaps close enough.

Reading more carefully, we are not asked to design a way to achieve an equal split. The Fairies are commanded, but we are not. It's on the fairies, it's their problem! Our challenge is to answer, "How can the second fairy verify..." A legitimate answer could be, "She cannot."

And fairy #2 gets to keep the pretty bottle anyway, probably, or return it for a deposit.

Ah, and since all fairy godmothers hate war, perhaps this is a clever scheme to spur a negotiated peace process. That always works! It's magic.

Ilya said...

Here is one way to find out involving use of some sort of a marker (or a rubber-band). Mark the water level line with a marker on both sides of the bottle (or a rubber-band around it). Then turn the bottle upside down carefully and attempt to line up the water level with the marked line. If it is possible to line it up closely then the bottle is about half-empty (or half-full if you are an optimist!).

The proof is easy: if it is possible to line up on both side when upside down, then the volume of water must be equal to the volume of air (which used to occupy the same space when in upright position).

Turn the bottle upside-down (secure the lid!) and see if the perfume level sits at the same level. It will only if the bottle is half full.

Jerome said...

There are ways of doing this problem if the shape is 2 dimensional, but I take it that we are looking at a 3 dimensional object.

The problem has a widespread problem dating back to about 1850 when Lord Kelvin first proposed it.

He discussed symmetrical shapes and simple ones at that. Here is one of the references. I think it is an original publication.

http://zapatopi.net/kelvin/papers/on_the_division_of_space.html

Here is another which is slightly less complex.

http://zapatopi.net/kelvin/papers/on_homogeneous_division_of_space.html

I can think of no easy way to do this problem except by symmetrical objects like a cylinder. To get 1/2 the volume of a cylinder without makes is easy: just tip the cylinder until the base is just covered and the half volume extends to the lip of the container at the same time. I don't think any such easy solution exists for just any container.

I'm wondering if, since we are dealing with fairies who are quite magical beings, if we couldn't use a temperature gradient? Calculate the temperature at which the liquid would be half the size it is at room temperature and mark the container at that point. When the perfume reaches that point, give the container to the other fairy. Course a million things are wrong with this idea including the fact that the perfume might solidify. It's really all I can come up with.

Annie said...

Assuming the fairy doesn't have any magical powers to ascertain the amounts in the bottle, I would say she should weigh the bottle when it is full, weigh it when she gets it, and weigh it when it is empty. The weight when she gets it should be half way between full and empty. This method has a few drawbacks such as the first fairy partially filling the bottle up with water and, in either case, it would be too late to recover the used fragrance. (And a more difficult question would be how would the first fairy know when she has used half! I'll have to think about that.)

anne-marie said...

Let's imagine that the bottle is full to the top.
After a couple of time that the first person used it, she can put one finger on the bottle to mark the level of perfume and then, turn the bottle upside down and the level should be at the finger if half of the bottle have been used.

Maria said...

I am awarding a point to everyone who dared to express their opinion, no matter how crazy it was :)

The intended and fascinating solution is marking the level of the liquid, then turning the bottle upside-down and comparing the levels. The important assumption (stated in the puzzle) is that the bottle was originally filled to the top.

Cool trick to teach your kids!

Jerome said...

I don't think the intended solution works. One that might (since these are magical beings) is to convert the shape to a symmetirical one where it would be easy to tell where the 1/2 way point is.

GB said...

I think you can find a measuring cup, say 2 cups and fill it 1 cup full. Immerse the bottle in the water. Find out the displacement. Estimate the thickness of the bottle what percentage of the displacement the bottle represents. Subtract the bottle displacement from the estimation of the bottle thickness and you get the contents. Divide by two to get the amount each one gets.

Tom said...

Yes, by gosh, Maria and others are right again. I overlooked the possibility that the fairies were allowed to mark a line on the bottle. At 50/50 the liquid level will be the same upside down. That is, if there's no cheating. Do fairies cheat?

Jerome said...

In my past lifetime, I taught Physics and I couldn't get this.

Shame!!!! (on me)

On a more serious note, this is only the second question I have seen that seems to be gender specific. Maria, you have asked the other one. As a bonus question, sometime during the term I would ask my class to explain acceleration to an 8 year old.

The guys never got it right. (Or very very seldom.) They'd go into m/s^2 and change in velocity (actually using the word) never once thinking that the average 8 year old would likely not know how to divide or even what division meant, let alone velocity or worse a change in velocity.

The ladies in my classes (particularly mothers) all hit on the notion of stop and go at a light or stop sign. It was amazing!!!

The same thing here. I think most of the females looked at something quite simple while the rest of us floundered. I'm still not very happy about the answer, but it is the only practical one possible.

This is just a super question.

Jerome said...

One last thought. Simply turning this upside down is really not the best answer. If they are fairies, they should be able to turn the shape into a tumbler that has the same interior diameter on it's closed bottom as it has on the (closed) top. That way they can just keep tipping the container until the liquid just touches the top of the bottom and the bottom of the top (if that makes any sense). Sorry to be so persistent, but I cannot seem to leave the question alone.

Maria said...

Jerome - this is a great idea for a cylinder or box-shaped container. I will talk to the fairies :)

I have been thinking of supplying some snapshots demonstrating the flipping answer but can't find any transparent non-symmetrical container around me. Everything transparent seems to be symmetrical. And non-symmetrical containers (like rainboot) are not transparent.