A legend about a group of ancient priests somewhere in the world tells of them solving a very old puzzle called the Tower of Hanoi. (You may be familiar with this puzzle from the computer program "Dr. Quandary®.") The Tower of Hanoi is a set of 3 spindles or poles upon which are placed disks of descending sizes. (See graphic representation.) The puzzle or challenge is to move the tower of disks to a different spindle following 2 rules:
- You may move only 1 disk at a time.
- At no time can a larger disk be placed on a smaller disk.
How many moves does it take to move the tower? Last year's class made sets of these tower blocks having 5 blocks per set. How many moves does it take to move this tower? What if you had a tower with 6 blocks? 7 blocks? Extension for the Bold: According to one version of the legend, the priests' puzzle has 64 gold disks on 3 diamond spindles. How many moves will it take them using the rules above?
I was looking for a task that encouraged students to develop general rules from seeing patterns and this is such a task. Students had made several towers of wooden blocks in technology education and had been playing with them in the classroom before the task was given. There is also a segment in the computer program Dr. Quandary® that involves moving such a tower and the students had had experience with this also. There are tasks similar to this one in Dale Seymour's Powers of Two Guide and in Equals Math for Girls and Other Problem Solvers. You can purchase versions of this game at many puzzle and gift shops. You can also solve it using a collection of coins. This task would be of most benefit with students who were familiar with the powers of two. Otherwise, transferring the pattern to a general rule may be impossible for them. By counting the number of moves needed to transfer towers of few blocks and keeping the results in an organized table, students should see a pattern based on powers of two and be able to form a general rule based on these observations. They can then use this rule to determine the number of moves for larger towers. By playing the game and recording the number of moves in an organized fashion, the student should see patterns develop. They should be able to transfer this information into a general rule to solve for towers too tall to physically move, while keeping track of the number of moves. I suggest having the games out for several weeks in advance of giving the task so students have knowledge of how the game works and can concentrate on counting the moves. The actual task can be done in one, 1 1/2 hours of class time with additional time outside of class for some students to complete the write-up. Students can construct these towers in technology education. My students made sets of five blocks from 1-1/2 inch to 6 1/2-inch square pieces of 3/4-inch pine boards. Encouraging students to "solve a simpler problem" of moving a single block, a tower of two and a tower of three will help them to develop the pattern if they are having difficulty. Beware of the first student developing the general rule and spreading the information to others (see Apprentice Benchmark).
- *Towers of Hanoi
- Calculators - preferably scientific
*You could use blocks, commercially purchased or stacks of coins (dimes, pennies, nickels, quarters, half-dollars). Stacks of squares of colored paper in ascending sizes will also work. Students definitely need some manipulative materials to solve this task. The general rule is 2n - 1 = number of moves needed to move the tower, with n being the number of blocks in the tower.
This student does understand the basic premise of solving the tower problem for a tower of three blocks. What happens when s/he switches to coins? There is no record of results gathered. The general rule appears and had s/he tested it against his/her clear results for a tower of three blocks, s/he would have seen that it did not work. S/he says that n = the number of rings, but then uses the number of moves for n in his/her example. The graph is extremely unclear and only confuses the reader. This child did not know what s/he was doing after the original tower of five worked and never tried to verify his/her work to see his/her errors.
My suspicion is that this student overheard the general rule being discussed and copied it down, as there is no evidence for how s/he "found" the rule. The diagrams start out all right, but closer examination shows two blocks moved between step four and five. S/he did a good job of keeping track of the moves in a table, but one wonders how many of those numbers were actually discovered through the use of the manipulatives and at what point the general rule was applied. There is sparse mathematical language, even when there was the opportunity (ex. subtract instead of "take away"). There was no reason for this student to give up on finding the number of moves for a tower of 64, as the rule had been established and there were calculators available. This student took a minimalist approach to solving this problem.
This student did a good job of documenting his/her thinking throughout the task. While s/he never discovers the general rule, s/he has a clear understanding of the pattern s/he found and is able to use this pattern to solve all the towers required in the task. S/he skips from 16 to 32 and then to 64 on his/her chart, and I believe his/her answer for 64 blocks has a few too many zeros. S/he has good use of representations throughout the solution to document his/her work. It is easy for the reader to follow his/her thinking without his/her having to write volumes. This student is working on more efficient write ups. I think this is a good example of documenting work efficiently.
This student was able to discover a pattern and use the pattern until s/he could develop the general rule. S/he then used this general rule to solve for a tower of 64 blocks. This sixth grade student has not studied scientific notation enough to fully understand how to interpret the calculator's response to 264, but does explain to the reader how it was derived. The solution is clear, concise, shows good mathematical reasoning, uses good mathematical language and notation and adequate representation to inform the reader of the student's solution without being overly verbose. This was the best response I got to this task from my group of 23 sixth and seventh grade students.
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