Tower of Hanoi

On Thursday and Friday I worked on a Tower of Hanoi puzzle.  The tower of Hanoi is a mathematical puzzle that consists of three rods and a certain number of disks that are in ascending order on one rod.  The goal of this puzzle is to move all the disks to the third rod without placing a larger disk on top of a smaller disk.  For example if you have three disks on the first rod the goal would be to rearrange those disks on the third rod.  Although it may sound easy it gets very complicated very fast.  In our group we determined that to move 2 disks it would take 3 moves, 3 disks: 7 moves, 4 disks: 15 moves, 5 disks: 31 moves, 6 disks: 63 moves...etc.  We also found the general formula for this puzzle which is Tn=2n-1 which means that however many disks there are you subtract that by one and raise 2 to the power of that.  For the recursive formula we found that the formula is Tn-1=(Tn(2))+1 which means that you take the number of disks and subtract that by one and then multiply that by 2 and add 1 which will get you the number of moves.  We then used mathematical induction to prove the general formula by assuming true for n=k which made the formula Tk=2k-1 and then we showed true for k+1 which made the formula Tk+1=2(k+1)-1.  When you do the algebra the equation works out to equal 2k+1-1 on both sides.  Because they equal each other this shows that mathematical induction works.  We know that mathematical induction works to prove something when you can get both sides to equal the same thing.  

Sequences and series

Sequences are shown as things like a1, a2 a3 a4......an with n being the last term.  An arithmetic sequences is a sequence that is made by adding or subtracting. A geometric sequence is a sequence that is made by multiplying or dividing.  The general equation of an arithmetic sequence is an=a1+(n-1)d and the recursive equation is an+1=an+d where d is the common difference between an+1 and an.  The general equation of a geometric equation is an=a1rn-1 and the recursive equation is an+1=anr where r is the common ration of an+1/an.  Another thing we looked at is series which uses the symbol ∑.  This is used for summation notation and on top of the symbol is n (where you end) to the right of the symbol is the equation and the bottom of the equation is where you start.  To use summation notation you start at the number it tells you to and plug the numbers into the equation until you reach the number at the top.  Here is an example of summation notation.

Graphing inequalities

In order to graph systems of inequalities you first graph the equation like you normally would.  For example, if your equation is 2x-3y>12 you would graph the line 2x-3y=12.  You can also graph parabolas using the equation y=(x-h)^2+k and circles using the equation x^2+y^2=r^2.  After you graph you pick a test point that is not on the line, for example (0, 0) as long as your line/graph does not pass through that point.  Plug the test point into the equation and if the point satisfies the equation then shade the plane that contains the test point.  If the point does not satisfy the equation then shade the plane that does not contain the test point.  The shaded portion is the answer.  Here is an example of a graph of an inequality.  

 

Cramers rule

Cramer's rule is another way to solve a system of equations without using substitution or elimination.  The equation for cramer's rule is x=Dx/d y=Dy/d z=Dz/d.  To use this equation you have to first find D which is the determinant of the equation.  To find x you substitute the answers into the x slot and then find the determinant and divide that by D.  You do this to all of the variables to find them.  If D=0 then the equation is inconsistent and has no answers.  Here is an example of Cramer's rule. 

Systems of equations

Systems of Equations are two equations that need to be solved together.  There are two types of answers to the equations; inconsistent or consistent.  Inconsistent equations have no solution and are usually parallel lines.  Consistent equations have solutions and if there is only 1 solution it is independent and if there are infinite solutions than it is dependent.  You can solve these equations through either substitution or elimination.  With substitution you first solve 1 equation for 1 variable and then substitute that into the other equation.   Then you solve for the variable and substitute back to find the 2nd variable. With elimination you interchange any two equations and multiply by a constant.  Then you add one equation to the other to eliminate a variable. Here is an example of this kind of problem. 

Polar coordinates

Polar coordinates are similar to coordinates on a unit circle but a little different.  To graph polar coordinates you need to use r and theta.  Theta is the angle of the coordinate and r is equal to the radius.  There are four different ways to graph polar coordinates with a positive r and theta, negative r and theta, positive r negative theta, negative r positive theta.  You can also convert rectangular points to polar points using  the formula r^2=x^2+y^2; tantheta=y/x.  To convert polar coordinates to rectangular use the formula x=rcostheta y=rsintheta.  Here is a picture of a polar coordinate graph.  

Graphs of polar equations

There are several different graphs and equations for the polar graph.  For circles at the origin the equation is r=a and |a|=radius and the equation for spirals is r=atheta.  For circles with a center on an axis the equations are r=asintheta; r=acostheta and a=diameter.  The equations for cardiods are r=a+asintheta; r=a-sintheta; r=a+acostheta; r=a-costheta. Limacons have a similar equation with it being r=a+bsintheta; r=a-bsintheta; r=a+bcostheta; r=a-bcostheta when a/b is less than 1.  The equation for rose curves are r=asin(n)theta and r=acos(n)theta and an odd n means n is the number of petal while an even n means that 2n equals the number of petals.  Lastly the equations for lemniscates are r^2=acos2theta and r^2=asin2theta.  Here is a picture of some of the graphs of polar equations.