Tower Of Hanoi Research Articles

Supervised structure learning

This paper concerns structure learning or discovery of discrete generative models. It focuses on Bayesian model selection and the assimilation of training data or content, with a special emphasis on the order in which data are ingested. A key move—in the ensuing schemes—is to place priors on the selection of models, based upon expected free energy. In this setting, expected free energy reduces to a constrained mutual information, where the constraints inherit from priors over outcomes (i.e., preferred outcomes). The resulting scheme is first used to perform image classification on the MNIST dataset to illustrate the basic idea, and then tested on a more challenging problem of discovering models with dynamics, using a simple sprite-based visual disentanglement paradigm and the Tower of Hanoi (cf., blocks world) problem. In these examples, generative models are constructed autodidactically to recover (i.e., disentangle) the factorial structure of latent states—and their characteristic paths or dynamics.

Different delayed consequences of attaining a plateau phase in practicing a simple (finger-tapping sequence learning) and a complex (Tower of Hanoi puzzle) task.

In practicing a new task, the initial performance gains, across consecutive trials, decrease; in the following phase, performance tends to plateau. However, after a long delay additional performance improvements may emerge (delayed/ "offline" gains). It has been suggested that the attainment of the plateau phase is a necessary condition for the triggering of skill consolidation processes that lead to the expression of delayed gains. Here we compared the effect of a long-delay (24-48h) interval following each of the two within-session phases, on performance in a simple motor task, the finger-tapping sequence learning (FTSL), and in a conceptually complex task, the Tower of Hanoi puzzle (TOHP). In Experiment 1 we determined the amount of practice leading to the plateau phase within a single practice session (long practice), in each task. Experiment 2 consisted of three consecutive sessions with long-delay intervals in between; in the first session, participants underwent a short practice without attaining the plateau phase, but in the next two sessions, participants received long practice, attaining the plateau phase. In the FTSL, short practice resulted in no delayed gains after the long delay, but after 24-48h following long practice, task performance was further improved. In contrast, no delayed gains evolved in the TOHP during the 24- to 48-h delay following long practice. We propose that the attainment of a plateau phase can indicate either the attainment of a comprehensive task solution routine (achievable for simple tasks) or a preservation of work-in-progress task solution routine (complex tasks); performance after a long post-practice interval can differentiate these two states.

Acute Effect of Simultaneous Exercise and Cognitive Tasks on Cognitive Functions in Elderly Individuals with Mild Cognitive Impairment.

The increasing prevalence of age-related cognitive decline, alongside the aging global population, underscores the urgent need for innovative and effective preventative strategies. While the advantages of combining physical and cognitive exercises have been recognized as a promising approach to address these socioeconomic challenges, the acute effects of such interventions on cognitive functions remain understudied. This study aimed to investigate whether simultaneous physical and cognitive exercise has a greater beneficial impact on the cognitive functions of older adults with mild cognitive impairment (MCI) than physical exercise alone or reading activities. A total of 44 MCI patients (75% females aged between 65 and 75 years) were randomly assigned to one of three groups: aerobic exercise alone (EG group, n = 15), aerobic combined with cognitive exercises (CEG group, n = 15), or a reading task for controls (CG group, n = 14). Attention, memory, and problem solving were assessed before and after the acute intervention using the Tower of Hanoi, Digit Span, and Stroop tasks, respectively. Statistical analysis revealed that both of the experimental interventions appeared to enhance cognitive function scores (p < 0.05), except for the number of moves in the Tower of Hanoi task, where no improvement was noted. In contrast, no significant differences in any cognitive performance measures were observed following the reading session. Notably, the CEG group exhibited a more pronounced positive impact, especially on working memory. This advantage was specifically evident in the digit span tasks, where significantly greater percentage gains were found in the CEG than in the CG (p = 0.02), while no significant difference existed between the EG and CG. Simultaneous combined exercise has proven to be a more effective method than aerobic physical exercise alone for improving cognitive function. The results of this study are recommended for inclusion in clinical practice guidelines to maintain the mental health of older adults, as simultaneous exercise seems to offer a time-efficient strategy to enhance cognitive performance in adults with MCI.

