a long but quite interesting adding and deleting balls problem

Question

suppose you have infinite black and white balls and note black balls with B and white balls with W. You are given some presets before the game( i.e BW, WB ,WWW, etc). You are also given some balls at the starting of the game and required to move the balls to a particular set. Now you need to add or delete the presets to reach the "particular set" you need. A example of the game is shown below:

1. You have a ball set of WB at beginning.
2. You are told that you need to transform the WB to BW.
3. You are given presets of BB,WW and BWBW and you are able to add or delete them but the balls must be together if they were to be deleted.
4. Now perform the following action: add BB in front of WB to get BBWB, and WW at the end of BBWB to get BBWBWW. Now delete BWBW between the first letter and the last letter to get BW.

Now prove:

1. If the presets given were BB, WWW and WBWB and you are given WB at the beginning, show that it is not possible to go from WB to BW in anyway.

2. Prove that we can use the presets given in part 1 to remove every ball in a sequence like BBB.....WWWWW.... where the black and white balls are all separated.

I know the description may sound really strange-- this is not like a formal problem but rather an extension of showing the example. My work on problem 1 right now:

1.We call applying preset BB be an action A. Call applying preset WWW be action B. Call applying preset WBWB be action C. Also denote removing the balls as a negative action (removing 2 times of BB= applying -2 times of action A) Now we count the number of balls after applying unknown times of those actions (Remember that we have WB at the beginning.): Blue ball$=1+2(A+C)$ Black ball$=1+(3B+2C)$ Now because we want $2(A+C)=0$ and $(3B+2C)=0$. However I don't know how to continue from here. Some suggestions or even answer will be greatly appreciated... ( sorry for such a long question)

Edit: I think there aren't much things to do with the number of balls; i think there is a way to argue that it is impossible by inversions? I don't really know much about it so if someone can tell me if inversions do do something here it will be very much appreciated...

Answer 1

This problem you have posted is almost identical to a SUMaC 2018 admissions exam problem (problem 7, specifically). Given that I see that you have already tried getting solutions on this site to problem 5 (and potentially also to problem 9), I doubt that this is a coincidence. Stop posting these questions, they violate the honor code you signed when you submitted your application.