This problem comes from a book called Enigma (Fabrice Mazza and Sylvain Lhullier), that was gifted to my stepfather (a maths teacher) recently. A particularly deluding puzzle is one entitled 'Water In The Wine.' Although I think it is well known, I've never actually tried to solve it before. The problem start:
"You have two completely identical tankards; one contains 150 ml wine, the other 150 ml water. You take a spoonful of liquid from the tankard of water, empty it into the tankard of wine and mix it in well. Then, using the same spoon, you take a spoonful of this mixture and empty it back into the first tankard of water. Thus each tankard once again contains 150ml liquid. Is there more water in the wine, or more wine in the water?"
All most everyone would follow their intuitive instinct, which might say something like 'Surely a full teaspoon of water was put in the wine, yet less wine was taken back to the water? Of course the aim of this puzzle is to trick the puzzler in exactly this manner, and as perhaps some clever chicken reading this has already spotted, the intuitive statement is true. But it's not the whole truth. You see a full teaspoon of water was poured into the wine, but it didn't stay there, since, when less wine was taken, the rest of the teaspoon was taken up by some of the water. In Enigma, the solutions a the back explains the phenomenon such:
"Suppose the tankard contains 80% water and 20% wine at the end of the process. The missing 20% of water can only be in the other tankard just as the missing 80% wine must also be there..."
Of course, the power of intuition is mighty indeed, and these explanations were still not enough to conquer the minds of my family members. So I devised a few demonstrations in order to attempt to use the art of persuasion to sway their attitudes. Fortunately I had an impressionable youth handy, and acquiring there assistance in insistence was not difficult. The first method of demonstration was a geometrical one.
Fig. 32 Geometry of the Water in the Wine
This diagram represents the two vessels after the water from the first to the second. The first vessel contains 150 ml - 1 tsp of water, and the second contains 150 ml wine + 1 tsp water. The division on the right is the tsp of liquid that will be poured back into the first vessel. Since it is equal in volume to the amount of water in the vessel and the small portion of water in the top right corner is common to the two, the volume of water left in the vessel and the volume of wine taken to be put in the first vessel must be equal.
While I may have convinced some, there are still a few who hold onto the idea that my logic is missing something. Such was the reaction of my subjects who continued their heated objection to my so obviously flawed logic. My next approach seemed a little more powerful though, and it was this that converted my younger sibling to the cause. It is a diagrammatic approach that gets to the roots of the problem by representing the liquids as a group of smaller volumes. These can be likened to the molecules that make up the substances, which adds a sense of fundamental truth to an already convincing argument, if I do say so myself (which I do!).
Fig. 33 Particles of the of Water in the Wine
Again, this diagram shows the result of the first transfer. Since an equal amount is taken back as was put in, we must take three of the particles from vessel two and put them into the first. If we take three wine particles there will be three of each the wrong vessel. If the three water particles are taken back, the starting state is reached in which no water is in the wine, and no wine is in the water. More likely, some of each is taken. If two wine particles are taken with one water particle, there will be two particles of each left in the wrong vessel. In fact, the less the number of wine particles taken, the more water is taken back and therefore the less that remains in the wine. This has the added advantage of showing that no matter how much stirring occurs, there will still be equal amounts in the wrong vessels.
Despite a couple of protestations my subjects were fairly convinced at this stage (with some aid from my younger sibling). To push my advantage I adapted this concept to rigorously prove the result beyond all doubt. How you ask? With ... (drumroll)... Algebra! First we establish the volumes
Let the volume of Vessel 1 be V ml, and the volume of Vessel 2 be W ml
Now, we perform the transfers.
Take p ml of water from Vessel 1 and pour it into Vessel 2.
Now Vessel 1 contains V - p ml water and Vessel 2 contains W ml wine and p ml water
If we take, in a teaspoon, q ml wine and p - q ml water from Vessel 2, we have taken a total of q + (p - q) = p ml.
Now Vessel 2 contains W - q ml wine and p - (p - q) = q ml water.
Pouring the contents of the teaspoon into Vessel 1 gives (V - p) + (p - q) = V - q ml water and q ml wine.
∴ There is q ml wine in the water = q ml water in the wine.
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