Lewis Carroll's Milk / Water Puzzle

by Tim Darling

You are given two glasses. One contains 100 parts milk, the other 100 parts water. Take one tablespoon of milk and mix it with the water. Now take one tablespoon of the water/milk mixture and mix it with the pure milk to obtain a milk/water mixture. Is there more water in the milk/water mixture or more milk in the water/milk mixture?

This is a famous puzzle that is not hard, but can be tricky. I haven't seen a good explanation of it on the web, so here's one. The easiest approach is probably to use this notation: x:y = x parts of milk + y parts of water. Asumme a tablespoon is 1/10 of the starting amounts per glass.

Glass 1Glass 2
Move 1/10 of Glass 1 to 290:010:100 (=110 total parts)
Move 1/11 of Glass 2 to 190:0 + (10/11):(100/11)(100/11):(1000/11)
Simplify(990/11):0 + (10/11):(100/11)(100/11):(1000/11)
Simplify more(1000/11):(100/11)(100/11):(1000/11)

End result: there is the same amount of milk in Glass 2 as there is water in Glass 1.

This problem and solution can be generalized for more glasses. For example, consider three glasses with red, green, and blue liquid in them respectively. In the first step, move 1/10 of the red liquid to the green glass. Then move 1/11 of the liquid from the green glass to the blue one and 1/11 back to green and 1/11 back to red. The end result: there will be as much red in the blue liquid as there will be blue in the red. Also, there will be as much red in green as green in red and similarly with green and blue.

Glass 1Glass 2Glass 3
End *1,209 : 1,110 : 11111 : 1,110 : 11011 : 110 : 1,210

(* Multiply all values by 100/1331).

All text and pictures copyright © 2005-2006 Tim Darling.