· Activity Name: Patterns: Towers of Hanoi

· Objectives:

The students will explore the Towers of Hanoi problem to discover the number of moves necessary for a given number of disks and a pattern for disk movements.

· EALR/Standards:

1.3 understand and apply concepts and procedures from geometric sense.

1.5 understand and apply concepts and procedures from algebraic sense.

2.1 investigate situations.

2.3 construct solutions.

3.1 analyze information.

3.3 draw conclusions and verify results.

4.1 gather information.

4.2 organize and interpret information.

· Materials:

Towers of Hanoi puzzles (see teacher notes) one per student is best

Worksheets

Scrap paper that is blank on one side –Many pages per student

· Teacher Notes

o Prerequisites for the learner:

None.

o Teacher hints for the activity:

This activity is best as a multi-day homework assignment with minimal time in class for questions and clarifications throughout the time span.

It works well in conjunction with the dragon curves and ABACABA lessons.

To make the puzzles cheaply, the students can simply cut out different size squares and pile them on three different papers. To make them a little more durable, use dowels stuck into styrofoam or wood, and them buy washers from the local hardware store (Some stores may even donate them especially if you return them when finished).

The scrap paper is used to record the moves for each trial. Depending on how they attack the problem, some students will use MANY pieces of paper.

o Assessment suggestions:

Collect the worksheets.

· The Activity:

Begin by handing out the worksheets and the Towers of Hanoi puzzles. If the students are to make their own, hand out the materials for making them. Allow the students to read through the problem, and to ask any clarifying questions.

Possible questions: Do we have to move all 19 disks?

**You need to know how to do it

Can we just do the 19 disks and skip the rest?

**Yes, but you must write out the moves, and don’t forget, it is due in ____ days

Have the students begin working or take it home.

· Extensions:

Have students explore ABACABA or Dragon curves.

Towers of NWMI

(Hanoi)

In a summer workshop many years ago, there was once a group

of teachers who wanted to get college credit without doing all of the assigned work. They asked the trusted grant writer if there was another way for them to get the credit. Thinking hard, Dr. King replied “ I have an assignment that will test your abilities as mathematicians. Solving this problem will determine if you receive the credit.”

The Conditions:

**You have 19 disks stacked on a pole.

**The disks are staked according to size: A larger disk may never rest upon a smaller one.

**You have three poles

**You have 2 days to complete the task

The problem:

You must move the disks from the current pole to either of the other two poles so that the finished product will once again have two empty poles and one pole with 19 disks.

Like the teachers, you have two choices for this assignment:

Choice 1: Your job is to solve the problem for the teachers. You need to document your strategy(ies) and your results thoroughly in order to receive full credit.

Choice2: You will need to discover the number of moves it will take for any given number of rings, and explain what the pattern would be for moving the rings (i.e. move A to 2 then B to 3 etc.) You may use the following chart to help with these questions.

Number Moves Moves Number Difference Difference

of Disk tower of between n as a power

Disks Labels numbers moves moves and of 2

(Separate with comas) (Separate with comas) (n-1) moves

1

____________________________________________________________________________________

2

____________________________________________________________________________________

3

____________________________________________________________________________________

4

____________________________________________________________________________________

5

____________________________________________________________________________________

6

____________________________________________________________________________________

7

____________________________________________________________________________________

8

Number Moves Moves Number Difference Difference

of Disk tower of between n as a power

Disks Labels numbers moves moves and of 2

(Separate with comas) (Separate with comas) (n-1) moves

1 A 2 1 1 2⁰

____________________________________________________________________________________

2 A,B,A 2,3,3 3 2 2¹

____________________________________________________________________________________

3 A,B,A,C,A,B,A 2,3,3,2,1,2,2 7 4 2²

____________________________________________________________________________________

4 A,B,A,C,A,B,A,D,A,B,A 2,3,3,2,1,2,2,3,3,1, 15 8 2³

C,A,B,AD,A,B,A 1,3,2,3,3

____________________________________________________________________________________N ABACABA continued Match up the letters 2ⁿ-1 2^n-1 2^n-1

with the numbers to

see how each letter

moves in order either

counting up (1,2,3)

or down (3,2,1).