A box contains 731 black balls and 2000 white balls. The following process is to be repeated as long as possible. Arbitrarily select two balls from the box. If they are of the same color, throw them out and put a black ball into the box (enough extra black balls are available to do this) if they are of different color, place the white ball back into the box and throw the black ball away. Which of the following statements is correct?

  1. The process can be applied indefinitely without any prior bond
  2. The process will stop with a single white ball in the box
  3. The process will stop with a single black ball in the box
  4. The process will stop with the box empty
  5. None of the above

How do I approach this question? Any idea?


The number of balls decreases by $1$ each time. Thus after $2730$ steps we will have $1$ ball in the box, and we must stop.

At each step, the parity of the number of black balls changes (from odd to even or from even to odd). At the beginning there is an odd number of black balls in the box. So after $2730$ steps $\dots$ (you can finish the argument).


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