Throwback Fact: Mancala

An example of a modern Mancala board from Europe
Source: https://en.wikipedia.org/wiki/File:Bao_europe.jpg

        Many historians believe that Mancala is the oldest board game in existence. Since Mancala is one of our favorite games in the office we decided that for the throwback fact of the week we would take a look at Mancala.

        Mancala is a/the term used to describe a collection of similar games, of which there are over 800 around the world. Some of the variations include Bao, which is played across the east coast of Africa, Oware, which is played in the Caribbean, and Kalah, which is the modern game played in the US and Europe, which we call Mancala. It is unclear exactly where Mancala originated but many historians believe it was in Africa where players used to carve holes in the ground and use pebbles to play.

Bao players in Zanzibar 
Source: https://en.wikipedia.org/wiki/File:Bao_players_in_stone_town_zanzibar.jpg

        There are many different theories as to what the original purpose of the game was for ancient people. One theory is that it was a record keeping system used to manage debit and credits, which were represented by the two sides of the board. Another theory is that Mancala was a ritual at funerals, weddings and other ceremonies. It could also have just originated as a game, which is its main use now.

        Whatever the origins of Mancala one thing is for sure, ancient people needed to count to play, just like we do now. Mancala is a great game for teaching kids how to count and work on strategic thinking skills as they age. 

Tell us, what are some of your favorite board games?

Source: 1, 2, 3

TeachingCenter - From Comedy Central's Key & Peele

Key & Peele - TeachingCenter

Just added to the Center of Math Liked Videos Playlist

Imagine a world in which society places as much importance on teachers and academics as it does athletes and athletics.

For more math videos visit us at youtube.com/centerofmath

Advanced Knowledge Problem of the Week

Here's a new differential equations Advanced Knowledge Problem of the Week, which we've put up on all our social media.

As always, you can check out the solution below.

Problem of the Week

Here's a new Problem of the Week from the Center of Math, which is a bit harder than some other ladders-resting-on-walls problems you might know.

Figured out the answer? Find out if you have it right  with the solution below...

Throwback Fact: Ada Lovelace and the Analytical Engine

Portrait of Ada Lovelace
image: commons.wikimedia.org

Ada Lovelace was born in London, England in 1815 to a poet father and math-loving mother. After her father left the family when she was just four, Ada’s mother raised her on a strict regiment of science, logic, and mathematics. Due to this she developed an early love for machines and technology, a passion that would lead her down a path of scientific discovery. 

Relationship with Charles Babbage:

In 1833, at around seventeen, Lovelace was introduced to Charles Babbage, a mathematician and inventor who was famous for his work with calculating machines. Babbage became a mentor to Lovelace who was captivated by his innovative work. Soon, Babbage would ask Lovelace to work with him on his newest invention called the Analytical Engine, which was meant to perform mathematical calculations just like a computer.

Lovelace’s Work:

Babbage asked Lovelace to translate an article on the analytical machine written by Italian engineer Luigi Federico Menabrea. While translating the text from French to English, she also added her own thoughts and ideas on the machine. She theorized that the machine could repeat a series of instruction, a process known as “looping” that is still used today by computer programmers. She also created codes so that the device could handle numbers, letters, and symbols. Due to these findings and theories, Lovelace is often considered the first computer programmer. 

Lovelace’s Legacy:

Ada Lovelace envisioned a world where machines would be an integral part of human imagination. At the time, these ideas where dismissed by the scientific community because they were too difficult to comprehend. Just like many other visionaries of her era, only time and knowledge could lead to the acceptance of her work. In the 1970’s, engineers at the United States Department of Defense, inspired by her work on the calculation of Bernoulli numbers, named a computer programing language Ada in her honor. Today, Lovelace also serves as an inspiration to many women in the STEM fields, who see her as a visionary who lived before her time. They celebrate Ada Lovelace Day on the second Tuesday of October in her honor. 

