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Recent questions tagged kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 8.1 Question 43 (Page No. 512)
Question $38-45$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the Frame-Stewart algorithm for moving the disks from peg $1$ to peg $4$ so that no disk is ... then $R(n) = \displaystyle{}\sum_{i = 1}^{k} i2^{i−1} − (t_{k} − n)2^{k−1}.$
Question $38–45$ involve the Reve’s puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe th...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 42 (Page No. 512)
Question $38-45$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the Frame-Stewart algorithm for moving the disks from peg $1$ to peg $4$ so that no disk is ... that if $k$ is as chosen in question $41,$ then $R(n) − R(n − 1) = 2^{k−1}.$
Question $38–45$ involve the Reve’s puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe th...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 41 (Page No. 512)
Question $38-45$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the Frame-Stewart algorithm for moving the disks from peg $1$ to peg $4$ so that no disk is ever on top ... $R(0) = 0\:\text{and}\: R(1) = 1.$
Question $38–45$ involve the Reve’s puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe th...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 40 (Page No. 512)
Question $38-45$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the Frame-Stewart algorithm for moving the disks from peg $1$ to peg $4$ ... $5$ disks. $6$ disks. $7$ disks. $8$ disks.
Question $38–45$ involve the Reve’s puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe th...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 39 (Page No. 512)
Question $38-45$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the Frame-Stewart algorithm for moving the disks from peg $1$ ... disks are moved. Show that the Reve's puzzle with four disks can be solved using nine, and no fewer, moves
Question $38–45$ involve the Reve’s puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe th...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 38 (Page No. 512)
Question $38-45$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the Frame-Stewart algorithm for moving the disks from peg $1$ to ... are moved. Show that the Reve's puzzle with three disks can be solved using five, and no fewer, moves.
Question $38–45$ involve the Reve’s puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe th...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 37 (Page No. 512)
Question $33-37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on an account by the historian Flavius Josephus, who was part of a band of $41$ ... $J (100), J (1000),\: \text{and}\: J (10,000)$ from your formula for $J (n).$
Question $33–37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on a...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 36 (Page No. 512)
Question $33-37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on an account by the historian Flavius ... induction to prove the formula you conjectured in question $34,$ making use of the recurrence relation from question $35.$
Question $33–37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on a...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 35 (Page No. 512)
Question $33-37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on an account by the historian Flavius Josephus, who was part of a band of $41$ Jewish rebels ...
Question $33–37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on a...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 34 (Page No. 512)
Question $33-37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on an account by the historian Flavius Josephus, who was part of a band ... $m$ is a nonnegative integer and $k$ is a nonnegative integer less than $2m.]$
Question $33–37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on a...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 33 (Page No. 512)
Question $33-37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on an account by the historian Flavius Josephus, who was part of a band of $41$ ... by $J (n).$ Determine the value of $J (n)$ for each integer $n$ with $1 \leq n \leq 16.$
Question $33–37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on a...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 32 (Page No. 512)
In the Tower of Hanoi puzzle, suppose our goal is to transfer all $n$ disks from peg $1$ to peg $3,$ but we cannot move a disk directly between pegs $1$ and $3.$ Each move of a disk must be a move ... top of a smaller disk? Show that every allowable arrangement of the n disks occurs in the solution of this variation of the puzzle.
In the Tower of Hanoi puzzle, suppose our goal is to transfer all $n$ disks from peg $1$ to peg $3,$ but we cannot move a disk directly between pegs $1$ and $3.$ Each mov...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 31 (Page No. 512)
Use the recurrence relation developed in Example $5$ to determine $C_{5},$ the number of ways to parenthesize the product of six numbers so as to determine the order of multiplication. Check your result with the closed formula for $C_{5}$ mentioned in the solution of Example $5.$
Use the recurrence relation developed in Example $5$ to determine $C_{5},$ the number of ways to parenthesize the product of six numbers so as to determine the order of m...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 30 (Page No. 512)
Write out all the ways the product $x_{0} \cdot x_{1} \cdot x_{2} \cdot x_{3} \cdot x_{4}$ can be parenthesized to determine the order of multiplication. Use the recurrence relation developed in Example $5$ to calculate $C_{4},$ ... $(B)$ by finding $C_{4},$ using the closed formula for $C_{n}$ mentioned in the solution of Example $5.$
Write out all the ways the product $x_{0} \cdot x_{1} \cdot x_{2} \cdot x_{3} \cdot x_{4}$ can be parenthesized to determine the order of multiplication.Use the recurrenc...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 29 (Page No. 512)
Let $S(m, n)$ denote the number of onto functions from a set with m elements to a set with $n$ elements. Show that $S(m, n)$ satisfies the recurrence relation $S(m, n) = n^{m} − \sum_{k=1 }^{n−1} C(n, k)S(m, k)$ whenever $m \geq n$ and $n > 1,$ with the initial condition $S(m, 1) = 1.$
Let $S(m, n)$ denote the number of onto functions from a set with m elements to a set with $n$ elements. Show that $S(m, n)$ satisfies the recurrence relation$$S(m, n) = ...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 28 (Page No. 512)
Show that the Fibonacci numbers satisfy the recurrence relation $f_{n} = 5f_{n−4} + 3f_{n−5} \:\text{for}\: n = 5, 6, 7,\dots,$ together with the initial conditions $f_{0} = 0, f_{1} = 1, f_{2} = 1, f_{3} = 2, \:\text{and}\: f4 = 3.$ Use this recurrence relation to show that $f_{5n}$ is divisible by $5$, for $n = 1, 2, 3,\dots .$
Show that the Fibonacci numbers satisfy the recurrence relation $f_{n} = 5f_{n−4} + 3f_{n−5} \:\text{for}\: n = 5, 6, 7,\dots,$ together with the initial conditions $...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 27 (Page No. 512)
Find a recurrence relation for the number of ways to lay out a walkway with slate tiles if the tiles are red, green, or gray, so that no two red tiles are adjacent and tiles of the same color are considered indistinguishable. What are the ... $(A)?$ How many ways are there to lay out a path of seven tiles as described in part $(A)?$
Find a recurrence relation for the number of ways to lay out a walkway with slate tiles if the tiles are red, green, or gray, so that no two red tiles are adjacent and ti...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 26 (Page No. 512)
Find a recurrence relation for the number of ways to completely cover a $2 \times n$ checkerboard with $1 \times 2$ dominoes. [Hint: Consider separately the coverings where the position in the top right corner of the checkerboard ... How many ways are there to completely cover a $2 \times 17$ checkerboard with $1 \times 2$ dominoes?