Active Equalization for Lithium-Iron Battery Pack Based on Reduced-Order Solving Strategy for the Hanoi Tower Problem

In order to address the energy imbalance issue of a series-connected lithium-iron battery pack, this paper proposes an active equalization method based on a reduced-order solving strategy for the Hanoi Tower problem. The proposed scheme utilizes a combined structure of a switching-network circuit and a bidirectional Cuk converter and leverages an ultracapacitor cell as the energy-transfer carrier. Simulation and comparison demonstrate that the exchange of unbalanced energy within the battery pack can be achieved. The proposed approach can effectively achieve various balancing modes such as cell-to-cell, cell-to-string, string-to-cell, and string-to-string with a relatively fast balancing speed.

Spindle-slow wave coupling and problem-solving skills: impact of age.

We examined how aging affects the role of sleep in the consolidation of newly learned cognitive strategies. Forty healthy young adults (20-35 years) and 30 healthy older adults (60-85 years) were included. Participants were trained on the Tower of Hanoi (ToH) task, then, half of each age group were assigned to either the 90-minute nap condition, or stayed awake, before retesting. The temporal co-occurrence between slow waves (SW) and sleep spindles (SP) during non-rapid eye movement sleep was examined as a function of age in relation to memory consolidation of problem-solving skills. We found that despite intact learning, older adults derived a reduced benefit of sleep for problem-solving skills relative to younger adults. As expected, the percentage of coupled spindles was lower in older compared to younger individuals from control to testing sessions. Furthermore, coupled spindles in young adults were more strongly coupled to the SW upstate compared to older individuals. Coupled spindles in older individuals were lower in amplitude (mean area under the curve; μV) compared to the young group. Lastly, there was a significant relationship between offline gains in accuracy on the ToH and percent change of spindles coupled to the upstate of the slow wave in older, but not younger adults. Multiple regression revealed that age accounted for differences in offline gains in accuracy, as did spindle coupling during the upstate. These results suggest that with aging, spindle-slow wave coupling decreases. However, the degree of the preservation of coupling with age correlates with the extent of problem-solving skill consolidation during sleep.

The use of the tower of Hanoi in vocational education of youths and adults

The use of the tower of Hanoi in vocational education of youths and adults

בדיקת תוקף של מטלה ביצועית בלתי מובנית (UPT), למדידת תפקודים ניהוליים בקרב ילדים

Background: Executive Functions (EFs) are assessed by performance-based measures in clinical settings and rating scales in ecological settings. Research has shown a lack of a correlation between both methods. In order to bridge the gap, the Unstructured Performance-Based Task (UPT) was developed. This assessment includes 42 questions randomly presented on a large piece of paper and is accompanied by a behavioral checklist. Aim: To examine the construct validity of the UPT and the behavioral checklist by examining group differences in UPT scores between children with and without Attention Deficit Hyper Disorder (ADHD) and EF deficits A further aim is to examine associations between the UPT and performance-based measures and rating scales. Method: The UPT was translated and culturally adapted for Hebrew speaking children in Israel. Participants were 42 children (aged 8-11 years), 21 with ADHD and EF deficits and 21 without a diagnosis or EF deficit. Parents completed a sociodemographic questionnaire, the Performance Skills Questionnaire, and the Behavior Rating Inventory of Executive Function (BRIEF) to assess EF deficits in ecological settings. Children completed the UPT and other widely used assessments: Digit Span Forward and Backward, Verbal Working Memory, Trail Making Test, and Tower of Hanoi. Results: Significant differences were found between groups in some of the UPT measures and the behavioral checklist. Associations were found between some UPT measures and the behavioral checklist and performance-based EF measures, as well as the BRIEF rating scale. Conclusions: The measures of Total Correct Items, Total Blank Items, and completion time in UPT may be valid for EFs assessment. The behavioral checklist may be a valid assessment for behavior in ecological settings. The UPT may assist in identifying EF deficits in children and setting goals in Occupational Therapy interventions.

