A box contains 2 white marbles, 3 blue marbles, 7 black marbles and 9 green marbles. Juan takes one marble at time out of the box and without looking at its color puts it in another box. What is the least number of marbles Juan must transfer to the second box to be certain that two of the marbles in the second box are the same color?
Please explain??????
Since there are 4 colors, even if the 1st 4 marbles are all different, the 5th marble must be the same color as one of them.
To determine the minimum number of marbles Juan must transfer to be certain that two of them are the same color, we need to consider the worst-case scenario. In this case, the worst-case scenario is that Juan will keep transferring marbles without getting two of the same color until there are no more marbles left for him to transfer.
We can solve this problem by using the Pigeonhole Principle, which states that if you have n+1 objects distributed into n pigeonholes, then there must exist at least two objects in the same pigeonhole.
Let's apply this principle to our problem:
1. Initially, Juan has 2 white marbles, 3 blue marbles, 7 black marbles, and 9 green marbles in the first box.
2. If Juan transfers one marble to the second box, there will be 1 marble of that color in the second box.
3. If Juan transfers another marble to the second box, there will be at most 2 marbles of that color in the second box.
4. If Juan transfers a third marble of that color to the second box, there will be at least 3 marbles of that color in the second box.
Since we need to be certain that two marbles in the second box are the same color, the worst-case scenario occurs when Juan transfers marbles of different colors until he has no more marbles left to transfer.
Let's see how this worst-case scenario plays out:
1. Juan transfers the first marble to the second box. No marbles of the same color exist in the second box yet.
2. Juan transfers a second marble to the second box. No marbles of the same color exist in the second box yet.
3. Juan transfers a third marble to the second box. No marbles of the same color exist in the second box yet.
4. Juan transfers a fourth marble to the second box. No marbles of the same color exist in the second box yet.
5. Juan transfers a fifth marble to the second box. No marbles of the same color exist in the second box yet.
6. Juan transfers a sixth marble to the second box. No marbles of the same color exist in the second box yet.
7. Juan transfers a seventh marble to the second box. No marbles of the same color exist in the second box yet.
At this point, Juan has transferred 7 marbles to the second box, and there are no marbles of the same color in the second box. However, if Juan transfers just one more marble (the eighth marble), by the Pigeonhole Principle, there must already be two marbles of the same color in the second box.
Therefore, the least number of marbles Juan must transfer to the second box to be certain that two of the marbles in the second box are the same color is 8 marbles.
Note: The actual colors of the marbles are irrelevant to this problem. The only important factor is the total number of marbles available in the first box.