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 solution still applies no matter how many cups of any sizes and compositions are exchanged, or how little or much stirring at any point in time is done to any barrel, as long as at the end each barrel has the same amount of liquid.

For the similarly-named probability paradox, see Wine/water paradox.

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 wineBarrel originally with water
Wine: V0Water: V0
→

Move V1 of wine right

Wine: V0 – V1Water: V0 , Wine: V1
←

Move V1 of mixture (comprising V2 wine and V1 – V2 water) left

Wine: V0 – V1 + V2 , Water: (V1 – V2) Water: V0 – (V1 – V2), Wine: V1 – V2
Purity of wine = V0 – V1 + V2/(V0 – V1 + V2) + (V1 – V2)

  = V0 – V1 + V2/V0

 Purity of water = V0 – (V1 – V2)/(V0 – (V1 – V2)) + (V1 – V2)

  = V0 – V1 + V2/V0

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 containerWhite 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)

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
  1. Gamow, George; Stern, Marvin (1958), Puzzle math, New York: Viking Press, pp. 103–104
  2. Gardner, Martin (1988). "Chapter 10". Hexaflexagons And Other Mathematical Diversions: the first Scientific American book of puzzles & games. Chicago and London: The University of Chicago Press. ISBN 978-0-226-28254-1.
  3. David Singmaster (2005). "Mathematical Recreations And Problems Of Past And Present Times". In Ivor Grattan-Guinness; Roger Cooke; et al. (eds.). Landmark Writings in Western Mathematics 1640–1940. Elsevier. p. 662. ISBN 978-0-444-50871-3.
  4. W. W. Rouse Ball (1896). Mathematical recreations and problems of past and present times (3rd ed.). London and New York: Macmillan. ASIN B0006AHNJU. OCLC 2948122.
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