Young children’s transfer of learning on a touchscreen tablet is determined by similarities between tasks and between digital contexts

Transfer of learning is influenced by transfer distance, i.e., the degree of correspondence of contexts as well as of the content to be transferred between a main task and a transfer task (Barnett & Ceci, 2002; Klahr & Chen, 2011). A long tradition of research has used either similarity of tasks or similarity of contexts designs to test transfer distance in problem-solving in children. To our knowledge, no study to date has experimentally varied the degree of similarities between tasks and between contexts jointly, which is the aim of this study. In the present study, we varied the degree of similarity between tasks, by presenting different versions of the Tower of Hanoi (ToH), and between digital learning contexts, by presenting different graphical interfaces. One hundred children aged 6 to 7 years old were divided into four conditions, High similarity of tasks/High similarity of contexts (H/H); High similarity of tasks/Low similarity of contexts (H/L); Low similarity of tasks/High similarity of contexts (L/H); Low similarity of tasks/Low similarity of contexts (L/L). Inhibitory control was also measured, as it may be involved in transfer of learning (Clerc, Miller, & Cosnefroy, 2014; Thibaut, French, Vezneva, Gérard, & Glady, 2011). Transfer of the solving procedure was accompanied by a performance decrement in the L/L and H/L conditions, whereas no such decrement was observed in the L/H and H/H conditions, showing the role of the similarity of digital contexts. Furthermore, inhibition scores predicted transfer performance. Results are discussed considering the theoretical framework of similarities between tasks and between contexts, and the role of inhibition.

Entropy and Multi-Fractal Analysis in Complex Fractal Systems Using Graph Theory

In 1997, Sierpinski graphs, S(n,k), were obtained by Klavzar and Milutinovic. The graph S(1,k) represents the complete graph Kk and S(n,3) is known as the graph of the Tower of Hanoi. Through generalizing the notion of a Sierpinski graph, a graph named a generalized Sierpinski graph, denoted by Sie(Λ,t), already exists in the literature. For every graph, numerous polynomials are being studied, such as chromatic polynomials, matching polynomials, independence polynomials, and the M-polynomial. For every polynomial there is an underlying geometrical object which extracts everything that is hidden in a polynomial of a common framework. Now, we describe the steps by which we complete our task. In the first step, we generate an M-polynomial for a generalized Sierpinski graph Sie(Λ,t). In the second step, we extract some degree-based indices of a generalized Sierpinski graph Sie(Λ,t) using the M-polynomial generated in step 1. In step 3, we generate the entropy of a generalized Sierpinski graph Sie(Λ,t) by using the Randić index.

Interference Effects on Procedural Memory

Three experiments were conducted assessing interference effects acting on procedural task learning and performance using the Tower of Hanoi puzzle as the target procedural task. Experiment 1 found that participants in the no retroactive interference (RI) group completed the Tower of Hanoi task faster than the RI groups, with no differences between the type of the tactile interference. These results indicate that RI may hinder procedural task performance, but does not differentiate between consolidation and retrieval. Experiment 2 addressed RI placement by manipulating the delay prior to the interference task. Subsequent analyses suggested that interfering with only consolidation or recall does not produce interference effects, but qualitative feedback regarding emotion raised further questions about the effects of affect induction. Experiment 3 attempted to address the influence of a proactive and retroactive affect induction technique, but was unsuccessful in producing a dichotomous happy versus neutral effect. However, it was noted that alertness changed across the task acquisition stage, and the change in alertness may account for enhanced procedural memory acquisition. Overall, the series of experiments conducted provides conceptual replication of some previous research while contradicting other findings. This research not only extends our current understanding but also highlights the need to replicate research and continue to explore factors that may influence the practice or acquisition of novel procedural tasks.

A tour of general Hanoi graphs

The Tower of Hanoi puzzle has fascinated researchers in mathematics and theoretical computer science for over a hundred years. Many variants of the classical puzzle have been posed, such as allowing more than 3 pegs, and adding restrictions on the possible moves between the pegs. It is natural to view the pegs and allowed moves by means of a graph. The pegs are represented by the vertices of the graph, and the allowed moves by the edges. For each n, this graph yields a graph on the set of all legal configurations of n disks. Thus, the questions about the original puzzle may be reformulated as questions about connectivity and shortest paths in the graph of all configurations. Moreover, it was shown that classical Hanoi graphs are related to several interesting and useful structures such as the Sierpiński gasket and Gray codes, and thus several properties of these graphs were studied, including Hamiltonicity and planarity. In this paper we study these properties, and several others – chromatic number, clique number, and girth – for general Hanoi graphs.

