Tower of Hanoi-Mathematical Induction

The Tower of Hanoi (also called the Tower of Brahma or Lucas' Tower,[1] and sometimes pluralized) is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape. The objective of the puzzle is to move the entire stack to another rod, obeying the following simple rules: Only one disk can be moved at a time. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack i.e. a disk can only be moved if it is the uppermost disk on a stack. No disk may be placed on top of a smaller disk. With three disks, the puzzle can be solved in seven moves. The minimum number of moves required to solve a Tower of Hanoi puzzle is 2n - 1, where n is the number of disks.

But you cannot place a larger disk onto a smaller disk. The equation for the tower of Hanoi is 2^n -1. The equation 2^n-1 can be used to solve with any number of discs. Given 12 discs, this would require 4,095 moves.

As disc continue to stack, more moves are required to move them from pole A to C. I also learned that after finding the recursion formula, I was able to plug it into the mathematical induction to make both sides equal.

2nd Semester Summary

It's been really confusing this semester, but it was so much easier than 1st semester. I feel really sad that now I will no longer be with Ms. V. And now I might have to move up to Calc AB and be with Pelletier. I am so scared because my grades were not that grades this year and so that might affect next year, especially knowing how tough Ms. Pelletier might be. D:

Trig Review

In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem's "a2 + b2 = c2" for right triangles. There are loads of trigonometric identities, but the following are the ones you're most likely to see and use.

Notice how a "co-(something)" trig ratio is always the reciprocal of some "non-co" ratio. You can use this fact to help you keep straight that cosecant goes with sine and secant goes with cosine. sin2(t) + cos2(t) = 1 tan2(t) + 1 = sec2(t) 1 + cot2(t) = csc2(t) The above, because they involve squaring and the number 1, are the "Pythagorean" identities. You can see this clearly if you consider the unit circle, where sin(t) = y, cos(t) = x, and the hypotenuse is 1. sin(–t) = –sin(t) cos(–t) = cos(t) tan(–t) = –tan(t) Notice in particular that sine and tangent are odd functions, while cosine is an even function.

Repeating Decimals

A repeating or recurring decimal is a way of representing rational numbers in base 10 arithmetic. The decimal representation of a number is said to be repeating if it becomes periodic (repeating its values at regular intervals) and the infinitely-repeated portion is not zero. For example, the decimal representation of ⅓ becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333…. A more complicated example is 3227 / 555 , whose decimal becomes periodic after the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144…. At present, there is no single universally accepted notation or phrasing for repeating decimals.

Any number that cannot be expressed as a ratio of two integers is said to be irrational. Their decimal representation neither terminates nor infinitely repeats but extends forever without regular repetition. Examples of such irrational numbers are the square root of 2 and pi.