(ii)ĝuring any move, only smaller disk can be placed above larger disk whereas, no larger disk can be placed above smaller. (i) Only one peg can be taken at a time which is located at top of that peg. A move refers to shifting of disk from a peg to another under the following two conditions: For doing this, we need to move the disks. The task of this mathematical puzzle is to transfer all the disks from the first peg to either second or third. Circular disks of different radius were stacked in one of the pegs in such a way that the larger disk lies at the bottom, the next larger lies above the largest one, until the smallest disk lies at the top as shown in Figure 1. Tower of Hanoi Problem In this interesting mathematical game problem, three identical pegs were placed in a stand. Let £ a„ represent a divergent series of real Finally a geometrical interpretation was provided to have better understanding of the result derived. After doing this, I will determine the Ramanujan summation for the solution obtained. This problem has fascinating solution which I will derive in this paper. In 1883, French mathematician Edward Lucas introduced a wonderful mathematical combinatorial game called Tower of Hanoi Problem (Tower of Brahma in India), named after the Hanoi, the capital city of Vietnam. This idea created plenty of new results and generalizations in analytic number theory. Srinivasa Ramanujan introduced the concept of Ramanujan summation in connection with Bernoulli numbers and Riemann zeta function. ![]() Keywords: Ramanujan Summation, Tower of Hanoi Problem, Definite Integral, Recurrence Relation, Mersenne Numbers. In this paper, after briefly discussing Tower of Hanoi problem and deriving its solution, I had determined Ramanujan summation for the divergent series whose terms represent the solution of Tower of Hanoi problem. Similarly, one of the fascinating combinatorial mathematical games was called Tower of Hanoi Problem. The concept of Ramanujan Summation which is a novel way of assigning particular value to given divergent series was introduced by one of the great Indian mathematicians Srinivasa Ramanujan in early part of 20th Century. ![]() Independent Research Scholar African Moon University, South West Africa and USA I.e MoveDisks ( N - 1, aux_peg, source_peg, dest_peg ).RAMANUJAN SUMMATION FOR TOWER OF HANOI PROBLEM Now that we have N - 1 disks on aux_peg, we could move them to dest_peg using source_peg ( as auxiliary peg ) Move the bottom-most disk N left on the source_peg to dest_peg.ĥ. I.e MoveDisks ( N - 1, source_peg, dest_peg, aux_peg ).Ĥ. First move the top ( N - 1 ) disks from source_peg to aux_peg using dest_peg ( which is used as an auxiliary peg ). Thus the rules of the puzzle are obeyed and we get the below recursive algorithm for solving the puzzle of Tower Of Hanoi.Īlgorithm : MoveDisks ( Integer disks, String source_peg, String using_peg, String dest_peg ) The bottom-most disk that is now left on the source peg is then moved to the desitnation peg. Idea : The idea behind recursion is to move the top ( N - 1 ) disks from source peg to auxiliary peg. If N = 3, we could have 2 3 - 1 = 7 moves.If N = 2, we could have 2 2 - 1 = 3 movesġ: Move the smaller disk at the top from peg A to peg B.Ģ: Move the bigger disk at the bottom from A to peg C.ģ: Move the smaller disk from peg B to peg C. ![]() If N = 1, we could just moved the disk from peg A to peg C without using the auxiliary peg.It can be mathematically proved that the minumum number of moves required to move N disks is 2 N - 1. A bigger disk cannot be placed on a peg containing a smaller disk.īelow is the example of solving the Tower Of Hanoi puzzle with 3 disks.At a time only one disk can be moved and placed on other peg.Objective : The puzzle would be solved if the entire stack of disks on peg A is moved to peg B using some simple rules.I.e the bottom most disk is biggest in diameter and as we go to the top the diameter of the stacked disks decreases. Beginning : The puzzle starts with all the N disks stacked on top of each other in decreasing order of diameter on peg A.Tower of Hanoi is a mathematical puzzle consisting of 3 pegs / towers and some disks of varying diameter.
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