I am behind on posting activities. In the next few weeks, I will try to post short descriptions of a few of the activities we did in our math circle over the last semester.
This post covers a session we held in January 2015.
I began the session with a couple of warm-up problems that I thought would take about five minutes, and then we would move on to a different activity. However, the children found the warm-up problems interesting, and kept suggesting variations on the problem, so we ended up spending most of our time exploring different, related questions, and how the answers would change.
The setting: A bowl full of marbles is sitting in the back of a completely dark closet. Some of the marbles are black, and some are white. However, since there is no light in the closet, there is no way of telling which marbles are black and which are white while you are in the closet. You will be drawing marbles out of the bowl and taking them out of the closet, into the light, to check their colors.
Suppose there are 11 white marbles in the bowl and 17 black ones.
1. What is the smallest number of marbles you need to draw out of the bowl (and take out of the closet) to guarantee that when you leave the closet, you have at least two marbles of the same color?
2. What is the smallest number of marbles you can take out of the bowl that will guarantee that you have at least two white marbles when you leave the closet?
3. What is the smallest number to take to guarantee you have at least two black marbles?
Answers:
Some of the children didn't really understand the problem until we had explained the answer. If you try this activity, as the children make suggestions for the answers, try to get them to explain their answers, so you can clarify the setting and the situation if you need to. Remind them that they can't see what color the marbles are when they are in the closet, or determine the color by feel, so they have to choose marbles without knowing the color. Also, they can't go back and take more once they have left the closet. They get one chance to pull out the marbles of the correct color!
Here are the answers:
1. How to guarantee there are at least two marbles of the same color? Draw three marbles. Since there are only two colors, if you draw out three marbles, at least two of them will be the same color.
In our class, on the board, we went through all possibilities: three could be black (two of the same color), two could be black and one white, two white and one black, or three white. In all cases, at least two marbles have the same color.
2. How to get at least two white marbles? I had the students explain what might happen in the worst possible case. Maybe when you get into the closet, you first pick all the black marbles, and only pick your first white marble when you've taken out all the black ones. In this case, you would need to take 19 marbles out of the bowl to guarantee that at least two are white.
3. Similar reasoning gives that you need to take 13 marbles to ensure at least two are black.
Variations:
Now suppose there are 5 red marbles added to the bowl, so now there are three colors: 11 white, 17 black, 5 red. Ask the same three questions as above. How do the answers change?
1. How to guarantee you take two of the same color? Drawing three is not enough! Is four?
2. To guarantee you get two white marbles, you have to take more than before. Now in the worst case, you might only pull out black and red marbles at the beginning. You would need to take out 24 marbles to ensure that there are at least two white marbles, in case you draw 17 black + 5 red + 2 more.
3. To guarantee at least two black marbles, you need to take eighteen marbles.
I asked the children about other ways to change the problem, and they had good ideas, such as the following:
4. How many marbles to guarantee you have at least one of each color?
Answer: When there are only two colors of marbles (11 white, 17 black), you need to take 18. (Why doesn't 12 work?)
When there are three colors of marbles (5 red, 11 white, 17 black), you need to take 29! Why so many?
5. How many marbles to guarantee you have three of the same color?
6. How many to guarantee you have at least two of every color?
7. What if you put a fourth color into the bowl? How do the answers change?
(I'm leaving these questions for you to think about -- I can't take away all the fun by giving everything away.)
Conclusions:
I was surprised at the students' interest in these types of problems. Opening up the discussion to let them ask their own questions and change the problem seemed to be very successful for this group!
If I run this activity again, I might prepare a bowl of marbles in advance, say covered with a cloth so the children can draw marbles out of the bowl without seeing which color they are choosing at any time. Of course, it is unlikely for problems 2 and 3 above that the students end up in the worst case scenario. Perhaps it would be interesting to discuss "worst case" versus "average number"....
In any case, there seem to be lots of interesting variations on this type of problem.
