Mitan Ltd
Step 1
At the start there is "r" red liquid in glass R and "b" blue liquid in glass B.
The Two-Glass Explanation
. The amount of red liquid in glass B is exactly the same as the amount of blue liquid in glass R.
Short Explanation
We start and finish with the same amount of liquid in each glass, so the blue liquid that ends up in glass R must have displaced the same amount of red liquid, and the displaced red liquid must be in glass B. So the amount of red liquid in glass B must be the same as the amount of blue in glass R.
Detailed Explanation
For those comfortable with some algebra., the adjacent panels give a detailed proof, and also shows that the result holds even if the volume of red liquid is different from the volume of blue liquid. Each glass ends with the same volume it started with
Step 2
If the volume of red liquid moved from glass A top glass B is "x", then we end up with "r-x" red liquid in glass A and "x" red liquid in glass B alongside the "b" blue liquid which is all still in glass B.
Step 3
Mixing the red and blue in glass B does not change the amounts in the glasses.
Step4
The volume of (mixed) liquid transferred from glass B back to glass R must be "x", in order to end up with the glasses containing the same volume they started with. Assume that the volume transferred back is "y" blue liquid and "x-y" red liquid.
Result
Some simple algebra gives the answer
Download the Two-Glass program