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Debra Locke
Math 151 Summer Semester 00
Hanoi Tower Challenge Problem
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Help with essay on I dunno
The Tower of Hanoi Legend
Monks in a temple in ancient India have to me a pile of 64 sacred disks from one location to another. The disks are fragile; only one can be carried at a time. A disk may not be placed on a smaller, less valuable disk and there is only one other location in the temple (besides the original and destination locations) sacred enough that a pile of disks can be placed there.
The monks start moving disks back and forth, between the original pile, the pile at the new location, and the intermediate location, always keeping the piles in order (largest on the bottom, smallest on the top). Legend has it that before the monks make the final move to complete the new pile in the new location the temple will turn to dust and the world will end.
ProblemThere are n disks of different diameters on three pegs, labeled as Peg 1, Peg , as Peg .At the beginning, all n disks are on Peg 1, arranged in an increasing order of their diameter from top down.
The question is to move all disks to Pegaccording to the following rules
1.Each time, only one disk on top of a peg is moved to the top of another peg.
.No disk can be put on top of a smaller disk.
Solving Idea
The base caseIf there is no disk (n = 0), nothing has to be done.
ReductionNow suppose the number of disks is al least one.Ignore the largest disk, we have n1 disks.A smaller case.
RecursionThe smaller case can be solved by a recursive call to the same algorithm since this reduction can eventually be reduced to the base case.
Building up the solutionMove n1disks from the source peg to the intermediate peg, move one disk from the source peg to the destination peg, and finally, move n1 disks from the intermediate peg to the destination peg.
Equation
DisksMinimum MovesChange
00N/A
111
74
4158
5…..1…..16……
567,057,54,07,7,5= change for 55 rings
Therefore y = n 1 (Minimum moves)
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