Though the idea of game theory has been around for quite a while, It was first defined and described first in a book called *The Theory of Games and Economic Behavior* in 1944. Created by mathematician, John von Neumann and economist, Oskar Morgenstern.

Essentially, game theory studies how people will respond to rules, or “play the game” in situations where the motives of other players are involved.

A popular example of game theory analysis would be a hypothetical situation called the Prisoner’s Dilemma. One version of the prisoner’s dilemma is where 2 individuals – we can call them person A and person B – are faced with a decision separately. They can either stay silent(cooperate) or betray the other person(defect) for a total of 4 outcomes. If they both stay silent they both serve 5 years. If they both defect, however, they serve 10 years. In addition if one person defects while the other stays silent, the person who defects goes free while the other serves 20 years.

Thus both players are faced with choosing between self-interest and mutual-interest, where not one situation would be ideal to both the individual or the group. People who work in jobs involving Game theory, work with similar situations related to their profession.

Mathematicians in particular analyze the conditions of different situations and develop algorithms to predict outcomes. Economists also use game theory to model the flow of money in a market. Game theory also has applications in the world of Artificial Intelligence, used by researchers and scientists to understand deep learning in machines. Which has been used to model real life situations. And the increasing success of AI is attributed to its increasing involvement with game theory. Besides economics and AI, it also has involvement in various other subjects including chemistry, bioinformatics, engineering, and many more areas.

Game theory may seem a small area of mathematics but it has widespread effects on topics far different from games and often impacts research on complex analysis of machines and humans – it has changed economics, math and much of the world today.

]]>In his talk, he began by questioning the definition of line of best fit. Introducing a data set of cricket chirps and temperature, he questioned how to find the slope and intercept of a line of best fit such that the error between the actual values from the data set and predicted values is as small as possible. By representing the error, Professor Klee was able to plot the slope and intercept of the best fit line as a 3D model. In such a model, the height of the 3D shape represents the error.

Professor Klee demonstrated that the line of best fit will have a slope and intercept such that the height of the model is zero. This strategy to determine the line of best fit can be applied to many real life scenarios as well. However, in real life scenarios, the amount of data significantly increases. With alarge amount of data, the line of best fit may not be the best model. Thus, Professor Klee depicted how a quadratic fit, a degree five polynomial of best fit or even a degree seven polynomial of best fit can represent the data better than a line.

Professor Klee ended with discussing the skills of solving real world problems, which includes asking interesting questions, finding and manipulating data, understanding shapes of functions, finding the best fit, understanding quality of fit, and having the ability to communicate findings to the real world. These skills relate to different fields, including computer science, mathematics, engineering, statistics and marketing. He then explained that jobs where you solve these problems include Boeing, Microsoft, Amazon, Google, UPS, FedEx, Hulu Netflix, Pacific Northwest National Lab, PWC, the Government and the military. He described that studying math leads to a wide range of career options.

Overall, Professor Klee’s talk was very inspiring. He clearly explained the role of math in solving real world problems and the importance of math in many careers. Throughout his lecture, Professor Klee stressed the benefits of studying math in high school and college and continuing math further on as a career.

]]>One of the most played game was called Frontrunner. Kids were drawn to this game by its bright colors and rather simplistic looking design. To play, each kid picks a horse that corresponds with the numbers 2-7. Taking turns, each player roles 3 die and draws an equation card. On the card, there are three different equations, going from easy to difficult, each with blanks for the numbers. The player uses the numbers obtained from rolling the die to plug into the equation and solve for the answer. If any of the horses’ numbers are factors of the answer, then that horse get to proceed to the next spot. At first, I was confused because it seemed that horses with smaller numbers would have a greater advantage because the answer of the equation is more likely to be a multiple of 2 or 3, rather than 6 or 7. However, I realized that horses 2 and 3 had many more spots to move than 6 or 7 before reaching the end, a design aspect meant to make the game more fair.

Frontrunner is a great game for kids who are practicing their addition, subtraction, multiplication, and division skills. Because there are three different equations of varying difficulty on each card, the game also works to cater towards kids at all different math levels and allow them all to play together. My only reservation about the game is that each game takes a very long time to finish. Because it takes a while to solve equations and figure out the factors, kids will often get bored and lose interest about halfway through.

Another popular game was called Set. In Set, 24 cards are set out. Players must find sets of three cards that are share one or more common characteristics, either shape, number of shapes, color, or pattern. The player who finds the most sets of cards wins. Set can be played by kids of almost all ages, even if they haven’t learned addition or subtraction yet. Each game also goes by pretty quickly, as the objective is to find sets as quickly as possible in order to beat the other players.

