Calculus I Blog

Originally, this puzzle was used to model algorithms. It  is an interesting puzzle indeed. It is 4 x 4 and has 15 small tiles numbered in order from one to fifteen with the last two numbers (fourteen and fifteen) reversed. There is an empty tile positioned in the bottom right.  The goal is to  re-order the pieces so all of them are in the correct position. The empty place should be positioned bottom right still. Depending on how you position the tiles, this puzzle may seem impossible. But do you want to know the secret to solving this puzzle?  This puzzle can only be solved when the number of exchanges necessary to solve the puzzle is even. However, one should plan on using up to 80 moves. Some other popular names for this puzzle include: Gem Puzzle, Boss Puzzle, Game of Fifteen or Mystic Square.

According to Wikihow.com here are easy steps to BEAT the fifteen puzzle!

1. Put each tile in order, one by one, without disturbing the locations of the tiles that are already in place.
2. Locate the tile marked 1, and the empty space tile. Move the empty space around the 1 tile, so that it can be moved towards the top left hand corner. Move 1 into that space, then move the space back around the tile again so that 1 moves in the same direction again. Don’t worry about the other tiles
3. Eventually, using this system, you should get the 1 tile into the top left hand corner. Without disturbing it, do the same with 2 and 3.
4. Move the 4 tile into place without disturbing the order of the 1, 2 and 3 tiles. Rotate the top row to the side of the board with a placeholder where 4 will go, so that it can use two spaces to switch with the place holder. Then the row is switched back into place.
5. Move 4 directly below the space it will go into (top right hand corner). Have the empty space directly to the left of the four. Now, move the entire second row a step closer to 4 (moving the empty space under the 1). Then move 1 down, the entire top row left, 4 up, the tile left of the empty space right, and the tile next to the four down. Then move the rest of the top row next to 4, and the 1 up.
6. Repeat this for the next two rows.
7. The last row is now out of order, but all the others are in order. There is no way to swap the misplaced tiles, because they cannot be moved up. Roll the second to last row into a circle on the left, and rotate the last row pieces (13, 14, and 15) in the square of space on the right. This gives you a second row without disturbing the other pieces, and is accomplished by getting 9 into the second bottom space (from left), the 10 to the left of it, the 11 above this, and the 12 right of this (counterclockwise pattern). Rotate the remaining three pieces in the right four squares. Once you have them in order (thirteen on the bottom left, fourteen on the bottom right corner, fifteen above the fourteen, empty space above the thirteen), unroll the second to last row.
8. The following instructions describe which piece next to the empty space should be moved into it. Left, for instance, indicates that the tile to the left of the empty space should be moved in. Assuming that you have the tiles in the configuration described in step 9, move as follows: Left, Left, up, right, right, right, up, left, left, left, up, right, right, right.

The creator Noyes Chapman was a postmaster from New York who had a fascination with the theory of numbers and puzzles.

Did you know…a man named Bobby Fischer could solve the fifteen puzzle in 25 seconds and has even appeared on The Tonight Show Starrring Johnny Carson. Is anyone up for the challenge??

The Millennium problems, also known as the Millennium Prize Problems were started nine years ago at the turn of the century.  The Clay Mathematics Institute, a private not for profit organization, is dedicated to improve mathematical proficiency and knowledge levels. This Cambridge, MA based institute created seven challenge problems that have been considered “classic questions” that have simply been avoided and remained unsolved. Today there are still six problems with no solution. If someone is able to solve one of the problems first, they are awarded a $1,000,000 prize! The Clay Institute had a professional mathematician write up an official statement of the problem which will then be used when determining if an answer can measure up to what is specifically being challenged. The seven unique problems and their description are as follows:

1.       P versus NP

The question for P vs. NP is whether, for all problems for which a computer can verify a given solution quickly, it can also find that solution quickly.

 This is generally considered the most important open question in theoretical computer science as it has far-reaching consequences in the field of mathematics, philosophy and cryptography.

(The official statement of the problem was given by Steven Cook)

2.       The Hodge conjecture

The Hodge conjecture is for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.

