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PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications A computational phase transition for learning-to-sample from Ising models Covering vertices by sequential stars Fermi-Dirac machines as quantizations of neurons A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation A Tight Bound on Localization of Electrical Flows Optimal Dimension-Free Sampling for Regularized Classification Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs Beyond the Half-Approximation: Fair and Efficient Online Class Matching Efficient Uniform Sampling of Surjections via their Profiles Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth Fairness in Aggregation: Optimal Top-$k$ and Improved Full Ranking Learning-Augmented Online Scheduling with Parsimonious Preemption Entropy Equivalence Testing Lumberjack: Better Differentially Private Random Forests through Heavy Hitter Detection in Trees The Secretary Problem with a Stochastic Precursor Polynomial-Time Robust Multiclass Linear Classification under Gaussian Marginals Efficient Banzhaf-Based Data Valuation for $k$-Nearest Neighbors Classification Block-Sphere Vector Quantization An Approximation Algorithm for Graph Label Selection Iterative Chow Filtering for Learning with Distribution Shift Complexity of Non-Log-Concave Sampling in Fisher Information Stochastic Matching via Local Sparsification Finite Sample Bounds for Learning with Score Matching What is Learnable in Valiant's Theory of the Learnable? Provable Quantization with Randomized Hadamard Transform Min-Max Optimization Requires Exponentially Many Queries Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm A proximal gradient algorithm for composite log-concave sampling
Magnetic Towers of Hanoi and their Optimal Solutions
Uri Levy · 2010-11-17 · via cs.DS updates on arXiv.org

The Magnetic Tower of Hanoi puzzle - a modified "base 3" version of the classical Tower of Hanoi puzzle as described in earlier papers, is actually a small set of independent sister-puzzles, depending on the "pre-coloring" combination of the tower's posts. Starting with Red facing up on a Source post, working through an Intermediate - colored or Neutral post, and ending Blue facing up on a Destination post, we identify the different pre-coloring combinations in (S,I,D) order. The Tower's pre-coloring combinations are {[(R,B,B) / (R,R,B)] ; [(R,B,N) / (N,R,B)] ; [(N,B,N) / (N,R,N)] ; [R,N,B] ; [(R,N,N) / (N,N,B)] ; [N,N,N]}. In this paper we investigate these sister-puzzles, identify the algorithm that optimally solves each pre-colored puzzle, and prove its Optimality. As it turns out, five of the six algorithms, challenging on their own, are part of the algorithm solving the "natural", Free Magnetic Tower of Hanoi puzzle [N,N,N]. We start by showing that the N-disk Colored Tower [(R,B,B) / (R,R,B)] is solved by (3^N - 1)/2 moves. Defining "Algorithm Duration" as the ratio of number of algorithm-moves solving the puzzle to the number of algorithm-moves solving the Colored Tower, we find the Duration-Limits for all sister-puzzles. In the order of the list above they are {[1] ; [10/11] ; [10/11] ; [8/11] ; [7/11] ; [20/33]}. Thus, the Duration-Limit of the Optimal Algorithm solving the Free Magnetic Tower of Hanoi puzzle is 20/33 or 606 0/00. On the road to optimally solve this colorful Magnetic puzzle, we hit other "forward-moving" puzzle-solving algorithms. Overall we looked at 10 pairs of integer sequences. Of the twenty integer sequences, five are listed in the On-line Encyclopedia of Integer Sequences, the other fifteen - not yet. The large set of different solutions is a clear indication to the freedom-of-wondering that makes this Magnetic Tower of Hanoi puzzle so colorful.