The last game that the kids really seemed to enjoy was called Vinculum. Vinculum is also a rather straightforward game. To set up, players must first set out most of the tiles face-up, leaving a stack of them face-up. Each tile has a fraction on it. To play, one player will flip one tile from the stack over, and the rest of the players must look for an equivalent fraction in the tiles that are laid out. For example, if the tile turned over had 2/3 on it, a tile with 4/6 on it would be considered a match. The player who finds the most matches when the stack runs out wins. Vinculum is also a fast-paced game, ideal for kids who are just learning fractions.

Overall, math games are a fun and often overlooked way of practicing math and can be very valuable to a kid’s learning. Besides the three mentioned in this post, there are hundreds of more math-related games out there that cater to kids of all ages and skill levels that are worth checking out.

]]>The rule of the Fibonacci sequence can be viewed in the following way:

*F*_{0} = 0*, F*_{1} = 1*, F** _{n} *=

The Lucas numbers are very similar to the Fibonacci numbers – the only difference is the starting term. The Lucas numbers were studied by the mathematician Francois Edouard Anatole Lucas. Whereas the Fibonacci numbers start with 0, the Lucas numbers start with 2. However, both the Lucas and Fibonacci numbers have the property that the ratio between two consecutive terms converges to the golden ratio. The first ten Lucas numbers are: 2, 1, 3, 4, 7, 11, 18, 29, 47 and 76.

The rule of the Lucas sequence can be viewed in the following way:

*L*_{0} = 2*, L*_{1} = 1*, L** _{n} *=

The “period” of Fibonacci when reduced modulo n, or Ψ(n), for all integers greater than 1, is the number of terms in the Fibonacci sequence before the sequence repeats when reduced modulo n. One of the first patterns regarding the period is that the period of some composite numbers is equal to the product of the periods of its prime factors.

There are also many common patterns between the Lucas and Fibonacci sequences. Each Lucas number can be expressed as the sum of two Fibonacci numbers:

**Theorem 5.1. ***L _{n }*=

However, the equation can be generalized further.

**Conjecture 5.2. ***For any odd number k *∈ N*,* *L** _{n} *·

Numerical Examples:

*n *= 6 and *k *= 5: *L*_{6 }· *F*_{5+1 }= *F*_{6−5 }+ *F*_{6+5 }⇒ *L*_{6 }· *F*_{6 }= *F*_{1 }+ *F*_{11 }⇒ 11 · 5 = 0 + 55

*n *= 8 and *k *= 3: *L*_{8 }· *F*_{3+1 }= *F*_{8−3 }+ *F*_{8+3 }⇒ *L*_{8 }· *F*_{4 }= *F*_{5 }+ *F*_{11 }⇒ 29 · 2 = 3 + 55

*n *= 9 and *k *= 7: *L*_{9}·*F*_{7+1 }= *F*_{9−7}+*F*_{9+7 }⇒ *L*_{9}·*F*_{8 }= *F*_{2}+*F*_{16 }⇒ 47·13 = 1+610 ⇒ 611 = 611

What about for even differences?

**Conjecture 5.3. ***For any even number j *^{∈ }N*, L _{j}*

Numerical Examples:

*n *= 6 and *k *= 4: *L*_{4+1 }· *F*_{6 }= *F*_{6−4 }+ *F*_{6+4 }⇒ *L*_{5 }· *F*_{6 }= *F*_{2 }+ *F*_{10 }⇒ 7 · 5 = 1 + 34 ⇒ 35 = 35

*n *= 10 and *k *= 2: *L*_{2+1 }·*F*_{10} = *F*_{10−2 }+*F*_{10+2}⇒ *L*_{3}·*F*_{10} = *F*_{8}+*F*_{12} ⇒ 3·34 = 13+89 ⇒ 102 = 102

*n *= 7 and *k *= 6: *L*_{6+1}·*F*_{7 }= *F*_{7−6}+*F*_{7+6 }⇒ *L*_{7}·*F*_{7 }= *F*_{1}+*F*_{13 }⇒ 8·18 = 0+144 ⇒ 144 = 144

For some even natural number *k *− 2 (where *k > *2), we assume that:

*F** _{n−(k−2)}* +

We also know that *k *− 1 will be odd, and so:

*F** _{n−(k−1)}* +

Is it possible to prove that

*F** _{n−k }*+

and that

*F** _{n−(k+1)}* +

The Fibonacci and Lucas Sequences are two of the most powerful sequences in mathematics. Using these sequences, mathematicians and scientists have discovered patterns in bracts of a pine cone, number of petals in a flower and scales of a pineapple. In addition, these sequences have enhanced our understanding of the Golden Ratio and its significance in our lives.