(The official statement was given by Pierre Deligne)

3.       The Ponicaré Conjecture

In topology, a sphere with a 2-dimensional surface is essentially characterized by the fact that it is connected. It is also true that every 2-dimensional surface which is both simply connected and compact is topologically a sphere.

The The Ponicaré Conjecture says that this is also true for spheres with three-dimensional surfaces. The question had long been solved for all dimensions above three. Solving it for three is central to the problem of classifying three manifolds.

This is the only one of the Millennium problems that has been solved. A proof of it was done by Grigori Perelman in 2003, but he did not complete the requirements for the award, and declined a separate award.

(The official statement of the problem was given by John Milnor)

4.       The Riemann hypothesis

The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1/2. A proof or disproof of this would have far-reaching implications in number theory, especially for the distribution of prime numbers. This was Hilbert’s eighth problem, and is still considered an important open problem a century later.

(The official statement of the problem was given by Enrico Bombieri)

5.       Yang Mills Existence and Mass Gap

In physics, classical Yang-Mills theory is a generalization of the Maxwell theory of electromagnetism where the chromo-electromagnetic field itself carries charges. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe mass-less particles.

Another aspect of confinement is asymptotic freedom which makes it conceivable that Yang Mills theory exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum Yang-Mills theory and a mass gap.

(Official statement of the problem was given by Arthur Jaffe and Edward Witten)

6.       Navier–Stokes existence and smoothness

The Navier-Stokes equations describe the motion of both liquids and gases. Although they were found in the 19th century, they still are not well understood. The problem is to make progress toward a mathematical theory that will give us insight into these equations.

(The official statement of the problem was given by Charles Fefferman)

7.       The Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture deals a type of equation that defines elliptical curves over rational numbers. The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions.

(The official statement of the problem was given by Andrew Wiles.)

Hope this information is exciting for all you math wizzes out there… good luck solving!

References

Jaffe, Arthur M. “The Millennium Grand Challenge in Mathematics.” 2000. Pages: 657-660.

URL: http://www.ams.org/notices/200606/fea-jaffe.pdf

Millennium Prize Problems. Clay Mathematics Institute.2009.

URL: http://www.claymath.org/millennium/

A prime number is a natural number that can only be divided by two things, one and itself. It seems to be one of those annoyance numbers that when trying to figure out a problem, let alone be used for anything mathematical.
There are twenty-five prime numbers that fall between one and one hundred. Prime numbers still hold valid questions within the mathematical world such as, is there a formula as to what makes something prime? Or also, how are they distributed? These questions and others have been worked on for ages now. The first talk of prime numbers began in 300 BC by Euclid. It is still being discussed and researched today, considering recently this year it was found to be that the largest prime number has about twelve million decimal digits.
There are multiple ways of verifying that a number is prime. Most were developed by mathematicians and later named after them. The most famous one is Reimann’s hypothesis. In his studies he believes that there is a formula that exists to finding the regularity of prime numbers. His zeta function has been studied by multiple people and has been manipulated in many ways to try and crack the “code”. This code being the fact of finding out the real answer to this mathematical question that has been lurking for centuries.
That is mathematician’s greatest goal, to be able to solve Reimann’s hypothesis and figure out the unsolved mystery of prime numbers. As a current math student I find this goal fascinating. Personally, I think that it would be so frustrating and hard to keep up hope when trying to solve a century long mystery. This is also a feat for any person willing to take up the challenge due to the fact that it is a world known hypothesis and people trying to solve it each and every day.

http://en.wikipedia.org/wiki/Prime_number

http://www.timetoeternity.com/time_space_light/prime_time.htm

How to find the solution to antiderivative problem #46 on page 345

f”'(x)=cos x, f(0)=1, f'(0)=2, f”(0)=3

First find the antiderivative for f”(x)
To find the antiderivative, think about what cos x is the derivative of.
It is the derivative of sin x.
So f”(x)=sin x + C
Now find the antiderivative for f'(x) by observing sin x + C.
The derivative of cos x is -sin x. So the antiderivative of sin x is -cos x.
C is a constant. To take the antiderivative of a constant, multiply by x.
So f'(x)=-cos x + Cx + D
Next find the antiderivative for f(x)
We now know that cos x is the derivative of sin x. So take the antiderivative and multiply it by the constant: -1
Now what is a constant times x the derivative of? Take x to the second power and divide by two. So it is C/2*x^2.
D is a constant. To take the antiderivative of a constant, multiply by x.
So f(x)=-sin x + C/2*x^2 + Dx + E

