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Milk and Coffee

This week we will have only a logic/math puzzle. Here it is.

Question

You have a cup of coffee and a cup of milk.You pour a spoon of milk into the coffee.Then you pour a spoon of the diluted coffee back into the milk.Which is more: coffee in the milk cup or milk in the coffee cup?

Note that the cups are not necessarily the same size.

Scroll down for the answer.

Answer

They are the same.

Hard Solution

Let’s use algebra for the hard solution. Let’s call:

c = coffee in coffee cup initially.

m = milk in milk cup initially.

s = volume in a spoon.

Let’s make a table for how much of each in each cup. The left column shows the cup and the top row shows the liquids.

Cup Coffee Milk Total
Coffee c 0 c
Milk 0 m m
Total c m c+m
 

Here is the table after the first move.

Cup Coffee Milk Total
Coffee c 0 c
Milk s m-s m
Total c+s m-s c+m
 

Here is the table after the second move.

Cup Coffee Milk Total
Coffee c2/(c+s) cs/(c+s) c
Milk cs/(c+s) m-s+s2/(c+s) m
Total c m c+m
 

Note that milk in the coffee cup = coffee in the milk cup = cs/(c+s).

Easy Solution

After the second move, there is some amount of coffee in the milk cup. Let’s call is x. The milk cup has the same total liquid as it did initially. So, that coffee in the milk cup had to displace the same quantity of milk. Where else could that displaced milk go except for into the coffee cup?

This puzzle is asked and discussed in my forum at Wizard of Vegas

It is based on the video “Can you solve the milk and coffee mixing puzzle?” from the Mind Your Decisions YouTube channel (one of my favorites). In that video, the two cups are the same size.