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.