Wine/water mixing problem
In the wine/water mixing problem, one starts with two barrels, one holding wine and the other an equal volume of water. A cup of wine is taken from the wine barrel and added to the water. A cup of the wine/water mixture is then returned to the wine barrel, so that the volumes in the barrels are again equal. The question is then posed—which of the two mixtures is purer?^{[1]} The answer is that the mixtures will be of equal purity.
The problem can be solved with logic and without resorting to computation. It is not necessary to state the volumes of wine and water, as long as they are equal. The volume of the cup is irrelevant, as is any stirring of the mixtures. ^{[2]}
Solution
Conservation of substance implies that the volume of wine in the barrel holding mostly water has to be equal to the volume of water in the barrel holding mostly wine.^{[2]}
The mixtures can be visualised as separated into their water and wine components:
Barrel originally with wine  Barrel originally with water 

Wine: V_{0}  Water: V_{0} 
→
Move V_{1} of wine right 

Wine: V_{0} – V_{1}  Water: V_{0} , Wine: V_{1} 
←
Move V_{1} of mixture (comprising V_{2} wine and V_{1} – V_{2} water) left 

Wine: V_{0} – V_{1} + V_{2} , Water: (V_{1} – V_{2})  Water: V_{0} – (V_{1} – V_{2}), Wine: V_{1} – V_{2} 
Purity of wine = V_{0} – V_{1} + V_{2}/(V_{0} – V_{1} + V_{2}) + (V_{1} – V_{2})
= V_{0} – V_{1} + V_{2}/V_{0} 
Purity of water = V_{0} – (V_{1} – V_{2})/(V_{0} – (V_{1} – V_{2})) + (V_{1} – V_{2})
= V_{0} – V_{1} + V_{2}/V_{0} 
To help in grasping this, the wine and water may be represented by, say, 100 red and 100 white marbles, respectively. If 25, say, red marbles are mixed in with the white marbles, and 25 marbles of any color are returned to the red container, then there will again be 100 marbles in each container. If there are now x white marbles in the red container, then there must be x red marbles in the white container. The mixtures will therefore be of equal purity. An example is shown below.
Red Marble Container  White Marble Container 

100 (all red)  100 (all white) 
→
Move 25 (all red) right 

75 (all red)  125 (100 white, 25 red) 
←
Move 25 (20 white, 5 red) left 

100 (80 red, 20 white)  100 (80 white, 20 red) 
In a short essay, the physicist Hendrik Casimir stated that when you mix wine and water, the change in density is nonlinear; therefore, the volume of the mixture is less than the sum of the two original volumes, and any premise of conservation of volume is, strictly speaking, incorrect (as is the traditional solution to the problem).
History
This puzzle was mentioned by W. W. Rouse Ball in the third, 1896, edition of his book Mathematical Recreations And Problems Of Past And Present Times, and is said to have been a favorite problem of Lewis Carroll.^{[3]}^{[4]}
References
 ^ Gamow, George; Stern, Marvin (1958), Puzzle math, New York: Viking Press, pp. 103–104
 ^ ^{a} ^{b} Chapter 10, Hexaflexagons And Other Mathematical Diversions: the first Scientific American book of puzzles & games, Martin Gardner, Chicago and London: The University of Chicago Press, 1988. ISBN 0226282546.
 ^ p. 662, Mathematical Recreations And Problems Of Past And Present Times, David Singmaster, pp. 653–663 in Landmark Writings in Western Mathematics 16401940, edited by Ivor GrattanGuinness, Roger Cooke, et al., Elsevier: 2005, ISBN 0444508716.
 ^ Mathematical recreations and problems of past and present times, W. W. Rouse Ball, London and New York: Macmillan, 1896, 3rd ed.