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# Tower of Hanoi – Solutions

Introduction The Tower of Hanoi is a puzzle popularized in 1883 by Edouard Lucas, a French scientist famous for his study of the Fibonacci sequence. However, this puzzle’s roots are from an ancient legend of a Hindu temple. The legend states that there is a secret room in a hidden temple that contains three large pegs. One of these poles has 64 golden disks stacked upon it, each disk being smaller then the disk underneath it, with the biggest disk at the bottom. Since the beginning of time, monks have been trying to shift the 64 disks onto the third peg. The monks also can only transfer the disks if the following rules are followed. First, the monks can only move one disk at a time, and second, they cannot put larger disks on top of smaller disks. Once all 64 disks are shifted to the third peg, the world will end. After encountering this puzzle (in a simpler form) at a science fair a couple years ago and seeing it occasionally even today in an engineering classroom environment, I decided that this was the perfect opportunity to examine this puzzle at a deeper level. (A basic rendition of a 3 disk tower) Aim My aim is to explore the different patterns that lead to the answer to the legend: how much time would it take for the world to end? Finding an Optimal Strategy In order to get closer to solving this puzzle, the goal will be to find the most efficient way to get 64 disks onto the third peg. To better grasp the mathematical concepts and patterns when solving the tower, it would be easier to look at a simpler version of the puzzle, such as the following 3 disk example. Binary Code (Standard Gray Code) We can relate the pattern seen above to binary code, specifically Standard Gray Code. Standard Gray Code is a binary numeral system where two successive values differ in only one (binary) digit. Using this method may bring us closer to being able to solve the 64 disk tower. If we relate the example in Figure 1 with Standard Gray Code, using 3 binary digits, we are left with something like this: 000 Step 1 001 Step 2 011 Step 3 010 Step 4 110 Step 5 111 Step 6 101 Step 7 100 Step 8 For example, Step 1 is shown by the three digits 000. The next step is 001, changing the digit that corresponds with the smallest disk, which means disk 1 is the first disk to move in the solution. And to continue, Step 2 is 011, showing that now the middle (second) disk is being moved. This method could lead us to the solution of a 64 disk tower, as it would show which disk to move; however, the flaw in this method is that even though the binary digits can show which disk has moved, it does not show where to move it. There are always two possibilities for each disk, and when we factor in the 64 other disks, the calculations get extremely tedious and suboptimal as a solution. Recursive Pattern The next viable solution is finding a recursive pattern to determine how many moves it would take to solve the puzzle, depending on the number of disks. A recursive pattern uses information from the previous step to find the next. In order to move n amount of disks from peg 1 to peg 3, we can again refer to Figure 1. The first step is transferring n-1 disks from peg 1 to peg 3. We assign a variable to the number of moves this takes, in this case, M. Next, transfer the middle disk to peg 2 (step 3) and finally, transfer the remaining disks from peg 3 to peg 2 (step 4). When you move n amount of disks to any peg, the number of moves will be the same, no matter which direction you choose to go. From this, we can find an equation to finding the moves needed for any number of disks: 2M 1, where M equals the number of moves needed to transfer n-1 disks from peg 1 to peg 2. This brings up another flaw to the problem. In order to find how many moves needed to transfer 64 disks, we also need to calculate the number of moves for 63, 62, 61, etc amount of disks as well. Because of this, the recursive pattern cannot be used to find the time it takes before the world ends. However, what the recursive pattern can do is generate numbers that lead into a non recursive pattern. # of Disks # of Moves 2M 1 1 1 2(0) 1 = 1 2 3 2(1) 1 = 3 3 7 2(3) 1 = 7 4 15 2(7) 1 = 15 5 31 2(15) 1 = 31 From Table 2, we can see that the third column represents a geometric progression that can help us find a formula for a non-recursive pattern. Non-Recursive Pattern (Explicit Pattern) When looking back at Table 2, there is a direct correlation that can be made from the number of disks and the number of moves. Recognizing that there is a geometric progression, one could infer the pattern that is being used though the power of two. # of Disks # of Moves 1 21 – 1 = 1 2 22 – 1 = 3 3 23 – 1 = 7 4 24 – 1 =15 5 25 – 1 =31 Therefore the function to find the number of steps with any number of disks would be 2n – 1, with n being the # of disks. Just to further prove that 2n – 1 is the correct function, we can graph 2n – 1 and compare to the number of disks and moves in Table 2. It completely fits the data points, confirming the relation between the points and the function. Now we can just plug in the function: 264 – 1 = 590,000,000,000 years Conclusion In order to move 64 disks from the first peg to the third, the monks would need over 590 billion years, assuming that they can move one disk per second. The function 2n – 1 was found by recognizing the geometric progressions in the recursive formula and using it in an explicit pattern. This function can be used to find the most optimal number of moves it would take to move any number of disks to the third peg. Bibliography Bogomolny, Alexander. “Tower of Hanoi.” Tower of Hanoi. N.p., n.d. Web. 3 Mar. 2017. . Johnson, P. Sam, Recurrence Relations And Their Solutions (Problem : Tower Of Hanoi), 2015 December 26, and 1/1. “Information on Subsets of a Set.” N.p., n.d. Web. 3 Mar. 2017. . Longman, Addison Wesley. “Miller’s Mathematical Ideas, 9th Edition Web Site Chapter 4 — Internet Project.” Miller’s Mathematical Ideas, 9th Edition Web Site Chapter 4 — Internet Project. Pearson Education, n.d. Web. 3 Mar. 2017. Math, Dr. “Ask Dr. Math FAQ: Tower of Hanoi.” Mathforum.org. Drexel University, n.d. Web. 3 Mar. 2017.
ECOM 201 Saudi Electronic Robust Presence in Social Media Platforms Questions.