Correction to Mneimneh (2019)

In the article “Simple variations on the Tower of Hanoi: A study of recurrences and proofs by induction” by Saad Mneimneh (Teaching Mathematics and Computer Science, 2019, 17(2), 131–158. https://doi.org/10.5485/TMCS.2019.0459), there was an error in Table 1 (p. 155), and consequently, the first paragraph of Section 8 (p. 154) also needed correction.

The effects of arts and crafts therapy on post-stroke executive dysfunction: a pilot randomized control test

BackgroundExecutive function is an important determinant of independent living among stroke survivors. Patients with post-stroke executive dysfunction (PSED) have a lower engagement in therapy activities and reduced independent living abilities. One potential method for improving executive function and engagement is arts and crafts therapy (ACT). This study aimed to explore the effects of ACT on patients with PSED.MethodsThe study was a pilot randomized controlled trial (RCT) with two groups: an experimental intervention group receiving ACT combined with individual rehabilitation therapy (IRT) and an active control group receiving IRT only. Fifty-seven patients with PSED participated. Outcome measures included the Trail Making Test (parts A and B), the Stroop test, the Tower of Hanoi (TOH), and the Lawton–Brody Instrumental Activities of Daily Living Scale. Two groups were compared at baseline and 4 weeks after the completion of the training.ResultsThere were significant differences in the Trail Making Test part A (TMT-A; time; p &amp;lt; 0.01), the TMT part B (TMT-B; p &amp;lt; 0.05), the TMT-B (errors; p &amp;lt; 0.01), and the Stroop test (time; p &amp;lt; 0.01) between the experimental and the control group after 4 weeks. There were no significant differences in TMT-A (errors), Stroop test (errors), TOH, and Instrumental Activities of Daily Living Scale.ConclusionThe therapeutic use of arts and crafts could be an effective intervention to improve executive function and self-efficacy for stroke survivors.Clinical trial registrationhttp://www.chictr.org.cn; Identifier: ChiCTR2200063547.

Fibonacci-like Sequences for Variants of the Tower of Hanoi, with Corresponding Graphs and Gray Codes

Fibonacci-like Sequences for Variants of the Tower of Hanoi, with Corresponding Graphs and Gray Codes

The influence of cognitive training using mobile applications on attentional control and impulsivity among pre-service teachers

The purpose of this study was to investigate the influence of cognitive training (CT) using mobile applications on attentional control and impulsivity among pre-service teachers. Pre-service teachers were divided into two groups: experimental (n=25) and control (n=14) groups, they were selected from a large sample (n=718). Over 28 sessions, the training group engaged in CT tasks (the tower of Hanoi [TOH] and simple reaction time [SRT]), whereas the control group did not participate in training sessions or understand the main goal of the study. In the criterion tasks (matching familiar figures and numerical Stroop), all participants were pre- and post-tested. CT using mobile applications helps us see how the trained group’s attentional control and impulsivity had influenced. we also observed the progress of trained group as measured by number of moves or time for TOH, and reaction time (RT) for SRT. Other effects were observed in comparison to a control group that underwent no training. There were changes in impulsivity in post-test in favor of training group according to number of moves, and time component of matching familiar figures test. Corresponding to attentional control, the finding indicated that there were changes in (number and size) RT component in post-test in favor of training group.

Out-of-Distribution Generalization by Neural-Symbolic Joint Training

This paper develops a novel methodology to simultaneously learn a neural network and extract generalized logic rules. Different from prior neural-symbolic methods that require background knowledge and candidate logical rules to be provided, we aim to induce task semantics with minimal priors. This is achieved by a two-step learning framework that iterates between optimizing neural predictions of task labels and searching for a more accurate representation of the hidden task semantics. Notably, supervision works in both directions: (partially) induced task semantics guide the learning of the neural network and induced neural predictions admit an improved semantic representation. We demonstrate that our proposed framework is capable of achieving superior out-of-distribution generalization performance on two tasks: (i) learning multi-digit addition, where it is trained on short sequences of digits and tested on long sequences of digits; (ii) predicting the optimal action in the Tower of Hanoi, where the model is challenged to discover a policy independent of the number of disks in the puzzle.

