For the problem 4. Monkey Beans: At each move, the number of beans in the urn falls by 1. So, after 23 + 34 - 1 = 56 moves, the number of beans in the urn wi

For the problem 4. Monkey Beans: At each move, the number of beans in the urn falls by 1. So, after 23 + 34 - 1 = 56 moves, the number of beans in the urn will be 1. At each move, the number of white beans in the urn either stays the same or falls by 2. Since the initial number of white beans is 23 and odd, as long as there are any white beans in the urn, the number is odd. Since the number of beans in the urn falls to 1, the number of white beans must fall to 1. Suppose there is 1 white bean in the urn: If there are no black beans in the urn, then we are done and the last bean is white. Else we keep playing and observe that at each play the number of white beans in the urn remains 1. That is, there is no way to remove the last white bean from the urn. So, in all cases, the last bean in the urn is white.