Saturday, October 26, 2013

History of Math: Isaac Newton and the Schoolyard Bully

When talking about great mathematicians of the past, many will rank the top three as Archimedes, Gauss, and Newton. I have posted stories about the others, but none about Isaac Newton. So, I think now is a good time.

Isaac Newton
Newton is best known for his work in physics, but he also made huge contributions to calculus, algebra, geometry, and infinite series. Many mathematicians expand their expertise to different diverse branches of math, but Newton stuck to the things that applied the most to his physics and are currently ruling the American school system.

Let me tell you an interesting story about Newton. When he was a young student, he was very shy and not at all the genius that he is known as today. One day at recess, a bully came up to him and punched him in the stomach. Newton chose to fight back, and proceeded to shove his face in the mud. All of his classmates, who did not like this kid, cheered him on as he proved his superiority to the bully.

After this incident, he decided that physical prestige wasn't enough for him, and he wanted mental prestige as well. So, he started working much harder at his schoolwork, and soon after became top of the class, proving to everyone that he was smarter than the bully as well. This motivation could have been what turned him into one of the best scientists and mathematicians of all time.

I think this story shows that anyone who has drive and dedication can become a genius, and it also is a story themed around the negativity of bullying. I also like it because it is an interesting aspect about a mathematician's childhood, which help people get to know who is behind what they are learning and practicing.

Saturday, October 19, 2013

A Quick Way to Check Your Work

In school, the teacher is always on top of you for checking your work. When you do a subtraction problem, solve the reversed addition problem and make sure it is right, when you do an algebra problem, make sure you plug your solution back into the original equation. These are all things that are drilled into our heads, but never quite executed.

I did post a year and a half ago about checking your work in algebra problems: plugging the answer into the original equation (click here to see how to do that). But there is also a shortcut for checking work on plain arithmetic problems as well.

Let's take the problem 138 + 253. I would have went smaller, but the method will be easier to demonstrate with larger numbers.

+ 253

If we add that up normally, we would get:

+ 253

How do we know if that is correct? Well, we do something called mod sums. What that means is we add up the digits in the number, and then add up the digits in this sum, and keep going until we find a single digit number. This is called the number's mod sum or digital root.

So, what is the mod sum of 138? Well, we add up the digits.

1 + 3 + 8 = 12

1 + 2 = 3

So, the mod sum or digital root of 138 is 3. Let's find it for 253.

2 + 5 + 3 = 10

1 + 0 = 1

The mod sum of 253 is therefore 1. Let's find the mod sum of the total and see if you notice the pattern.

3 + 9 + 1 = 13

1 + 3 = 4

So, the two addends have mod sums of 3 and 1. The sum has a mod sum of 4. What is the pattern? That's right, the mod sum of the answer is the sum of the mod sums of the addends. What about a subtraction problem?

- 643

The answer to this problem is 281. But how do we confirm it?

The mod sum of 924 is 6 (9+2+4=15 and 1+5=6) and the mod sum of 643 is 4 (6+4+3=13 and 1+3=4). So, the mod sum of the difference must be the difference of the two mod sums. The mod sum of 281 is 2 (2+8+1=11 and 1+1=2), which is the difference of 6 and 4. So, the answer was correct.

What about a multiplication problem? Say 71 x 55. If you do the math, you will find that the answer is 3905. But let's check it with mod sums.

Mod Sum of 71 = 8
Mod Sum of 55 = 1
Mod Sum of 3905 = 8

8 x 1 = 8

So it is correct. There are some glitches in the technique, but this is the basis of it. You might run into scenarios that I didn't quite explain how to deal with, but feel free to comment. I will be happy to respond with some more specific pointers. Have fun actually checking your work now!

Saturday, October 12, 2013

Math in the News: Rota's Conjecture is Solved

One of the things that lots of people seem to be oblivious to is that mathematics is developing and innovating just as much as any other discipline, which I allude to in many of my presentations. There are many conjectures, or unsolved problems, out there that mathematicians are working on and trying to prove or solve.

Rota's Conjecture was a problem like this, in the branch of matroid theory. This is a diverse area of mathematics that isn't taught or mentioned in the American school system (another concept I allude to in my presentations). So when I read this article about Geoff Whittle solving the problem, I thought it would make for a great post. Here is the story:

Saturday, October 5, 2013

Chomp: A Proof in Game Theory

Game theory and proofs are two of my favorite areas of mathematics; game theory is practical and fun while proofs are interesting and insightful. So, when I learned about this problem that combines the two, I thought that it was definitely worth a post.

This game is called Chomp. It is normally played with just a table of squares, but I find it easier to understand by thinking of a chocolate bar.

The mouth-watering Chomp playing board
Chomp is played where the first player chooses a square on the board, and then takes away everything above and to the right of it (essentially taking a bite out of the top right corner of the chocolate bar). The second player would do the same thing with another remaining square. This process keeps continuing until all that remains is the bottom left square. Whoever is forced to take that square loses.

To better understand how the game works, click here to practice playing it. You will see how easy it is to play and understand.

At this point, any game theorist would be wondering if there is an optimal strategy for this game. From what we saw a couple weeks ago with Anti Tic-Tac-Toe, you might be wondering if symmetry is involved in this game. And yes, you can win this game by playing symmetrical moves in the end game. However, the board is not square, it is a rectangle. So, there cannot be full symmetry.

I do not know what the actual optimal strategy is. But, I do know that one exists that would enable player one to always force a win. I will demonstrate this by an "existence proof" where you prove it exists without finding the actual thing.

Pretend player one just took the top right corner square. This is either a good position or a bad position. If it is a good position, then by definition, player one can continue to play perfectly and force a win. If it is a bad position, then player two must have a responding move that will force them to win.

But, this responding move must be a square that player one could have hit on their first move. Since the top right square really doesn't have an effect on the rest of the board, this would not be a problem. So, player one could have played this strategy, which would allow them to force a win as well.

In either of these situations, player one wins. So, there is our proof. I find these existence proofs really interesting because you don't always think you can know if a statement is true without being able to see an example, but with mathematics, it can be done.