Lewis Carroll’s Milk and Water Puzzle: The Spoon Does the Math

Puzzle. You have a glass of milk and a bucket of water. Take one spoonful of milk from the glass and pour it into the bucket of water. Then take one spoonful of the mixture from the bucket and pour it back into the glass. Which will be greater: the amount of water in the glass or the amount of milk in the bucket?

Answer. Equal 

Fig. 1 This classic riddle is often attributed to Lewis Carroll.

Solution. You probably find the answer surprising because the containers are different sizes and the mixture in the bucket wasn't stirred to ensure uniformity. But those details are irrelevant. The important point is that the same spoon is used in both transfers. Therefore, the amount of milk added to the bucket equals the amount of mixture returned to the glass. That is all that matters. Let’s see how it works, using practical examples.

Suppose we added a spoonful of milk to the bucket and removed a spoonful of the mixture, containing 1 part water and 2 parts milk (Fig. 2). When we add this mixture to the glass, the glass receives 1 part water. At the time, since the spoon contains 2 parts milk, 1 part milk remained in the bucket. Therefore, we end up with 1 part water in the glass and 1 part milk in the bucket. Equal.

The same arithmetic works in every case. Whatever amount of milk the returning spoon contains, the rest of it is water. That water goes into the glass. But the milk that did not return to the glass must have been left behind in the bucket. So the amount of water added to the glass always matches the amount of milk left in the bucket.

Fig. 2. Diagram illustrating the transfer of 3 parts of milk to the bucket, with a mixture of 1 part water and 2 parts milk returning to the glass. As a result, the bucket contains 1 part milk, and the glass contains 1 part water.

Now, suppose the spoon returning from the bucket contains 6 parts water and 1 part milk (Fig. 3). The glass receives 6 parts water. And, since the spoon contains only 1 part milk, the remaining 6 parts milk must be in the bucket. Again, the amounts are equal: there are 6 parts water in the glass and 6 parts milk in the bucket.

The spoon does the math. It adds milk to the bucket and then removes the same amount of the mixture. The amount of water in that mixture replaces the amount of milk that remained in the bucket. So the spoon transfers as much water to the glass as it leaves milk in the bucket.

Fig. 3. Diagram illustrating the transfer of 7 parts of milk to the bucket, with the mixture of 6 parts of water and 1 part of milk returning to the glass. As a result, the bucket contains 6 parts milk, and the glass contains 6 parts water.

You have probably seen a solution based on the idea that the liquid volumes in the containers remain unchanged. However, unchanged volumes are not required. The puzzle is more interesting than that. We could add or remove milk from the glass any time we wanted, before, during, or after the exchange. And it would not matter. The glass would still receive exactly as much water as the spoon leaves milk behind in the bucket.

The puzzle is about the foreign liquid in each container: water in the milk, and milk in the water. The amount of native liquid — milk among milk, or water among water — is irrelevant. We could just as well add or remove water from the bucket, and the solution would still hold. This would change the mixture's concentration, but that is not the puzzle's focus. As long as the same spoonful volume is used in both exchanges, the amount of water transferred to the glass will always equal the amount of milk left in the bucket. This subtle detail is the true beauty of the puzzle that was overlooked.