Sources: 1, 2

Advanced Knowledge Problem of the Week

Here's a new Advanced Knowledge Problem of the Week from the Center of Math! This one's from the field of analysis, and deals with a very peculiar metric space, even though it's built out of some of the most basic ones. 

Think you've figured it out? Find out below...

Problem of the Week

Good morning from the Center of Math! Try not to lose your marbles with this probability-based Problem of the Week, which I've put up on our Facebook, Twitter, Google+, and Tumblr pages.

Think you've figured it out? Check out the solution below...

Advanced Knowledge Problem of the Week

Hello everyone! We're starting up a second Problem of the Week series, especially for those of you who want to exercise your more in-depth mathematical skills, from fields such as calculus, set theory, and many others. Don't worry, though-our regular Problems of the Week and their solutions will still be posted every Monday, with the Advanced Knowledge problem going up each Wednesday, and both will be here and across all our social media outlets. Without further ado, here's today's Problem, based on a generalized version of the von Koch fractal curve: 

Think you have the solution? Find out after the break. 

Problem of the Week

        Good morning from the Center of Math! Here's the latest Problem of the Week, which we'll have up on Facebook, Twitter, Tumblr, and Google+ shortly. 

        There are a number of ways to go about solving this problem; here's my method. I began by multiplying both sides of the definition of x by 2, then adding a to both sides and subtracting 2b:

        Rearranging this equation in two different ways yields:

        If x is divisible by 7, then so is 2x, that is, 2x = 7k for some integer k. Then, from equation (5),

        a and k are integers, so 3a - k is as well, thus a - 2b is divisible by 7: this is the "only if" in the problem statement. Working the other way,  if a - 2b is divisible by 7, then it equals 7j for some integer j: using equation (4), I found:

        So 2x must be divisible by 7, and since 2 isn't divisible by 7, x must be: this is the "if" in the statement, and since we've shown both the "if" and the "only if," we've proven the statement true. 

        As you might have guessed, this is a handy test for seeing if a base-10 integer is divisible by 7: just subtract twice the final digit from the number formed by the rest of the digits, and if the result is divisible by 7, so is the  original number. 

        Did you think of a different way to show that this formula works? Let us know in the comments, and be sure to check out the rest of our Problems of the Week. 

Problem of the Week

There are a number of different ways to approach this problem; the one I thought of uses "working backwards." I guessed, since the main complexities of x in the stated form are the radical signs, that x is of the form a+b*sqrt(c) for some rational numbers a and b and some positive integer c that isn't a perfect square. Then,

Since 3 is the only quantity in the radical in the RHS, I surmised that c = 3, and thus 2ab = 4. Because a, b, and c are all rational, neither a^2 nor (b^2)c can contribute to the sqrt(3) term in the RHS. Then, equating the rational terms, I solved for a using the quadratic formula:

As I required a to be rational, a = +/- 2, and thus b = +/- 1 from (5), with the signs on each the same. So, since the convention is to take the positive square root,

Did you think of a different solution? What did you think of my method? Let us know in the comments, and be sure to check out the rest of our Problems of the Week.

Throwback Fact: Stokes' Theorem

George Stokes (1819 - 1903)

The University of Cambridge, founded in 1209, has produced many of the greatest scientific minds of all time. It makes sense that Sir George Stokes, famous in both the fields of mathematics and physics, spent the entirety of his career within those halls.

On this day in 1850, Stokes' theorem appeared in writing for the first time. Stokes' theorem is used in multivariable calculus to relate a line integral to a surface integral. To view the notation used for Stokes' theorem that isn't supported in Blogger, click here; Mathworld did excellent work of describing the math behind the theorem. As a sidenote, this theorem appears as a topic in this Center of Math textbook.

A pictoral representation of Stokes' theorem

At the time that his most famous mathematical theorem was first discovered, Stokes was one year into his position as a professor of mathematics at Cambridge. He kept this position until his death 54 years later in 1903. During his career, Stokes made contributions many fields of science. He published several papers relating to fluid dynamics and motion, fluorescence, polarization, and light. His published, and unpublished, works have done much for the development of mathematical physics.