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Re: physics/pedagogy of coffee-mixing



The question is, why is this puzzle more difficult than it ought to be?
First my take on that... then my solution to the puzzle.

* * * *

First, let me remark that this puzzle has been around for a long time. I
first heard it in the 1960s, and I assume it was already old by that time.
I have used this puzzle many times over the years. Based on the answers
people have given me, I can tell you what I think are the most common
mistakes. It's interesting that the two most common errors give opposite
results.

Let's assume the first transfer is from coffee to tea.

(1) The solver concludes we end up with more coffee in the tea, because the
coffee-to-tea transfer moved pure coffee, but the tea-to-coffee transfer
moved diluted tea. So the two transfers moved more coffee to the tea cup
than they moved tea to the coffee cup. This, in fact, is true... but the
final conclusion is wrong. This solver recognized some coffee comes back
during the second transfer, but failed to consider that this
less-than-one-teaspoon of tea was transferred into less-than-one-cup of
coffee.

(2) The solver assumes the second transfer is pure tea. One this basis, the
one teaspoon of pure coffee was put into a full cup of tea during the first
transfer. But the second transfer put one teaspoon of pure tea (or
"near-enough" pure tea) into a cup of coffee that was not a full cup.
Therefore, the concentration of tea in the coffee ends up higher. This
solver recognized the second transfer was placed into a less-than-full cup
of coffee, but did not recognize the teaspoon was less-than-pure tea, or
perhaps concluded the coffee in the teaspoon was too small to be of any
consequence.

I find solvers give me both answers, (1) more coffee in tea, (2) more tea in
coffee, with equal frequency based upon the two types of reasoning described
above. The correct answer, that the concentration of tea in coffee is
exactly equal to the concentration of coffee in tea, is rarely given. I
think this is because approximation doesn't work well for this type of
problem. Suppose the solver correctly see both pitfalls mentioned above,
and has the inkling the mixtures might end up being the same ratio. At this
point the inkling is just that... a guess. They guess the two features
mentioned above, which lead to opposite conclusions, might
offset each other. Moving from a guess to a concrete answer requires
quantitative reasoning (using numbers). Unfortunately most people are very
shy of using numbers when they solve brain teasers.

* * * *

BTW, I do not think it works well to discuss balls, marbles, etc. because
they are too large and/or there are not enough of them. If we use that type
of discrete math to solve this problem we end up saying things like "...
this teaspoon contains one or two balls of..." when we really need to say
0.98 balls, etc. We need to be more realistic and think of a huge number of
small molecules. Stated better... It is reasonable to enumerate the
teaspoons in a cup, but it is not reasonable to enumerate the molecules of
tea or coffee... rather, we should speak of these as concentrations
expressed as percentages or ratios. Here is how I explain the problem.

Suppose there are 50 teaspoons in a cup. (This is approximately correct...
5 ml per teaspoon, 250 ml per cup.) The transfer of coffee to tea leaves 49
teaspoons of coffee in the coffee cup, and 51 teaspoons of mixture in the
teacup. The mixture in the tea cup is 50/51 tea and 1/51 coffee. When the
second teaspoon of mixture is removed from the "teacup" the teacup returns
to 50 teaspoons total, and the ratio remains 50/51 tea and 1/51 coffee.
That means the coffee in the tea is... 1/51 (ratio) times 50 teaspoons
(total volume) = 50/51 of a teaspoon of coffee in tea.

The teaspoon carried to the coffee cup brings that total volume back to 50
teaspoons. But that teaspoon was 50/51 tea and 1/51 coffee. At this point
you can continue to find the ratios in the coffee cup if you want, or you
can immediately see the answer. The immediate insight is to read the second
sentence of this paragraph again. The second transfer transferred 50/51 of
a teaspoon of tea into the coffee cup (which is brought back to full). The
full teacup has already been established to contain 50/51 teaspoon coffee.

Conclusion... the amount of tea in the coffee, and the amount of coffee in
the tea are exactly the same. In this example, there is 50/51 teaspoon
coffee mixed into the teacup, and there is 50/51 teaspoon tea mixed into the
coffee cup.

Those people who would like to chastise me for using specific numbers to
yield a "proof" can replace 50 with X, 51 with X+1, 49 with X-1,


Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817