The weighted Tower of Hanoi

The weighted Tower of Hanoi problem is a new generalization of the classical Tower of Hanoi problem, where a move of a disc between two pegs [Formula: see text] and [Formula: see text] is weighted by a positive real [Formula: see text]. This new problem generalizes the concept of finding the minimum number of moves to solve the Tower of Hanoi, to find a sequence of moves with the minimum total weight. We present an optimal dynamic algorithm to solve the weighted Tower of Hanoi problem, as well as some of its properties. We also show the link of this generalization to the Tower of Hanoi variants with restricted disc moves, and how this new problem can generalize the concept of restricting a disc move.

Influence of Age, Gender and Education Level on Executive Functions and Functioning in People with Stroke.

Alterations in mental functions are among the most frequent manifestations of stroke that have a direct impact on the patient's functionality. The objective of this study was to analyze the relationship of sociodemographic variables with the executive functions (EFs) of participants with right middle cerebral artery (MCA) stroke. A cross-sectional observational case-control study was conducted at the State Center for Brain Damage in Madrid, Spain. Fifty-eight subjects were recruited and divided into two groups. Each participant was administered the following: the FIM+FAM Functional Assessment Measure, the Lawton and Brody scale, The Trail-Making Test, the Zoo Map Test and the Hanoi Tower. Statistically significant differences (p < 0.05) were identified between participants with ischemic stroke and control in functional and EF functions, as well as between participants with hemorrhagic stroke and control. No statistically significant differences were found in the experimental group between subjects who had sustained ischemic and hemorrhagic stroke. No significant associations were identified between the variables age, gender and education level in relation to functionality and executive functions (p > 0.05) in people with stroke. People who have suffered a right cerebral artery stroke have deficiencies in the EFS, resulting in poorer performance of the activity of daily living, compared to healthy subjects of the same age, gender and education level. In the correlational analysis of the stroke participants, no significant associations were identified between the variables gender, age and education level in relation to functionality and EF.

Eye movements during phasic vs. tonic REM sleep are biomarkers of dissociable EEG processes for the consolidation of novel problem-solving skills.

The hallmark eye movement (EM) bursts that occur during Rapid Eye Movement (REM) sleep are markers of consolidation for procedural memory involving novel cognitive strategies and problem-solving skills. Examination of the brain activity associated with EMs during REM sleep might elucidate the processes involved in memory consolidation, and may uncover the functional significance of REM sleep and EMs themselves. Participants performed a REM-dependent, novel procedural problem-solving task (i.e., the Tower of Hanoi; ToH) before and after intervals of either overnight sleep (n=20) or a daytime 8-hour wake period (n=20). In addition, event-related spectral perturbation (ERSP) of the electroencephalogram (EEG) time-locked to EMs occurring either in bursts (i.e., phasic REM), or in isolation (i.e., tonic REM), were compared to sleep on a non-learning control night. ToH improvement was greater following sleep compared to wakefulness. During sleep, frontal-central theta (~2-8 Hz) and central-parietal-occipital sensorimotor rhythm (SMR) activity (~8-16 Hz) time-locked to EMs, were greater on the ToH night vs. control night, and during phasic REM sleep, were both positively correlated with overnight memory improvements. Furthermore, SMR power during tonic REM increased significantly from the control night to ToH night, but was relatively stable from night-to-night during phasic REM. These results suggest that EMs are markers of learning-related increases in theta and SMR during phasic and tonic REM sleep. Phasic and tonic REM sleep may be functionally distinct in terms of their contribution to procedural memory consolidation.

On Parking Functions and The Tower of Hanoi

The displacement of a parking function measures the total difference between where cars want to park and where they ultimately park. In this article, we prove that the set of parking functions of length n with displacement one is in bijection with the set of ideal states in the famous Tower of Hanoi game with n + 1 disks and n + 1 pegs, both sets being enumerated by the Lah numbers.