Imagine that you have before you two beakers, one containing a liter of water; the other, a liter of wine. One cubic centimeter of water is transferred to the beaker of wine and the wine and water mixed thoroughly. Then a cubic centimeter of the mixture is transferred back to the water. Is there now more water in the wine than wine in the water? Or vice versa?
The answer is that there is just as much wine in the water as water in the wine. The amusing thing about this problem is the extraordinary amount of irrelevant information involved. It is not necessary to know how much liquid there is in each beaker, how much is transferred, or how many transfers are made. It does not matter whether the mixtures are thoroughly stirred or not. It is not even essential that the two vessels hold equal amounts of liquid at the start! The only significant condition is that at the end each beaker must hold exactly as much liquid as it did at the beginning. When this obtains, then obviously if x amount of wine is missing from the wine beaker, the space previously occupied by this wine must now be filled with x amount of water.
If the reader is troubled by this reasoning, he can quickly clarify it with a deck of cards. Place 26 cards face down on the table to represent wine. Beside them put 26 cards face up to represent water. Now you may transfer cards back and forth in any manner you please from any part of one pile to any part of the other, provided you finish with exactly 26 in each pile. You will then find that the number of face-down cards in either pile will match the number of face-up cards in the other pile.
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A version of this puzzle originally appeared in the September 1957 issue of Scientific American. Read Gardner's entire column, about the mathematics behind card tricks, here.