This is an individual project, which is part from your course score. It requires effort and critical thinking.Use the given cover page below. One mark will be deducted if there is no cover page.Your assignment must be supported by evidence and resources. Otherwise, your answer will not be valid. Use font Times New Roman, Calibri or Arial.Use 1.5 or double line spacing with left Justify all paragraphs.Use the footer function to insert page number.Ensure that you follow the APA style in your project. Your project report length should be between 400 to 500 words.Useful links:http://www.nottingham.ac.uk/studentservices/documents/planning-and-preparing-to-write-assignments.pdfAPA reference system https://student.unsw.edu.au/apaAbout plagiarism http://wts.indiana.edu/pamphlets/plagiarism.shtmlAbout plagiarism https://en.wikipedia.org/wiki/Plagiarism
ECOM 201 Saudi Electronic Robust Presence in Social Media Platforms Questions

ASU MIS Analysis in Decision Making and Strategic Planning Analysis Paper.

In your role as Director of Operations, you have collaborated with department heads to create a cross-functional and collaborative organization, good work. Thus far, Management Information Systems (MIS) has been supporting the organization’s activities, and assisted in organizational management. Now it is time for you to continue with the next phases of the organization’s strategic management plan. In your meeting with the board of directors, you must present on many aspects of the organization, including technology. Specifically, the board needs to understand how your current and future technology implementations will facilitate the decisionmaking process. Please present a detailed analysis of the following topics in the context of decision-making and strategic planning:● Data Mining in the context of predictive analytics & Big Data, as a competitive strategy ● CRM as a source of sustaining competitive advantage ● Hardware & Software in terms of disaster recovery & business process continuity (BPC)● Enterprise Resource Planning (ERP) in terms of sustaining a cross-functional organization ● Telecommunications in terms of using networks & social media to compete on the global stage ● Systems implemented to mitigate the effects of risk on information security and the ethical use of information ● Summary of how to integrate all of these systems
ASU MIS Analysis in Decision Making and Strategic Planning Analysis Paper

## ECON 620 SFSU Financial Crisis and Developing Countries Research Paper

ECON 620 SFSU Financial Crisis and Developing Countries Research Paper.

I’m working on a economics project and need support to help me understand better.

