30, August 2025

Computational Generation and Analysis of Magic Squares in the Age of Artificial Intelligence

VOLUME-11; ISSUE-8; AUG-2025

Author(s): 1. Rajpal Singh, 2. Kalpit

Authors Affiliations:

1.Faculty, GPS Tatarpur( Pataudi) ,Gurugram, India 

2.Student,Central University of Haryana, India

DOIs:10.2015/IJIRMF/202508035     |     Paper ID: IJIRMF202508035


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Magic squares, ancient mathematical constructs where numbers in a grid sum identically across rows, columns and  diagonals, have entered a new era of discovery through artificial intelligence. This review explores how modern computational approaches—including machine learning, evolutionary algorithms, and reinforcement learning—have transformed magic square generation, analysis, and application. While classical methods (e.g., Siamese, Strachey) remain foundational for small orders, AI techniques now efficiently construct high-order squares (n > 10) and discover previously unknown variants, such as pandiagonal and multimagic squares. Recent advances demonstrate neural networks predicting valid 7×7 squares with 92% accuracy, genetic algorithms optimizing 15×15 configurations, and reinforcement learning agents mastering construction strategies. These AI-generated squares enable novel applications in cryptography (28% stronger S-box resistance), optimization (40% improved load balancing in cloud scheduling), and generative art. However, challenges persist in scalability (exponential complexity for n ≥ 30), interpretability ("black box" AI models), and dataset scarcity. Emerging quantum computing approaches show promise, with early experiments achieving 200× speedups for 4×4 squares. Ethical considerations, including cryptographic dual-use risks and environmental costs (3.2 kWh per 20×20 square), necessitate governance frameworks. Interdisciplinary collaborations are unlocking further potential, from metamaterial design to adaptive math education (35% learning gains). As AI continues bridging ancient mathematics with cutting-edge computation, magic squares evolve from recreational puzzles to tools for scientific and industrial innovation, while raising profound questions about AI's role in mathematical discovery. This synthesis highlights breakthroughs, unresolved challenges, and future directions at this intersection of tradition and technology.

Magic squares, Artificial intelligence, Computational mathematics, Cryptography, Constraint optimization

Rajpal Singh,  Kalpit (2025); Computational Generation and Analysis of Magic Squares in the Age of Artificial Intelligence, International Journal for Innovative Research in Multidisciplinary Field, ISSN(O): 2455-0620, Vol-11, Issue-8, Pp. 267-275.         Available on –   https://www.ijirmf.com/

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