scholarly journals What is the least number of moves needed to solve the k-peg Towers of Hanoi problem?

2019 ◽  
Vol 11 (01) ◽  
pp. 1930001 ◽  
Author(s):  
Roberto Demontis
Keyword(s):  
Lower Bound ◽  
Tower Of Hanoi ◽  
Towers Of Hanoi ◽  
Minimal Sequence

We prove that the solutions to the [Formula: see text]-peg Tower of Hanoi problem given by Frame and Stewart are minimal. The proof relies on first identifying that for any [Formula: see text]-disk, [Formula: see text]-peg problem, there is at least one minimal sequence is symmetric. We show that if we order the number moved required for the disks in the minimal symmetric sequence in an increasing manner and obtain the sequence [Formula: see text], then [Formula: see text]. We also prove that the maximum number of disks that can be moved using [Formula: see text] steps is [Formula: see text]. We use these to lower bound the telescopic sum (2) that is a lower bound on the number of moves required for any minimal symmetric sequence. This gives us the required result.

scholarly journals A Note on the Frame-Stewart Conjecture on the Generalized Tower of Hanoi Problem

2019 ◽  
Vol 43 (1) ◽  
pp. 79-83
Author(s):  
AAK Majumdar
Keyword(s):  
Lower Bound ◽  
Tower Of Hanoi ◽  
Minimum Number ◽  
Academy Of Sciences

The generalized Tower of Hanoi with p (≥ 3) pegs and n (≥ 1) discs, proposed by Stewart (1939) is well-known. To solve the problem, the scheme followed is : First, move the tower of the topmost i (smallest, consecutive) discs (optimally) to one of the auxiliary pegs in a tower, using the p pegs; next, move the remaining n – i (largest) discs (optimally) to the destination peg in a tower, using the p – 1 pegs available; and finally, transfer the discs on the auxiliary peg to the destination peg (optimally) in a tower. This is the so-called Frame-Stewart conjecture, which remains to be settled. The minimum number of moves under the scheme is denoted by SF(n, p). Chen and Shen (2004) have re-considered the Frame-Stewart conjecture in more detail, and claimed that SF(n, p) is of the order of 2 [ n ( p 2 )!] 1 / ( p 2 ). This paper gives a better lower bound of SF(n, p), which shows that the claim of Chen et al. (2004) is not correct. Journal of Bangladesh Academy of Sciences, Vol. 43, No. 1, 79-83, 2019

Towers of Hanoi and London: Is the Nonshared Variance Due to Differences in Task Administration?

2000 ◽  
Vol 90 (2) ◽  
pp. 562-572 ◽  
Author(s):  
Marilyn C. Welsh ◽  
Veronica Revilla ◽  
Dawn Strongin ◽  
Michelle Kepler
Keyword(s):  
Executive Function ◽  
Executive Functions ◽  
Internal Consistency ◽  
Repeated Testing ◽  
Tower Of London ◽  
Tower Of Hanoi ◽  
Towers Of Hanoi ◽  
The One ◽  
Normal Adults ◽  
Task Administration

Although it has been assumed that the Tower of Hanoi and Tower of London are more or less interchangeable tasks dependent on executive function, a series of studies in our laboratory have indicated substantial nonshared variance between the performances on the two tasks. The purpose of the present study was to examine how much methods of administration, such as number of trials per problem, contribute to this nonshared variance. A new one-trial version of the Tower of Hanoi was developed to be identical to the Tower of London in four procedural characteristics. The one-trial version of the Tower of Hanoi was administered to 39 normal adults along with the traditional Tower of Hanoi and the Tower of London–Revised in two test sessions 5–7 weeks apart. The correlations between the two tasks were in the same range as found previously with the traditional task, indicating that administration differences do not account for the nonshared variance between the tasks. A reliability analysis of the one-trial tasks showed poor internal consistency. Also, the internal consistency of the 6-trial tower was artificially inflated by aspects of the administration and scoring procedures. Moreover, this task exhibited a ceiling effect on repeated testing. These results suggest that it would be of value to redesign the one-trial Tower of Hanoi systematically to increase its reliability and, potentially, its validity as a measure of executive functions.

scholarly journals A New Lower Bound for the Towers of Hanoi Problem

10.37236/5503 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Codruƫ Grosu
Keyword(s):  
Lower Bound ◽  
General Problem ◽  
Towers Of Hanoi ◽  
Minimum Number ◽  
New Ideas

More than a century after its proposal, the Towers of Hanoi puzzle with 4 pegs was solved by Thierry Bousch in a breakthrough paper in 2014. The general problem with $p$ pegs is still open, with the best lower bound on the minimum number of moves due to Chen and Shen. We use some of Bousch's new ideas to obtain an asymptotic improvement on this bound for all $p \geq 5$.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

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