Wednesday, July 10, 2013

Permutation or Combination?

This morning one of the proctors lead a group on a 3 mile run. I decided last night that I would participate, so I went down to the lobby at 7:00 AM. I was surprised, and so was the proctor: There were over 10 students participating. The proctor said that in Session 2 (9th and 10th graders), he only had two runners. We ended up not running a full three miles. The goal was for everyone to keep up, but whenever there was a corner, the proctor would wait for everyone else. After about ten minutes, one of the students could not keep up, and had to walk the rest. The proctor could not send him back by himself, so we ended up all walking back on a shorter route. The person who could not keep up said he hadn't run in over a year. He was also wearing jeans. This did not make sense to me. Who exercise in jeans? Anyway, after the run, I took a shower and went to breakfast.

Today's puzzler was a logic type puzzle. The more correct name for it is an indirect proof, or a proof by contradiction. Here's the problem... After just one day, a boy at summer camp is out of money. He does not even have any money to make a phone call (Assume he does not have a cell phone). He finds a postcard with a stamp on it and writes to his parents "Send more money." Instead of writing it as a sentence, he writes it like this.
 + M O R E

Surprisingly, his parents get the message, and send him the amount of money he requested. (The letters represent numbers) It turns out, there is only one solution. Each letter stands for a unique digit (0-9), and M is a nonzero number. Find the values of each of the letters. As usual, STOP reading if you want to solve it yourself.

First, you want to figure out what M is. Notice that we are adding a two 4 digit numbers to get a 5 digit number. If we add the biggest possible 4 digit numbers (9999+9999) we get 19998. Since this is the max value, M must equal 1, as we cannot get to 20,000. We also know that S+M is 10 or greater (so it carries the one over.) Since M is 1, the only values for S are 9 without a carry from the previous column, or 8 with a carry. Also, you cannot have a 9 and a carry, because that makes 11. M and O must be different numbers. Therefore, O is 0.

Next let's look at the 100's column. E + 0 = N. Since E does not equal N, there must be a carry from the 10s column. In order for us to carry a one from the 100s to the 1000s, E would have to be nine. If that were the case, 1 + E + 0 = 10. This cannot be, because N cannot equal zero. Therefore, S is 9. 

Since we have to carry from the 10s to the 100s, N + R = E + 10. (10 is the carry.) But we do not know if there is a carry from the ones column, so the equation is really (1+) N + R = E +10. We also know that E + 1 = N. We can rewrite that as E = N - 1. We can then substitute it into the first equation, yielding N + R = N - 1 + 10. The N's cancel out, and -1 and 10 make 9 resulting in (1+) R = 9. This means that R is eight with a carry or 9 without. Nine is already in use, so it must be eight, and we must get a carry over from the ones column.

Now the only numbers left are 2, 3, 4, 5, 6, and 7. We know that D + E = Y +10. The only way to get over 10 with the remaining values for D and E are 7 and 6 or 7 and 5. E cannot equal 6, because N - 1 must equal E. Therefore, D is 7, and E is 5. The only letter left in N. N must equal 6. Therefore, the solutions are 0 is O, 1 is M, 2, 3 and 4 are unused, 5 is E, 6 is N, 7 is D, 8 is R, and 9 is S. Therefore, the equation is:
9 5 6 7 
+ 1 0 8 5 
1 0 6 5 2

One can only assume he meant $106.52, not over $10,652. After the Puzzler, we did Independent Study. I got almost all the way through the second section, so I will be taking the quiz soon. After Independent Study (and our 15 minute break), we learned about Combinatorics. There are two types of Combinatorics: Permutation and Combination. Both have to do with calculating probabilities. In Permutation, the order of the set matters, in Combinatorics, the order does not matter. For example, if you were asked the number of possible ways you could arrange a group of letters, that's permutation. If you were asked how many possible groups of 2 people you can make with a initial group of 10, you would use Combinatorics. There are various formulas for each, depending on the conditions. The lecture ended just before lunch.

After lunch, we continued with independent study for just 30 minutes. I was able to finish all but the last problem in Section 2, so I'll be taking the quiz tomorrow, after I finish up the last one. Next we played our afternoon game. This time we were divided into 3 teams of 4. He gave us 16 Combinatorics problems. At the end of the worksheet, you had to plug your answers into a formula to get the ultimate answer. For the first 45 minutes, we could only put two guesses of the ultimate answer on the board. During this time period, he would tell you how many you got correct out of 16. After the first 45, none of the groups had all the answers correct. In the remaining time, he told us which questions we got wrong so we could fix them. Our group got 9 right initially, and we couldn't figure out why they weren't all correct. It turned out that we were using Permutation instead of Combination on some of them. We ended up losing because of this, but it was a good learning experience. Without this experience, I would have thought I was doing it right the whole time. I really enjoy Dawson's afternoon activity/game. It kind of gives us a break from lecture, gives us a chance to show what we know, and is fun. After the game ended, and Dawson went over the answers and how you get them, we had study hall. Today was the last day to work on our presentations. I finished mine just in time, but a few people were not done. Unless they brought their laptop, I don't know what their going to do. I was glad to finish in time, so I don't have to worry about doing it now.

Today in Arête, we made a spinning top. This required three pieces of paper, all folded in different ways. Mine did not turn out that great, because I used the wrong paper. The teacher had laid out two types of paper. The regular kind that is the same color on both sides, and a special kind with different colors on each side. The white parts of my top are supposed to be green, as they are the same piece of paper. Also, I messed up some of the folds, so it did not come out very cleanly.

This evening, I experienced my first Tennessee thunderstorm. It started during dinner, and by the time dinner was over, it was really pouring. Everyone decided to wait it out in the dining hall, until if finally stopped at 7:45. Instead of going out for SOFT night, I participated in one of the Vanderbilt activities. We made tie-dye shirts for the neon dance on Friday. I had never done tie-dye before, so it was a unique experience. Now I know to wear glove when you do it, or you get dye all over your hands, and it doesn't wash off.

Tomorrow, we will present our PowerPoints in study hall. I am interested to learn more about mathematicians I know like Pascal, and Euler, but even more interested to learn about the mathematicians I have never heard of. I have no idea what the Puzzler will be, but I can guarantee it will be puzzling...

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