Explanation Tower of Hanoi is a mathematical puzzle which consists of three pegs and n number of disks of different sizes, which can slide onto any peg. The objective of the puzzle is to move all the stack to another peg with some rules like one disk can be move at a time, in each move we can take the upper disk from one peg and place it on the top of another peg and no larger disk can be placed on smaller disk. Tower of Hanoi puzzle is all based on three pegs with n number of disks, if we have 3 disks then the minimum number of moves required to solve this puzzle will be 7 and the minimum number of moves required for n disks is 2 n − 1 . Although the tower of Hanoi is three pegs puzzle but there are some variations that were proposed to it like tower of Hanoi with four pegs, it is called Reve’s puzzle and the optimal solution for Reve’s puzzle is given by Frame-stewart algorithm. The objective of the frame-stewart algorithm is same as the tower of Hanoi puzzle but with four pegs. when there is no disks then R ( 0 ) = 0 and if there is only one disk then R ( 1 ) = 1 , where R ( n ) represents the number of moves by frame-stewart algorithm to solve Reve’s puzzle, for n disks R ( n ) = 2 R ( n − k ) + 2 k − 1 where k is chosen to be the smallest integer with n ≤ k ( k + 1 ) / 2

Discrete Mathematics and Its Applications

8th Edition

ISBN: 9781260501759

Author: ROSEN

Publisher: MCG

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Chapter 8, Problem 3WP

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The different variations of the tower of Hanoi puzzle, including those with more than three pegs, those where the disk moves are restricted and those where disks may have the same size.

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Chapter 8, Problem 2WP

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Problem 11 (a) A tank is discharging water through an orifice at a depth of T meter below the surface of the water whose area is A m². The following are the values of a for the corresponding values of A: A 1.257 1.390 x 1.50 1.65 1.520 1.650 1.809 1.962 2.123 2.295 2.462|2.650 1.80 1.95 2.10 2.25 2.40 2.55 2.70 2.85 Using the formula -3.0 (0.018)T = dx. calculate T, the time in seconds for the level of the water to drop from 3.0 m to 1.5 m above the orifice. (b) The velocity of a train which starts from rest is given by the fol- lowing table, the time being reckoned in minutes from the start and the speed in km/hour: | † (minutes) |2|4 6 8 10 12 14 16 18 20 v (km/hr) 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0 Estimate approximately the total distance ran in 20 minutes.

- Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p − 1)/2 multiple of n, i.e. n mod p, 2n mod p, ..., p-1 2 -n mod p. Let T be the subset of S consisting of those residues which exceed p/2. Find the set T, and hence compute the Legendre symbol (7|23). 23 32 how come? The first 11 multiples of 7 reduced mod 23 are 7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8. The set T is the subset of these residues exceeding So T = {12, 14, 17, 19, 21}. By Gauss' lemma (Apostol Theorem 9.6), (7|23) = (−1)|T| = (−1)5 = −1.

Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2 multiple of n, i.e. n mod p, 2n mod p, ..., 2 p-1 -n mod p. Let T be the subset of S consisting of those residues which exceed p/2. Find the set T, and hence compute the Legendre symbol (7|23). The first 11 multiples of 7 reduced mod 23 are 7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8. 23 The set T is the subset of these residues exceeding 2° So T = {12, 14, 17, 19, 21}. By Gauss' lemma (Apostol Theorem 9.6), (7|23) = (−1)|T| = (−1)5 = −1. how come?

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Discrete Mathematics and Its Applications

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The different variations of the tower of Hanoi puzzle, including those with more than three pegs, those where the disk moves are restricted and those where disks may have the same size.

Solution Summary: The author explains the different variations of the tower of Hanoi puzzle, including those with more than three pegs and n number of disks of different sizes.

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To explain: The different variations of the tower of Hanoi puzzle, including those with more than three pegs, those where the disk moves are restricted and those where disks may have the same size.

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Chapter 8 Solutions

To explain:

The different variations of the tower of Hanoi puzzle, including those with more than three pegs, those where the disk moves are restricted and those where disks may have the same size.