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Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS. (English) Zbl 1149.05302

Summary: All 8th order Franklin bent diagonal squares with distinct elements \(1,\dots ,64\) have been constructed by an exact backtracking method. Our count of 1,105,920 dramatically increases the handful of known examples, and is some eight orders of magnitude less than a recent upper bound. Exactly one-third of these squares are pandiagonal, and therefore magic. Moreover, these pandiagonal Franklin squares have the same population count as the eighth order \` complete\', or \` most-perfect pandiagonal magic\', squares. However, while distinct, both types of squares are related by a simple transformation. The situation for other orders is also discussed.

MSC:

05A99 Enumerative combinatorics
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[1] Ahmed, M.M. 2004 How many squares are there, Mr. Franklin?. <i>Am. Math. Mon.</i>&nbsp;<b>111</b>, 394–410. · Zbl 1055.05502
[2] Ahmed, M. M. 2004<i>b</i> Algebraic combinatorics of magic squares. Ph.D. dissertation, University of California, Davis.
[3] Frost, A.H. 1878 On the general properties of Nasik squares. <i>Q. J. Math.</i>&nbsp;<b>15</b>, 34–49. · JFM 09.0124.05
[4] Jacobs, C.J. 1971 A reexamination of the Franklin square. <i>Math. Teach.</i>&nbsp;<b>64</b>, 55–62.
[5] Loly, P.D. & Steeds, M.J. 2005 A new class of pandiagonal squares. <i>Int. J. Math. Educ. Sci. Tech.</i>&nbsp;<b>36</b>, 375–388.
[6] McClintock, E. 1897 On the most perfect forms of magic squares, with methods for their production. <i>Am. J. Math.</i>&nbsp;<b>19</b>, 99–120. · JFM 28.0198.03
[7] Ollerenshaw, Dame K. 1986 On ’most perfect’ or ’complete’ 8{\(\times\)}8 pandiagonal magic squares. <i>Proc. R. Soc. A</i>&nbsp;<b>407</b>, 259–281. · Zbl 0602.05022
[8] Ollerenshaw, K. & Brée, D.S. 1998 Most-perfect pandiagonal magic squares: their construction and their enumeration. Southend-on-Sea: The Institute of Mathematics and its Applications, Foreword by Sir Hermann Bondi, F.R.S. · Zbl 0941.05003
[9] Pasles, P.C. 2001 The lost squares of Dr. Franklin: Ben Franklin’s missing squares and the secret of the magic circle. <i>Am. Math. Mon.</i>&nbsp;<b>108</b>, 489–511. · Zbl 0992.05028
[10] Pasles, P.C. 2003 Franklin’s other 8-square. <i>J. Recreational Math.</i>&nbsp;<b>31</b>, 161–166.
[11] Pinn, K. & Wieczerkowski, C. 1998 Number of magic squares from parallel tempering Monte Carlo. <i>Int. J. Mod. Phys. C</i>&nbsp;<b>9</b>, 541–546, (doi:10.1142/S0129183198000443).
[12] Schroeppel, R. 1971 <i>The order 5 magic squares</i>. Written by Michael Beeler with assistance from Schroeppel–see M. Gardner’s column on mathematical games, <i>Sci. Am.</i> 1976 <b>234</b>, 118–123.
[13] Trump, W. 2003 Estimate of the number of magic squares of order 6. See http://www.trump.de/magic-squares/. <a target=”_blank” href=’http://www.trump.de/magic-squares/’>http://www.trump.de/magic-squares/</a>
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