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[Update: Look for another (slicker) solution I found after coming up with the first one.]

My friend, John, asked me today if I had a solution to the following (well-known) problem which may be found, among other sources, in Chapter Zero (!) of the very famous book, Mathematical Circles (Russian Experience).

Three tablespoons of milk from a glass of milk are poured into a glass of tea, and the liquid is thoroughly mixed. Then three tablespoons of this mixture are poured back into the glass of milk. Which is greater now: the percentage of milk in the tea or the percentage of tea in the milk?

Note that there is nothing special about transferring three tablespoons of milk and/or tea from one glass to another – the problem doesn’t really change if we transfer one tablespoon of milk/tea instead, and that there is nothing special about transferring “volumes” – we could instead keep a count of, say, the number of molecules transferred.  We may, therefore, pose ourselves the following analogous “discrete” problem whose solution provides more “insight” into what’s really going on.

Jar W contains $n$ white objects (and no other objects) and jar B contains $n$ black objects (and no other objects.) We transfer $k$ objects from jar W to jar B. We then thoroughly mix –  in fact, we don’t have to – the contents of jar B, following which we transfer $k$ objects, this time, from jar B to jar W. Which is greater now: the percentage of black objects in jar W or the percentage of white objects in jar B?

Solution 1: Let us keep track of the number of black and white objects in both the jars before and after the transfers of $k$ objects from one jar to another. So, initially, in jar W,

# of white objects = $n$, and # of black objects = $0 \,$.

Also, in jar B,

# of white objects = $0 \,$, and # of black objects = $n$.

Now, we transfer $k$ objects from jar W to jar B. So, in jar W,

# of white objects = $n-k$, and # of black objects = $0 \,$.

Also, in jar B,

# of white objects = $k$, and # of black objects = $n$.

Finally, we transfer $k$ objects from jar B to jar W. Let the number of white objects out of those $k$ objects be $k_1$. Then, the number of black objects transferred equals $k-k_1$. Therefore, now, in jar W,

# of white objects = $n-k + k_1$, and # of black objects = $k-k_1$.

Also, in jar B,

# of white objects = $k-k_1$, and # of black objects = $n - (k-k_1)$.

From here, it is easy to see that the percentage of black objects in jar W is the same as the percentage of white objects in jar B! And, we are done.

Solution 2: (I think this is a slicker one, and I found it after pondering a little over the first solution I wrote above!) This one uses the idea of invariants, and there are, in fact, two of ’em in this problem! Note that at any given time,

# of white objects = # of black objects = $n$.

The above is the first invariant.

Also, note that after we transfer $k$ objects from jar W to jar B and then $k$ objects from jar B to jar W, the number of objects in each jar is also $n$. This is the second invariant. And, now the problem is almost solved!

Suppose, after we do the transfers of $k$ objects from jar W to jar B and then from jar B to jar W, the # of white objects in jar W is $k$. Then it is easy to see that the # of black objects in jar W is $n-k$ (using the second invariant mentioned above.) Similarly, using the first invariant, the # of white objects in jar B = $n-k$. Therefore, using the second invariant again, the # of black objects in jar B = $n - (n-k) = k$. And, from this the conclusion immediately follows!

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