One problem that many developing countries face is the risk of financial
shocks and financial crises. In this class we have studied several financial
crises.
a. What are the characteristics of most developing countries that make
them more susceptible to financial crises?
b. Describe the general process that occurs when a developing country
goes through a financial crisis. Please use at least two examples.
c. What policies would help developing countries be more resilient to
inevitable financial crises?
d. What similarities and differences are there between the financial
crises faced by developed countries and the crises faced by
developed countries.
ECON 620 SFSU Financial Crisis and Developing Countries Research Paper

## Pediatrics and family psychiatry

best essay writers Pediatrics and family psychiatry. Paper details 5 base slides one header one reference total of 7 I can do the first slide header WebEx Class Discussion Assignment Instructions OVERVIEW: The WebEx class discussion provides students with the opportunity to have discussion on the assigned topic for the week/module. Students will also share clinical encounters as it relates to the topic. The WebEx class discussion will allows students to interact, share experience, discuss assigned topic for the week and ask questions relating to practicum experience. Topic: • PMHNP Role in Pediatric Psychiatric Mental Health and Family INSTRUCTIONS: The discussion points will include: a. Evaluation of developmental stage b. Assessment and diagnosis c. Case management and collaboration d. Treatment plan consideration e. Engaging the family or legal guardian The students will come prepared to class to discuss the topic. Students must attend and participate in the discussion. The assignment grade will include a physical presence, participation, and valuable contribution during the class. WEB EX CLASS DISCUSSION ASSIGNMENT GRADING RUBRIC Criteria Levels of Achievement Met Unmet Ability to Participate, Knowledge of Content, Contribution and Attendance 2 – 25 points The student attended the Web-Ex class and was prepared to contribute to the discussion. Student demonstrated knowledge of content and participated in the overall discussion. 1 points The student did not attend the Web-Ex class and did not contribute to the discussion. The student did not demonstrate knowledge of content and did not participate in the overall discussion. Pediatrics and family psychiatry

## final exam Essay

final exam Essay. Problem: You are a sports analyst keeping up with the NHL Hockey teams. You want to track the standings of each team in the eastern conference. Instructions: Download and Open the following file. EX4-Data_4.xlsx Save it as yourname_Q4.xlsx before going on to the next step. Format the worksheet title to the TITLE STYLE. Merge and Center the worksheet title across columns A through E. Format the rage A2:E2 as Heading 2 Use a formula to calculate the points. The points are calculated in the following manner: 2 points for every win 1 point for every OTL (Overtime Losses) 0 points for each loss One example of the formula in column E is: =(2*B3) (1*C3) (0) Calculate the Total, Max, Min and average in each column. Set up the average with 2 decimal places. Center the data in cells B2:E14 and apply all borders. Change the page setup to landscape. Rename the worksheet Statistics and color the tab RED. Apply a conditional format to each team that has less than 70 total points. (Make the cell RED with a bold white font). Add a 3-D Column chart comparing all of the Hockey Teams on the list. Show their wins, losses and over time losses (OTL). Move the chart underneath the table. Format the table to your liking. Chose an color, borders, etc. Save your changes, close the document and submit the workbook.final exam Essay

## PHIL 1001 University of Waterloo Why Does Socrates Not Fear Death Paper

PHIL 1001 University of Waterloo Why Does Socrates Not Fear Death Paper.

I’m working on a philosophy writing question and need a sample draft to help me study.

I want my assignment done asap. It is philosophy assignment. You can read all the description in the attached file. Please do not use any other text than the following information for this assignment:

Socratic Conceptions of the Life Worth Living
? Plato (1966). Apology, H.N. Fowler (trans.), Harvard University Press: Cambridge.
Read the whole dialogue by left-clicking on TEXT under “View Text Chunked By” at:
http://data.perseus.org/texts/urn:cts:greekLit:tlg…
? Aristotle (1999). Nicomachean Ethics, W.D. Ross (trans.), Batoche Books:
Kitchener.
Read Sections 1-5, 7-9, and 13 of Book I, and Sections 1-5 of Book VIII at:
https://socialsciences.mcmaster.ca/econ/ugcm/3ll3/…

PHIL 1001 University of Waterloo Why Does Socrates Not Fear Death Paper

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