Roxbury Math Epsilon Club Lecture4

Lecture 4

Roxbury Math Epsilon Club

Today, we will examine a completely new problem: Marble Game In front of each of you are 2 bags of marbles: one filled with 20 black marbles and the other one filled with 20 white marbles. Our game is very simple: 1) Pick up 5 marbles from white bag and put all of them into black bag. 2) Mix up marbles in black bag 3) Pick up 5 marbles from the mixed bag and move them back into white bag. Now, let’s count the number of white marbles in black bag and the number of black marbles in the white bag… Something seems strange: it seems to be the same number. Perhaps, this is just an accident and we should do this same experiment again just to make sure… Once again: the number of foreign marbles in each of the bags seems to be always the same. Why is this happening? Let’s take a step back and forget about “how” marbles go from one bag to the other. Instead, let’s just look at what we had in the beginning and what we end up in the end. We always start out with 2 bags of marbles: 20 white and 20 black. We always end up with the same 2 bags of marbles: 20 “mostly” white and 20 “mostly” black. Obviously, we did not add and remove any marbles from our original group. So, obviously, if “some” white marbles ended up in a black bag, then they have to have been replaced by the exact same number of black marbles in a white bag. That’s all there is to it! Sometimes, when you solve a math problem, it’s better to forget about how something changes --- all you need to solve it is to realize that nothing was “lost” in the process. If you will, this is like the “law of conservation of marbles”. As you study later, you will see that there are many other examples of conversation laws in math and in physics. Laws of conservation of energy, conservation of momentum, conservation of mass are just a few examples of such conservation laws. Here is another way to think about this problem, in case you were not convinced using the conservation argument. Suppose, we start with 40 marbles = 20 white + 20 black and randomly put them into 2 bags. Clearly, the sum of each color of marbles has to be 20 for both black marbles and for white marbles… Well, this is the same thing as saying that the number black marbles in white bag is the same as the number of white marbles in black bag.

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Lecture 4

Roxbury Math Epsilon Club

Then, how is what we just did before different from this new experiment. The only difference is that we held 15 white marbles “out of the mix”, while we distributed the remaining 20 black marbles with 5 white marbles. So, we must end up with at least 15 white marbles in white bag + some other 5 marbles… but still the number of foreign marbles in each bag must be the same.

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Lecture 4

Roxbury Math Epsilon Club

Homework… I would like everybody to think of really clever solutions to these very formidable math problems… 1) How many ways are there to choose 1 white marble from 20 white marbles? 2) How many ways are there to choose 2 white marbles from 20 white marbles? 3) How many ways are there to first choose 1 white marble and then 1 black marble from 15 white marbles and 5 black marbles? 4) How many ways are there to first choose 1 black marble and then 1 white marble from 5 black marbles and 15 white marbles?

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