We can solve for f”(0)=3

Plug in 0 for x in the function f”(x) and solve for the constant C.
We can now substitute 3 in for C.

f”(0)=sin (0) + C=3
f”(0)= 0 + C=3, so C=3
f”(x)=sin x + 3

We can solve for f'(0)=2

Plug in 0 for x in the function f'(x) and solve for the constant D.
-cos 0=-1 and the rest is simple algebra
We can now substitute 3 in for D.

f'(0)=-cos (0) + 3 (0) + D=2
f'(0)= -1 + 0 + D=2, So D=3
f'(x)=-cos x + 3x + 3

We can solve for f(0)=1

Plug in 0 for x in the function f(x) and solve for the constant E.
We can now substitute 1 in for E.

f(0)=-sin (0) + 3/2*(0)^2 + 3 (0) + E=1
f(0)= 0 + 0 + 0 + E=1, So E=1
f(x)=-sin x + 3/2*x^2 + 3x + 1

And there you go, how to answer a simple antiderivative problem.

Patrick Heusmann

Truth Table

The truth table is an issue in math that I had never heard of before today.  So what is a truth Table? The truth table is a mathematical table used in logic.  The tables are usually connected to Boolean algebra, boolean functions, and propositional calculus.  They use the tables to help find the functional values of logical expressions on each of their functional arguments.  Truth tables can also be used to tell whether an expression is logically valid.  Truth tables can only have to possible logical answer those being true or false sometimes written as t or F and also 0 or 1.  Besides true and false there are several other symbols I that are important to know before getting any further in truth tables.  Those being:

  = AND (logical conjunction)

= OR (logical disjunction)

= XOR (exclusive or)

= XNOR (exclusive nor)

= conditional “if-then”

= conditional “(then)-if”

biconditional or “if-and-only-if” is logically equivalent to : XNOR (exclusive nor).

These symbols are important to understand and recognize while using one of the many different types of truth tables.  I’m going to talk eight different types of truth tables.  The first being:

• 1. Logical negation– is an operation on one logical value that produces a value of true if its operand is false and a value of false if its operand is true.
• 2. Logical conjunction -is an operation on two logical values, it has a value of false if both operand are true.
• 3. Logical disjunction– is an operation on two logical values it has a false value if both operand are false.
• 4. Logical implication– deals with to logical values and is false if the first operand is true and the second one is false.
• 5. Logical equality (also known as biconditional) – is an operation on two logical values has true value if both of the operand are either true or both are false.
• 6. Exclusive disjunction -is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if one but not both of its operands is true.
• 7. The logical NAND– is an operation on two logical values is false if both open operands are true or true if any of the operand is false.
• 8. The logical NOR– is an operation on two logical values does the opposite thing as the logical NAND, meaning it is true if both operand are false or false if any operand are true.

Truth tables are used a lot in finding if a expression is logically true or false.  I feel that to truly understand a truth table you need an example so I am going to put the most simple one a Logical negation truth table.

Truth tables are an will always be an important mathematical tool used to help solve and find the logical answer to expressions.

• Enderton, H. (2001). A Mathematical Introduction to Logic, second edition, Harcourt Academic Press. ISBN 0-12-238452-0
• Quine, W.V. (1982), Methods of Logic, 4th edition, Harvard University Press, Cambridge, MA.

Mathematics in Gambling

Gambling gives people the ability to turn little or no money into large sums of cash in a very short amount of time.  Throughout the world gambling is an industry worth hundreds of billions of dollars.  In 2007 alone, the gross revenue for legal gambling in the United States was over 92.27 billion.  I say legal gambling because that figure doesn’t take into amount any sort of wagers made between friends, family, or bookies.  It also does not include internet gambling, which is a relatively new but ever expanding venture.  That figure also does not take into account how much money was actually spent and gambled with, it only represents the amount wagered minus the winnings returned to players.

