Exercises 38-45 involve the Reve's puzzle, the variation of...

Exercises $38-45$ 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 of a smaller one. This algorithm, given the number of disks $n$ as input, depends on a choice of an integer $k$ with $1 \leq k \leq n .$ When there is only one disk, move it from peg 1 to peg 4 and stop. For $n>1$ , the algorithm proceeds recursively, using these three steps. Recursively move the stack of the $n-k$ smallest disks from peg 1 to peg $2,$ using all four pegs. Next move the stack of the $k$ largest disks from peg 1 to peg $4,$ using the three-peg algorithm from the Tower of Hanoi puzzle without using the peg holding the $n-k$ smallest disks. Finally, recursively move the smallest $n-k$ disks to peg $4,$ using all four pegs. Frame and Stewart showed that to produce the fewest moves using their algorithm, $k$ should be chosen to be the smallest integer such that $n$ does not exceed $t_{k}=k(k+1) / 2,$ the $k$ th triangular number, that is, $t_{k-1}<n \leq t_{k}$ . The long-standing conjecture, known as Frame's conjecture, that this algorithm uses the fewest number of moves required to solve the puzzle, was proved by Thierry Bousch in 2014 . Describe the moves made by the Frame-Stewart algorithm, with $k$ chosen so that the fewest moves are required, for $\begin{array}{llll}{\text { a) } 5 \text { disks. }} & {\text { b) } 6 \text { disks. }} & {\text { c) } 7 \text { disks. }} & {\text { d) } 8 \text { disks. }}\end{array}$

Exercises $38-45$ 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 of a smaller one. This algorithm, given the number of disks $n$ as input, depends on a choice of an integer $k$ with $1 \leq k \leq n .$ When there is only one disk, move it from peg 1 to peg 4 and stop. For $n>1$ , the algorithm proceeds recursively, using these three steps. Recursively move the stack of the $n-k$ smallest disks from peg 1 to peg $2,$ using all four pegs. Next move the stack of the $k$ largest disks from peg 1 to peg $4,$ using the three-peg algorithm from the Tower of Hanoi puzzle without using the peg holding the $n-k$ smallest disks. Finally, recursively move the smallest $n-k$ disks to peg $4,$ using all four pegs. Frame and Stewart showed that to produce the fewest moves using their algorithm, $k$ should be chosen to be the smallest integer such that $n$ does not exceed $t_{k}=k(k+1) / 2,$ the $k$ th triangular number, that is, $t_{k-1}<n \leq t_{k}$ . The long-standing conjecture, known as Frame's conjecture, that this algorithm uses the fewest number of moves required to solve the puzzle, was proved by Thierry Bousch in 2014 .
Describe the moves made by the Frame-Stewart algorithm, with $k$ chosen so that the fewest moves are required, for
$\begin{array}{llll}{\text { a) } 5 \text { disks. }} & {\text { b) } 6 \text { disks. }} & {\text { c) } 7 \text { disks. }} & {\text { d) } 8 \text { disks. }}\end{array}$

Key Concepts

Frame's Conjecture

Frame's conjecture asserts that the Frame-Stewart algorithm executes the fewest possible moves required to solve Reve's puzzle. This conjectural optimality was confirmed by Thierry Bousch in 2014, thereby establishing that the recursively structured approach indeed yields the minimal move solution.

Triangular Numbers

Triangular numbers, defined as t_k = k(k+1)/2, play a critical role in the Frame-Stewart algorithm by guiding the optimal choice of k. The algorithm employs these numbers to determine the smallest integer k for which the number of disks n satisfies the inequality t_(k-1) < n ? t_k, ensuring that the number of moves is minimized.

Reve's Puzzle

Reve's puzzle is a variation on the classic Tower of Hanoi problem, distinguished by its use of four pegs instead of three. This alteration introduces added complexity in devising the optimal strategy for moving a stack of disks so that no larger disk ever sits atop a smaller one, leading to a richer recursive structure and more intricate solution methods.

Frame-Stewart Algorithm

The Frame-Stewart algorithm is a recursive solution technique formulated to optimally solve Reve's puzzle. It divides the problem by selecting an integer k and first moving the smallest n-k disks to an intermediate peg, then using the classic three-peg method to relocate the k largest disks to the destination peg, and finally moving the n-k disks onto the larger stack. This algorithm strategically combines recursive moves with the traditional Tower of Hanoi steps.

Recursion

Recursion is a fundamental concept in solving puzzles like the Tower of Hanoi and its variants, including Reve's puzzle. The strategy involves breaking down a large problem into smaller, similar subproblems that are easier to manage and solve, and then combining these solutions to form a complete solution to the original problem.

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