Further to the discoveries of the area magic squares of order-3, the first linear area magic square of order-4 was found by Walter Trump on the 14th January 2017:

On the 15th January 2017, Walter Trump's computer program was able to produce thousands of 4x4 linear area magic squares. Meanwhile, he was joined by fellow mathematician and programmer Hans-Bernhard Meyer, who on the 16th January 2017, found a different type of 4x4 linear area magic square (L-AMS). From then on Walter and Hans-Bernhard worked in collaboration, and they have kindly authorised me to illustrate samples of their findings below:

The linear area magic squares (L-AMS) of order-4 with vertical or horizontal lines have not only interesting geometric properties, but also surprising arithmetic features that are even more exciting. On the 8th February Hans-Bernhard Meyer informed me that the number and the structure of the fourth-order L-AMS with 3 horizontal or vertical lines had been totally clarified by parameterisations with 3 parameters, although this was not yet the case for L-AMS with only 2 horizontal or vertical lines. For full details of the mathematics behind these area magic squares please refer to the related links at the foot of this page.

On the 12th February 2017, in an e-mail correspondence, I asked Hans-Bernhard Meyer if it was possible to create a L-AMS of order-4 that had a perpendicular intersection of slanted lines. He responded on the 13th February 2017, sending the following L-AMS:

Congratulations to Hans-Bernhard for this interesting square!

Concerning the possible existence of L-AMS of order-4 without vertical or horizontal lines, on the 12th February 2017 Hans-Bernhard informed me that this question was still quite difficult to resolve.

Many other types of fourth-order AMS are also possible, and the following squares are area magic interpretations of some of the classical Frénicle squares of order-4:

On the 12th February 2017, in an e-mail correspondence, I asked Hans-Bernhard Meyer if it was possible to create a L-AMS of order-4 that had a perpendicular intersection of slanted lines. He responded on the 13th February 2017, sending the following L-AMS:

Congratulations to Hans-Bernhard for this interesting square!

Concerning the possible existence of L-AMS of order-4 without vertical or horizontal lines, on the 12th February 2017 Hans-Bernhard informed me that this question was still quite difficult to resolve.

Many other types of fourth-order AMS are also possible, and the following squares are area magic interpretations of some of the classical Frénicle squares of order-4:

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__Related links__

Since the 3rd February 2017, full details of the findings of Walter Trump can be found in the chapter "Area Magic Squares" of his website: Notes on Magic Squares and Cubes.

Since the 25th January 2017, full details of the findings of Hans-Bernhard Meyer can be found in the article "Observations on 4x4 Area Magic Squares with Vertical Lines" of his website: Math'-pages.

Since the 25th January 2017, "Area Magic Squares of Order-6" relates the first findings of area magic squares of the sixth-order.

Since the 13th January 2017, "Area Magic Squares and Tori of Order-3" relates the first findings of area magic squares of the third-order.

In the N° 487 2018 May issue of "Pour La Science" (the French edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "Les Carrés Magiques d'Aires."

In the N° 487 2018 May issue of "Pour La Science" (the French edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "Les Carrés Magiques d'Aires."

First image has typo — 29 29 should be 29 39

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I have now corrected the image, and I wish to thank you for pointing out this out!

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