If I'm not mistaken, we end up going through every rational number between 0 and 1. So let's start with a collection of balls, with all of the real numbers in (0,1).

Now let's divide them up into two piles, one with all of the balls with rational numbers Q = Q∩(0,1) and the other with irrational numbers R = Q

^{c}∩(0,1). Do you agree that the intersection of Q and R is the null set? Do you agree that their union is (0,1)?

Now let's paint all of the balls in the Q pile blue, and all of the balls in the R pile red. Do you agree that it follows that every ball is painted with exactly one color?

Now let's do the process you described. You asserted that in the end, you have a jug filled with irrational-numbered balls. So then all of the balls must be red. Yet, throughout the process, you inserted and removed only blue balls. In fact, at any given step, the net number of blue balls in the jug went up by 9.

So you are claiming that something about the process does two things:

1. At midnight, all blue balls have been removed, despite the number of blue balls in the jug monotonically increasing.

2. At midnight, all red balls end up in the jug, despite all being outside of the jug at any time before midnight.

I agree with (1), for exactly the same reason that the jug is empty in the original puzzle. Lets talk about (2). Can you think about the following questions? I don't necessarily want an answer/explanation for each one, but your thoughts on any of them that you find particularly enlightening would be useful.

a. Is there any time in which there is both a red ball and a blue ball in the jug?

b. Is there any step in which there is both a red ball and a blue ball in the jug?

c. If the red balls are all destroyed before the game begins, is there any time in which the game would halt (due to being unable to follow the instructions) while there are still blue balls in the jug?

d. If the red balls are all destroyed before the game begins, is there any step in which the game would halt while there are still blue balls left in the jug?

e. If the red balls are all destroyed before the game begins, and the game halts due to error but only after there are no blue balls left in the jug, at what time/step does this occur?

f. If the red balls are all destroyed before the game begins, and the game does not halt due to error, are any balls left in the jug at midnight?

g. If the answer to (f) is yes, what color are they?

h. If the answer to (f) is no, is this not identical to the original puzzle?