Brahma Tower
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Brahma Tower Keith Thackery Class:
David Tuttle
Liz Murphy
Venkat Ellur
Algebra or pre-algebra.
Materials: Three towers (called Brahma or math towers) marked X, Y, and Z; disks marked A, B, C, D, ... of different radii but the same thickness; paper and pencil. Goals: Develop skills in problem solving, pattern recognition, and formula design. Time Required: Background: formulas.
One or two class periods.
The knowledge of sequences and corresponding
Setting: The following setting is for three disks. Disks A, B, and C are place on tower X in order from smallest (C on top) to largest (A on bottom). In each movement to be made, a larger disk is never placed on top of a smaller disk. After a series of movements, disks A, B, and C must all be on either tower Y or Z in the original order of sizes (smallest on top to largest on bottom). One disk movement constitutes a "move." The same basic problem can be used for 4, 5, or any number of disks. Problem: Move the disks from one tower to another in the minimum number of moves. In listing the positions of A, B, and C on X, Y, and Z, make sure to list them from bottom to top as shown in the sample solution. Keep a record of your trials using the chart provided. After finding the minimum number of moves for 3, 4, 5, 6, ... disks, devise a formula for the minimum number of moves for n disks. Sample Solution: The table below shows the postions of the three disks on the three towers at the completion of each move. Movement #
Tower X
Tower Y
Tower Z
0
A, B, C
------
------
1
A, B
C
------
2
A
C
B
3
A
------
B, C
4
------
A
B, C
5
C
A
B
6
C
A, B
------
7
------
A, B, C
------
Funded in part by the National Science Foundation and Indiana University 1995
Evaluation: Each team will submit a report with charts for their trials properly completed and will be prepared to present their solutions orally. The minimum number of movements for each number of disks and the formula for the minimum number of moves for n disks must be listed to get full credit. Extensions: Teams can find the minimum number of moves for any given number of n disks using four towers. Teacher Notes: Students have to check whether they have done the movements in proper order and the entries are made correctly. They may have to redo the activity to confirm the numbers.
Funded in part by the National Science Foundation and Indiana University 1995
Brahma Tower Student Data Sheet Record moves made to move four disks from Tower X to either Tower Y or Z. Movement #
Tower X
Tower Y
Tower Z
0
A, B, C, D
------
------
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Brahma Tower Student Data Sheet Record moves made to move five disks from Tower X to either Tower Y or Z. Movement # 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
Tower X
Tower Y
Tower Z
A, B, C, D, E
------
------
David Tuttle
Liz Murphy
Venkat Ellur
Algebra or pre-algebra.
Materials: Three towers (called Brahma or math towers) marked X, Y, and Z; disks marked A, B, C, D, ... of different radii but the same thickness; paper and pencil. Goals: Develop skills in problem solving, pattern recognition, and formula design. Time Required: Background: formulas.
One or two class periods.
The knowledge of sequences and corresponding
Setting: The following setting is for three disks. Disks A, B, and C are place on tower X in order from smallest (C on top) to largest (A on bottom). In each movement to be made, a larger disk is never placed on top of a smaller disk. After a series of movements, disks A, B, and C must all be on either tower Y or Z in the original order of sizes (smallest on top to largest on bottom). One disk movement constitutes a "move." The same basic problem can be used for 4, 5, or any number of disks. Problem: Move the disks from one tower to another in the minimum number of moves. In listing the positions of A, B, and C on X, Y, and Z, make sure to list them from bottom to top as shown in the sample solution. Keep a record of your trials using the chart provided. After finding the minimum number of moves for 3, 4, 5, 6, ... disks, devise a formula for the minimum number of moves for n disks. Sample Solution: The table below shows the postions of the three disks on the three towers at the completion of each move. Movement #
Tower X
Tower Y
Tower Z
0
A, B, C
------
------
1
A, B
C
------
2
A
C
B
3
A
------
B, C
4
------
A
B, C
5
C
A
B
6
C
A, B
------
7
------
A, B, C
------
Funded in part by the National Science Foundation and Indiana University 1995
Evaluation: Each team will submit a report with charts for their trials properly completed and will be prepared to present their solutions orally. The minimum number of movements for each number of disks and the formula for the minimum number of moves for n disks must be listed to get full credit. Extensions: Teams can find the minimum number of moves for any given number of n disks using four towers. Teacher Notes: Students have to check whether they have done the movements in proper order and the entries are made correctly. They may have to redo the activity to confirm the numbers.
Funded in part by the National Science Foundation and Indiana University 1995
Brahma Tower Student Data Sheet Record moves made to move four disks from Tower X to either Tower Y or Z. Movement #
Tower X
Tower Y
Tower Z
0
A, B, C, D
------
------
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Brahma Tower Student Data Sheet Record moves made to move five disks from Tower X to either Tower Y or Z. Movement # 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
Tower X
Tower Y
Tower Z
A, B, C, D, E
------
------
