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Number of Magic Squares from Parallel Tempering Monte Carlo

Author

Listed:
  • K. Pinn

    (Institut für Theoretische Physik I, Universität Münster, Wilhelm–Klemm–Str. 9, D–48149 Münster, Germany)

  • C. Wieczerkowski

    (Institut für Theoretische Physik I, Universität Münster, Wilhelm–Klemm–Str. 9, D–48149 Münster, Germany)

Abstract

There are 880 magic squares of size 4 by 4, and 275 305 224 of size 5 by 5. It seems very difficult if not impossible to count exactly the number of higher order magic squares. We propose a method to estimate these numbers by Monte Carlo simulating magic squares at finite temperature. One is led to perform low temperature simulations of a system with many ground states that are separated by energy barriers. The Parallel Tempering Monte Carlo method turns out to be of great help here. Our estimate for the number of 6 by 6 magic squares is(0.17745± 0.00016)×1020.

Suggested Citation

  • K. Pinn & C. Wieczerkowski, 1998. "Number of Magic Squares from Parallel Tempering Monte Carlo," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 9(04), pages 541-546.
    • Handle: RePEc:wsi:ijmpcx:v:09:y:1998:i:04:n:s0129183198000443
    DOI: 10.1142/S0129183198000443
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    Cited by:

    1. Yukito Iba & Nen Saito & Akimasa Kitajima, 2014. "Multicanonical MCMC for sampling rare events: an illustrative review," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(3), pages 611-645, June.

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