[**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 white objects (and no other objects) and jar B contains black objects (and no other objects.) We transfer 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 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 objects from one jar to another. So, initially, in jar W,

# of white objects = , and # of black objects = .

Also, in jar B,

# of white objects = , and # of black objects = .

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

# of white objects = , and # of black objects = .

Also, in jar B,

# of white objects = , and # of black objects = .

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

# of white objects = , and # of black objects = .

Also, in jar B,

# of white objects = , and # of black objects = .

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 = .

The above is the* first* invariant.

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

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

## 3 comments

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April 15, 2008 at 9:35 am

SamanthaWell, I have taken quite a close look on your website and I must say that I find it extraordinarily interesting.

April 16, 2008 at 6:32 pm

Todd TrimbleI too find it extraordinarily interesting :-)

I’ve not looked at the book Mathematical Circles, but I’ve heard of it through a review by Andrei Toom, who strikes me as a very interesting mathematician and teacher, with a lot of thought-provoking ideas on the ills of mathematics education in America.

Regarding the solution, it assumes that the amount of milk in one glass is the same as the amount of tea in the other (before we do any mixing), but the problem as stated doesn’t assume that. I think maybe it had better!

April 16, 2008 at 7:39 pm

Vishal LamaYes, our website(?) is extraordinarily interesting! :-)

Indeed, I came across Andrei Toom’s web page as well as his thought-provoking ideas/articles on mathematics education, earlier. His ideas certainly resonate with me! You may definitely want to take a look at

Mathematical Circles. In the US, the book is used in almost all ‘math circles’ (e.g. it is included in the list of recommended books at the Berkeley Math Circle.) Though it is mostly used as an introductory book – I do feel ‘book’ is not the correct word – to expose students (interested in math olympiads) to a whole bunch of important ideas/principles in problem-solving, it can/should be used by regular university students to improve their own problem-solving skillsat home. There is much that can written about it, but I will save it for another time!And, you correctly pointed out that it should be explicitly mentioned that the amount of tea in one glass is the same as the amount of milk in another glass, though that condition is implicitly assumed to hold. In fact, there is also another problem with mixing tea and milk: volumes may not add up when we mix two liquids,

i.e.from elementary chemistry we know that volumes don’t necessarily follow the additive linear law! This is the reason why I think that working with the analogous discrete problem helps to throw light on what the original problem really meant us to ‘discover’.