Bimagic squares
A magic square is said to be bimagic (or 2-multimagic) if it remains magic after each of its numbers is replaced by its square. A frenchman, G. Pfeffermann, built the first bimagic square in 1890. Rather than publishing it integrally, he proposed it partially, in the form of a puzzle, in the fortnightly magazine Les Tablettes du Chercheur - Journal de Jeux d'Esprit et de Combinaisons, number 2 of January 15, 1891.
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WANTED!
Who was this G. Pfeffermann? We have never found any information
about him, even though he authored numerous articles on magic
squares published in France, mainly between 1890 and 1896. Even
his full firstname is unknown to us. We have only found a few times
this signature: "Gg. Pfeffermann". Probably Georges. Or
perhaps Grégoire? Already, in 1926, André Gérardin (Nancy)
was astonished about that in the Annales de la Société
Scientifique de Bruxelles: " It's a mathematician about which
we have very little bibliographic information, because few families
take care of the scientific memory of their parents or of the keeping
of their archives". |
Here is the square as it was published at that time. It is up to you to complete it with the remainder of the 64 numbers in order to obtain a bimagic square! A hint: the sums of the rows, columns and diagonals of the 8th-order square (n=8) have to be equal to 260 (=n(n²+1)/2), and the sums when the numbers are squared have to be equal to 11,180 (=n(n²+1)(2n²+1)/6).
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56 |
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8 |
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18 |
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9 |
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20 |
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48 |
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29 |
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10 |
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26 |
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13 |
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64 |
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4 |
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5 |
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30 |
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12 |
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60 |
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15 |
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63 |
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41 |
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50 |
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55 |
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11 |
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58 |
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45 |
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61 |
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42 |
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27 |
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39 |
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62 |
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37 |
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51 |
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3 |
Pfeffermann published the solution a fortnight later, in this same magazine. The editorial staff of the Tablettes extended to the author of this first bimagic square their « most sincere compliments for this real tour de force that he has just accomplished ». And the famous Edouard Lucas (1842-1891), who was a writer of articles in Les Tablettes, wrote that this first bimagic square was a « very remarkable square ».
Pfeffermann continued to publish, in the following issues, several other 8th-order bimagic squares, and also 9th-order bimagic squares. Today we know how to construct, by various methods, bimagic squares of various orders. And the smallest bimagic squares possible are very certainly of the same 8th-order as this one of Pfeffermann's.
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