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Every year, the organization plans and organizes the math competition for 5th-8th graders in the Puget Sound area. Teams of three or four register together and compete within their category (5th-6th or 7th-8th).

Math Bowl is composed of three different kinds of tests. The first test is a Creativity test where the teams are given 40 minutes to complete 5 questions. This test encourages teams to work together to solve unique problems by applying their math skills and knowledge in new ways.

The Time Attack test follows the Creativity test. Time Attack is composed of 24 questions and teams are given 36 minutes to complete the test. In addition to the points teams earn from answering the questions correctly, bonus points are offered for turning the test in early. If a team turns the test in within the first 6 minutes, they would earn 8 bonus points. For every block of 6 minutes that passes, the amount of bonus points the teams earn decreases by one.

The last event is the tournament rounds. Teams with similar scores compete against each other in a Swiss-style tournament bracket. Questions are read from slideshows and the first team to buzz in and answer the question correctly wins the point. New brackets are made after each round so teams can continue to advance. Awards are given to the highest scoring teams in each of the divisions.

Overall, Math Bowl is a great opportunity to develop teamwork and communication skills while enhancing mathematical knowledge. Volunteer opportunities are also available for students in high school that still wish to get involved in the math community. For more information on Math Bowl and other math-related events, visit the WSMA webpage. Make sure you keep your calendars open for the event next year!

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The following was the schedule for the day:

8:30-9:00 am: Registration

9:00 – 9:30 am: Welcome, Introductions, Review of General Rules

9:30 – 10:20 am: Sprint Round

10:20 – 10:30 am: Break

10:30 – 11:15 am: Target Round

11:15 – 11:25 am: Break

11:25 – 11:50 am: Team Round

12:00 – 12:45 pm: Lunch

12:45 – 1:15 pm: Special Presentation

1:15 – 1:45 pm: Announcements, Prize Raffle

1:45 – 2:15 pm: Countdown Round

2:15 – 2:45 pm: Team & Individual Awards

For the Sprint Round, competitors worked individually and had 40 minutes to complete 30 questions without a calculator. For the Target Round, competitors also worked individually and received 4 pairs of problems with 6 minutes to complete each, with the use of a calculator.

For the Team Round, competitors in official teams gathered to complete 10 math problems in 20 minutes with the use of a calculator. Solely individual competitors could also participate in unofficial teams.

For the Countdown Round, competitors had a maximum of 45 seconds to answer a question without the use of a calculator. Spectators were allowed after lunch, and during the Countdown Round the room was filled with family and friends supporting the competitors. This round determined the order of the top 10 individuals.

The top placing individuals were:

- David Lee
- Joy An
- Brandon Chen
- Daniel Hong
- Alex Wei
- Justin Chen
- Kevin Zhang
- Fahmid Rahman
- Robert Chen
- Jeremy Lin

The top placing teams are pictured below:

The top 4 individuals in each state will form a state team and compete in both team and individual events in the National Competition in May. Congratulations to them!

For more information about MATHCOUNTS, visit here or https://www.mathcounts.org/.

]]>Competitors are split into three different categories, depending on the level of math they are taking in school. Mu is for students currently taking Calculus, Alpha is for Pre-Calculus, and Theta is for Algebra 2 and below. The competition is mainly geared towards students in high school, but middle school students can participate as well.

There are 4 main tests. The first test is a ciphering, where competitors are given 10 rounds of 4 questions each. This portion of the test focuses on speed, as competitors are only given 4 minutes to answer all 4 problems. Following Ciphering, there are two Topic Tests. Competitors are allowed to chose from various math topics including Logs and Exponents, Geometry, Calculus, Matrices and Vectors, Areas and Volumes, and Probability. Each test focuses specifically on the topic given. Competitors are given 60 minutes to answer 40 multiple choice questions. The last test is the Individual test, which includes questions testing a mix of different math topics. During lunch, there are also a few optional events: Chalk Talk, Jeopardy, and Mental Math.

