$$2 CHAINZ$$ "My wrist deserve a shout out, 'I'm like what up wrist?' My stove deserve a shout out, 'I’m like what up stove?'"
When finding derivatives there is simple rule to solve problems. As problems get more complicated though, other rules must come into play. The chain rule is a formula used to find a derivative when a function is put inside of another function. The chain rule was quite the struggle for me.
"I stack my money so tall, that you might need a giraffe when you counting this cash"
When we first learned the chain rule I still didn’t understand the power rule for simple equations. Everyone else in the class was mastering these complicated equations while I was still struggling to solve the simple power rule problems. Everyone was throwing around f= and g= and h= and I had no clue what any of them meant. The method that was taught worked for the rest of the class, but I found myself lost in between f's and g's and sine and cosine. After trying to understand this process with countless practice problems and explanations, I realized this was not going to work. I sat down and figured out a process that worked for me. None of the other students understand my process but it works for me. I simplified the process and broke it down into steps that made sense to me.
I am fairly proud of myself for figuring this out. I am rarely the only one who doesn’t understand a concept in math and that was hard for me to accept. Once I was able to accept it though, I was could take it into my own hands and figure out a solution. I honestly really enjoy solving chain rule equations now that I have a method. It’s good stuff.
My chain rule dilemma was the first time I really had to figure a math concept out for myself. This has a opened a whole new door of possibilities. I now realize that people learn very differently and the popular method doesn’t necessarily work for everyone. I know now that I can solve things by myself and tailor a solution to what makes sense to me. "My favorite dish is turkey lasagne."
POW #9
Six People
Restate the Problem:
In any group of six people, prove that there are either three people who all know each other or three people who all do not know each other.
Process:
This question was very confusing at first because it seemed as if the answer would depend on each specific scenario. To try to wrap my mind around it I drew a hexagon, having each corner represent a person. I tried to map out all the possible scenarios for who knew who but I realized very quickly that it gets very complicated with six different people. I then realized Person 1 knowing everyone was the exact same scenario as if any other person knew everyone. I had then narrowed it down to ten scenarios: knowing 1, 2, 3, 4, or 5 people or not knowing 1, 2, 3, 4, or 5 people. This seemed much more manageable. Since I was having no luck proving the theory, I again drew out the scenarios but this time with the intent to try to disprove the theory. As I did this I was not able to disprove the theory but I was able to narrow down the scenarios yet again by realizing knowing four people is the same as not knowing one person which is reflected with knowing one person and not knowing four people. Now I was down to three scenarios and was able to map them out.
Solution:
In any scenario, a triangle must be made. The dotted line shows that either color drawn will form a triangle.
Justification:
There is no possible way to complete these scenarios without connecting the dotted line. These three scenarios cover every scenario possible. If the line colors are flipped the result is still the same, a triangle is unavoidable. In a group of six people either three people will know each other or three people will not know each other.
Pow #3
Card Trick
10/3/14 Avra Saslow and Becca Schaldach
Restate Problem:
A “handler” has 5 cards, and pulls one out to show to a group of people. The “handler” then puts that one card aside and asks a volunteer (who didn’t see the card) to find the identity of that particular card by looking at the patterns between the remaining 4 cards. How is this done and what are the patterns used in identifying the specific card (x).
Process:
At first, we tried a process in which a certain card determined x’s suit and another determined whether it was odd or even. The first card in the order of four cards, resolved x’s suit, because no matter what out of the five cards there must be 2 cards of the same suit.
Next we thought maybe the fourth card resolved if the card was even or odd and that the middle two cards determined the actual number of the card. We tried adding those two numbers up, or having them as a range between what x could be. However, after trying this pattern with a lot of different cards, we realized that having the two middle cards couldn’t determine what x’s identity is.
We knew the math to determine the value had to be simple because the handler did not have a hard time setting up the cards. It is difficult to deal with 13 different values but if you set up the cards in a circle they can only ever be, at most, 6 cards away from each other. This makes the math much simpler. We thought maybe the distance between each card could be added, multiplied, subtracted, or divided to get the value of the hidden card but this proved to be much too complicated.
We were aware there was a chart to find the hidden card. We created this chart according to value first and then suit:
We then created a chart that determined the order of the 3 remaining cards after the first initial “suit” card. Depending on what order those three cards are in relative to the chart above, determines how many numbers you add to the initial card:
First-Second-Third means add 1 First-Third-Second means add 2 Second-First-Third means add 3 Second-Third-First means add 4 Third-First-Second means add 5 Third-Second-First means add 6
For example, if you had a 7♦, Q♦, 8♠, K♣, the hidden card would be 10♦. This is because the 7♦ determines the suit, and the the order of the next three cards appear 2nd, 1st, then 3rd in the chart. This means you have to add three to the initial card, 7, to equal 10.
Solution:
Referring to the two tables in the “process” section, we can identify any card. By having the first card as the determiner for the suit of the unknown card, we can use the three remaining cards to calculate the value of the hidden card by referring to the chart that arranges the cards by number and suit. By looking at the arrangement of cards, and how they are ordered relative to if they come first, second, or third, we can determine how much to add to the initial card. This involves the second chart, which outlines how many numbers to add depending on which cards are placed first, second, and third. Once the number to add has been determined, it can be added to the initial card, to determine the number and suit of the unknown card.
Justification:
There are 13 different numbered cards in a deck, when arranged in a circle though, at any point one card can only be 6 places away from another. For that reason, we know that we need 6 different patterns to arrange for 6 different solutions where you either add 1, 2, 3, 4, 5, or 6 to a specific card. We also know that it is crucial to understand how each card is ordered in a deck of 52. After determining these two factors, we played around with the order of the 3 remaining cards and found that depending on how they are ordered, we can determine how much to add to the initial card, thus also determining the number and suit of the unknown card. Additionally, this method works for any set of 5 cards in the deck.