Math Is Fun Forum

In response to the Lemonink puzzle: I believe that the lemon glass will contain less ink than the ink glass will contain lemon. The following is based on two assumptions: the amount taken out of each glass is 10 mL and that the amount taken is allowed to mix thoroughly in the glass. Assuming that the lemon transferred to the ink glass will mix into the ink, only part of the lemon will be returned to the lemon glass and some of the ink transferred to the lemon glass will contain some lemon in it. If the lemon transferred to the ink glass is thoroughly mixed into the ink before the same amount is taken out of the ink glass and transferred back to the lemon glass there will be 99.0909...% lemon in the lemon glass with the rest being ink, and there will be 90.909...% ink in the ink glass with the rest being lemon. The only way that the amount of lemon in one glass could equal exactly the amount of ink in the other glass would be if somehow one avoided taking any lemon back to the lemon glass while collecting ink out of the ink glass. If the assumption about the quantity is not taken, there is another way that the amount of lemon in each glass could equal the amount of ink in the other glass; that is if instead of a spoonful the amount taken from the lemon glass was exactly half (50mL). Depending on the size of the spoonful, the lemon glass will always contain less ink than the ink glass does lemon up to the point that the spoon is half of the glass. I would not assume a spoonful to contain 50mL or more, but if the spoon were to hold more than 50mL then the converse would be true, the ink glass would contain less lemon than the lemon glass did ink after the transferring of the quantities from each glass.

Let me know what you think, and how I could have said this more clearly. I am sure that there must be a simply beautiful way to say this in "Math Speak", unless of course I am just wrong.