Recently, the film “21” explored the possibility that gambling can be fixed, and that using mathematics, players can “cheat” casinos for easy money.  Is this possible?  What role do mathematics play in gambling?

One thing that must first be examined is the understanding of dependent and interdependent events.  The flip of a coin can be considered an interdependent event, because no matter how many times you flip it, you still have a 50% chance of getting either heads or tails.  This is an interesting concept to consider when people play the lottery.  Some people believe that picking the same numbers every day will increase their chances, but in reality, you always have the same odds of winning no matter how many times you pick the same numbers.

An example of a dependent event would be pulling cards from a deck.  For example, there are 4 kings in a deck of cards, so each time you pull a king from the deck, you have a smaller and smaller percentage of pulling another king.  Dependent events are especially useful in card games where you can see some, or all of the opponents cards, like blackjack or poker.  When you see what an opponent has, you can make estimated guesses as to what cards are still left in the deck.  The more cards you can see, the easier the guesses become.  For instance, if you were playing blackjack with 23 people, and were one of the last players to receive their second card, it would be fairly easy to deduct what your next card could be, based on the 46 or so cards already placed on the table.  This is one of the reasons why casinos set a max limit on how many people can join one table, because of a tactic known as “counting cards”.

While it is definitely possible to count cards, the movie “21” depicts this action in a somewhat misleading light.  The film makes it seem like counting cards is a very easy skill to pick up.  It is not.  Many books and websites span hundreds of pages trying to explain and speculate on certain strategies.  The film also portrays the main characters as winning almost every hand while they count cards.  This is also inaccurate, because even in simple games like blackjack, counting cards does not automatically ensure a win, it only increases the players odds.  In reality, even a person who has mastered the craft enough to be considered a perfect card counter only increases their odds over the dealer by 1%.  This means that while the average players odds are 49.5% versus the house’s 50.5%, a card counter can expect a 50.5% chance of victory over the houses 49.5.  So realistically, the use of card counting and more broadly, mathematics, are not necessarily feasible ways to get ahead in the world of gambling.

I took Calculus I in high school and I thought it would be a good idea to review over a section that will be coming up soon in our class…antiderivatives.  An antiderivative of a function f is a function whose derivative is f. In other words, F is an antiderivative of f if F’ = f. To find an antiderivative for a function f, we can often reverse the process of differentiation.or example, if f = x⁴, then an antiderivative of f is F = x⁵, which can be found by reversing the power rule. Notice that not only is x⁵ an antiderivative of f, but so are x⁵ + 4, x⁵ + 6, etc. In fact, adding or subtracting any constant would be acceptable.  This should make sense algebraically, since the process of taking the derivative (i.e. going from F to f) eliminates the constant term of F.  Because a single continuous function has infinitely many antiderivatives, we do not refer to “the antiderivative”, but rather, a “family” of antiderivatives, each of which differs by a constant. So, if F is an antiderivative of f, then G = F + c is also an antiderivative of f, and F and G are in the same family of antiderivatives.  To find antiderivatives of basic functions, the following rules can be used:

1. xⁿdx = xⁿ⁺¹ + c as long as n does not equal -1. This is essentially the power rule for derivatives in reverse.

2. cf (x)dx = cf (x)dx. That is, a scalar can be pulled out of the integral.

3. cf (x)dx = cf (x)dx. That is, a scalar can be pulled out of the integral.

4. sin(x)dx = – cos(x) + c
    cos(x)dx = sin(x) + c
    sec²(x)dx = tan(x) + c

These are the opposite of the trigonometric derivatives.

Example Problems from the book: page 345

#1 f(x) = x – 3
We are trying to find the antiderivative of the function, or in other words, the original function before we took the derivative.  So the antiderivative of “x” is ½ x ² .  The antiderivative of “3” is 3 x, so when you put the whole equation together, you get

½ x ² – 3 x + C (always remember the plus C whenever you do the antiderivative, which is any arbitrary constant). 

References:

http://www.onpedia.com/encyclopedia/Antiderivative

http://en.wikipedia.org/wiki/Antiderivative