PUZZLE. We have a glass of milk and a bucket of water. We take a spoonful of milk out of the glass and transfer it to the bucket. Then we take a spoonful out of the bucket and transfer it to the glass. What will be more, water in the glass of milk or milk in the bucket of water?
ANSWER. Equal
SOLUTION. Unexpected, isn't it? Still, the puzzle has a simple logical solution. The sizes of the vessels are irrelevant. They can be equal or different. That won’t affect the answer. It doesn’t matter either whether we stir the spoon of milk in the bucket or not. The key factors responsible for the solution are:
1. The glass and the bucket have the same amount of foreign liquid, zero water and zero milk (0 and 0).
2. We use the same size spoon to swap the liquid between the vessels.
Indeed, after completion, the spoon took away as much liquid from the glass as it put back. So the liquid volume in the glass stays unchanged. That means it gave away as much milk as it received water. If the glass gave away more milk than received water, then the amount of liquid in it would decrease, which is impossible. Similarly, if the glass gave away less milk than it received water, the amount of liquid in the glass would increase, which is again impossible. The same applies to the bucket. So the amount of foreign liquid in the glass and bucket has to be equal. Providing it was equal from the start (0 and 0) and we used the same size spoon.
Let’s prove this statement with examples. We fill the glass with white and the bucket with black peas. The spoon holds 10 peas. We consider 3 possible scenarios. The spoon takes 10 white peas from the glass and transfers them to the bucket. After that the spoon:
1. Takes 10 black peas out of the bucket and transfers them to the glass. We got 10 white peas in the bucket and 10 black peas in the glass. Equal.
2. Takes 10 white peas out of the bucket and transfers them to the glass. We got 0 white peas in the bucket and 0 black peas in the glass. Equal.
3. Takes a mixture of 7 black and 3 white peas out of the bucket and transfers it to the glass. We got 7 black peas in the glass and +10 - 3 =7 white peas in the bucket. Equal.
We can prove it with milk and water. We transfer a spoonful of milk to the bucket and take out a spoonful of mixture.
1. The mixture contains 1 part of milk and 6 parts of water (1/7 + 6/7 = 7/7 = 1 spoon). It means we transferred 6 parts of water to the glass. And left 7 - 1 = 6 parts of milk in the bucket. Equal.
2. The mixture contains 3 parts of milk and 2 parts of water (3/5 + 2/5 = 5/5 = 1 spoon). It means we transferred 2 parts of water to the glass and left 5-3 = 2 parts of milk in the bucket. Equal.
Will concentration be equal in the vessels? No, the concentration of water in the glass will be higher than the concentration of milk in the bucket. The concentration will be equal only in the equal-sized vessels.
Will equality of water in the glass and milk in the bucket hold if we repeat the operation many times? Yes, it will. We can endlessly repeat the operation. The equality will hold, providing we use the same size spoon and finish the cycle where we started. For example, if we start with the glass (our case), we have to finish with the glass to keep the swap complete. If we start with the bucket (identical case), we have to finish with the bucket.
Example. Suppose the spoon transfers 3/7 water and 4/7 milk from the glass to the bucket and then transferrers 1/3 water and 2/3 milk from the bucket to the glass. The bucket added 4/7 milk and lost 2/3 milk, making total milk loss +4/7 - 2/3 = -2/21. The glass lost 3/7 water and added 1/3 water, making total water loss -3/7 + 1/3 = -2/21. The vessels lost the same -2/21 parts of foreign liquid, so their original equality holds.
What if the glass has 3 times more water than the bucket has milk? The spoon will still reduce (or increase) the foreign liquids equally. And therefore, it will maintain their difference and not the ratio. Example. We have 30 black peas in the glass and 90 white peas in the bucket. The difference is 90 - 30 =60. The ratio is 90:30=3. The spoon holds 10 peas.
1. We took 1 black and 9 white from the glass, transferred them to the bucket, took 7 black and 3 white peas from the bucket, and transferred them to the glass. The glass now has 30 - 1 + 7 = 36 black peas. The bucket now has 90 + 9 - 3 = 96 white peas. The number of foreign peas in the vessels increased by the same 6 peas. The difference 96 - 36 = 60 is preserved. The ratio 96:36 = 2,7 is not.
2. Now we have 36 black peas in the glass and 96 white peas in the bucket. We take 8 black and 2 white peas out of the glass, transfer them to the bucket, take 4 black and 6 white peas out of the bucket, and transfer them to the glass. We have 36 - 8 + 4 = 32 black peas in the glass and 96 + 2 - 6 = 92 white peas In the bucket. The number of foreign peas decreased by the same 4 peas. The difference 92 - 32=60 is preserved. The ratio 92:32 = 29 is not.
This puzzle illustrates the fundamental law of physics, the Conservation Law of mass, momentum and energy in closed systems. The law states that no mass (momentum, energy) is lost or made. It can only be transferred from one body to another. The transfer must obey the Conservation Law constraints. We'll discuss this subject more in future posts.
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