Find a recurrence relation for the number of ways to completely cover a $2 \times n$ checkerboard with $1 \times 2$ dominoes. [Hint: Consider separately the coverings whe...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 25 (Page No. 511)
How many bit sequences of length seven contain an even number of $0s?$
How many bit sequences of length seven contain an even number of $0s?$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 24 (Page No. 511)
Find a recurrence relation for the number of bit sequences of length $n$ with an even number of $0s.$
Find a recurrence relation for the number of bit sequences of length $n$ with an even number of $0s.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 23 (Page No. 511)
Find the recurrence relation satisfied by $S_{n},$ where $S_{n}$ is the number of regions into which three-dimensional space is divided by $n$ planes if every three of the planes meet in one point, but no four of the planes go through the same point. Find $S_{n}$ using iteration.
Find the recurrence relation satisfied by $S_{n},$ where $S_{n}$ is the number of regions into which three-dimensional space is divided by $n$ planes if every three of th...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 22 (Page No. 511)
a) Find the recurrence relation satisfied by $R_{n},$ where $R_{n}$ is the number of regions into which the surface of a sphere is divided by $n$ great circles (which are the intersections of the sphere and planes passing through ... sphere), if no three of the great circles go through the same point. b) Find $R_{n}$ using iteration.
a) Find the recurrence relation satisfied by $R_{n},$ where $R_{n}$ is the number of regions into which the surface of a sphere is divided by $n$ great circles (which are...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 21 (Page No. 511)
Find the recurrence relation satisfied by $R_{n},$ where $R_{n}$ is the number of regions that a plane is divided into by $n$ lines, if no two of the lines are parallel and no three of the lines go through the same point. Find $R_{n}$ using iteration.
Find the recurrence relation satisfied by $R_{n},$ where $R_{n}$ is the number of regions that a plane is divided into by $n$ lines, if no two of the lines are parallel a...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 20 (Page No. 511)
A bus driver pays all tolls, using only nickels and dimes, by throwing one coin at a time into the mechanical toll collector. Find a recurrence relation for the number of different ways the bus driver can pay a toll of $n$ cents ... which the coins are used matters). In how many different ways can the driver pay a toll of $45$ cents?
A bus driver pays all tolls, using only nickels and dimes, by throwing one coin at a time into the mechanical toll collector.Find a recurrence relation for the number of ...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 19 (Page No. 511)
Messages are transmitted over a communications channel using two signals. The transmittal of one signal requires $1$ microsecond, and the transmittal of the other signal requires $2$ microseconds. Find a recurrence relation ... initial conditions? How many different messages can be sent in $10$ microseconds using these two signals?
Messages are transmitted over a communications channel using two signals. The transmittal of one signal requires $1$ microsecond, and the transmittal of the other signal ...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 18 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length $n$ that contain two consecutive symbols that are the same. What are the initial conditions? How many ternary strings of length six contain consecutive symbols that are the same?
Find a recurrence relation for the number of ternary strings of length $n$ that contain two consecutive symbols that are the same.What are the initial conditions?How many...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 17 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length $n$ that do not contain consecutive symbols that are the same. What are the initial conditions? How many ternary strings of length six do not contain consecutive symbols that are the same?
Find a recurrence relation for the number of ternary strings of length $n$ that do not contain consecutive symbols that are the same.What are the initial conditions?How m...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 16 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length $n$ that contain either two consecutive $0s$ or two consecutive $1s.$ What are the initial conditions? How many ternary strings of length six contain two consecutive $0s$ or two consecutive $1s?$
Find a recurrence relation for the number of ternary strings of length $n$ that contain either two consecutive $0s$ or two consecutive $1s.$What are the initial condition...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 15 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive $0s$ or two consecutive $1s.$ What are the initial conditions? How many ternary strings of length six do not contain two consecutive $0s$ or two consecutive $1s?$
Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive $0s$ or two consecutive $1s.$ What are the initial conditions...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 14 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length n that contain two consecutive $0s.$ What are the initial conditions? How many ternary strings of length six contain two consecutive $0s?$
Find a recurrence relation for the number of ternary strings of length n that contain two consecutive $0s.$ What are the initial conditions?How many ternary strings of le...
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