Individual awards are given for each test and for overall performance as well. Trophies are awarded to the top 10 highest scorers for the Individual and Ciphering tests and the top 5 highest scorers for the Topic Tests. Awards for the top scoring schools are given as well. This year, Thomas Jefferson High School won first, Bellevue won second, and Pullman won third.

Overall, Mu Alpha Theta is a great opportunity for students to gain experience with math competitions and showcase their knowledge in specific math topics. For more information on other math-related events in the community, visit the WSMA webpage.

]]>Each school can send up to 10 individuals to the local MATHCOUNTS Chapter Competition. The highest scoring students from the Chapter Competition then progress to the state level. The top 4 individuals from the state then move on to the National Competition, where the students compete for the title of National Champion.

I’ve conducted a Q&A with former MATHCOUNTS participants to get some firsthand insight on what the competition series is really like. Shifa Somji qualified for the State level Competition and Richard Dong competed at the national level. Both have had great success with the competition.

Q&A

When did you participate in MATHCOUNTS?

S: From 2013 and 2014

R: From 2012 to 2014

What was your favorite part about MATHCOUNTS?

S: The sprint round was my favorite because it was a lot of fun since the questions in that round were the most doable.

R: Countdown was really exciting. I got all the questions wrong, of course, but my adrenaline was really pumping!

If you could do it again, what would you do differently?

S: I’d pick a better team. Our conversations would often go off topic and we’d get really distracted during the competition so it didn’t work out very well.

R: Wear warm clothing during the competition. The competition room was freezing and I had a terrible headache both the day of and the day after the competition.

What was your key to success?

S: Using my favorite pencil really helped me because it made me feel lucky and calmed me down when I was feeling anxious.

R: I took a lot of practice tests with my friends. We’d compare answers, and I was motivated to practice because I wanted to beat all my friends.

What are some important things you would like people to know about the competition?

S: Don’t be scared. If you have faith in yourself and your math skills you’ll do well.

R: MATHCOUNTS is a pretty serious competition, obviously. National qualifiers get a free vacation! But a lot of it is for fun, and meeting other mathletes from other teams. So don’t get too caught up in trying to win. Y’all should have a good time.

MATHCOUNTS is a great opportunity to get more involved in math competitions at a young age. Registration for the School Competition begins in mid-August and you can register here. For more information on math competitions like this, visit http://www.wastudentmath.org/.

]]>KPMT is hosted by Newport Math Club and is aimed towards 5th to 8th graders. The competition is split into two divisions, 5th-6th graders and 7th-8th graders, and it consists of six tests: Individual, Algebra and Operations, Geometry, Mental Math, Probability and Potpourri, and Block Math. While the Individual and Block Math tests are done individually, the other four are completed in teams of 3-4 competitors. Timing for each test varies, but competitors usually get one to two minutes per problem.

One unique aspect of KPMT is the Block Math portion. The Block Math test is divided into five blocks of questions, with each block consisting of three questions. The problems in each block are around the same difficulty, and the difficulty of the blocks increase as the test progresses. When the test is scored, the number of points the competitor has earned for the test is multiplied by a value corresponding to the most difficult block in which he/she completed 2 out of the three problems. For example, if a competitor completed 2 out of three problems in the 4th block, his total score for the test would be multiplied by 3.

Last year, nearly 450 students and 130 teams participated in the competition. Awards are given individually and as teams to those with the highest number of points.

If this kind of competition appeals to you, go to Newport’s KPMT website to register. Late registration is open until December 31st, and the fee is 54$ per team. This is also a great volunteering opportunity for those in high school who are interested in organizing and running competitions. You can sign up to be a proctor, grader, or runner.

For more information on KPMT and other similar competitions and events, visit WSMA’s test hub.

]]>Students competed against their own grade level peers in individual competitions, and teams were divided in two divisions, 9th/10th and 11th/12th. Middle school students are also allowed to participate in the 9th/10th division for teams and compete against 9th graders in the individual competitions.

The competition consisted of the following tests:

- Individual Ciphering – Ciphering had 10 rounds of two questions apiece with 4 minutes to solve each round.
- Individual Test – Each individual test consisted of 50 multiple choice questions
- Team Test – Each team test consisted of 20 multiple choice questions. Teams of up to 4 students competed against each other in the team competition.

The top ten scoring students in each grade for the Individual Test and Individual Ciphering were eligible for trophies. The top ten scoring teams in the 9th/10th and 11th/12th divisions were also awarded trophies.

Fall Classic is a great way to test your math skills, and if you are a current or upcoming high school student, you should consider participating next year! For more information, visit the WSMA hub.

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