This post remplaces an article that was previously published on Wednesday 25th January 2017.

After my previous post "Area Magic Squares and Tori of Order-3," Walter Trump continued his computer research of the linear area magic squares of order-4, now joined in this venture by a fellow mathematician and programmer Hans-Bernhard Meyer.

After my previous post "Area Magic Squares and Tori of Order-3," Walter Trump continued his computer research of the linear area magic squares of order-4, now joined in this venture by a fellow mathematician and programmer Hans-Bernhard Meyer.

Having already some insight into the linear area magic squares of the odd-order-3 and of the doubly-even-order-4, it was tempting to search for a singly-even-order example. I therefore decided to do some exploring, and on the 22nd January 2017, I found this first linear area magic square of order-6:

For the area magic square shown above, the magic constant of each row, column and main diagonal is 1296 (6

^{4}). The total areas are therefore 7776 (6

^{5}). The mean number of each row, column and main diagonal is 216 (6

^{3}). The last digits of the first and last numbers of the horizontal non-parallel rows are either 1 for odd number sequences, or 6 for even number sequences. Please note that the area cell 41 is not a triangle, but a trapezium (or trapezoid) like all the other cells. For those amongst you who may be interested in the mathematical construction, Hans-Bernhard Meyer has since very kindly produced the following parameterisation:

From then on using a computer program to find many more examples, on the 26th January 2017, Hans-Bernhard Meyer drew to my attention a new linear area magic square with not only parallel vertical lines, but also a central horizontal line. He kindly authorised me to publish this example illustrated below:

The magic constant of this linear area magic square (L-AMS) is 474, and the total areas are 2,844. The last digits of the first and last numbers of the non-parallel rows are either 4 for the even number sequences, or 9 for the odd number sequences. I noticed that in addition to the central horizontal line, there was also a pair of broken magic diagonals which would enable an easy transformation, as illustrated below:

It can be seen that these are two essentially different L-AMS viewpoints of a same area magic cylinder. Please note that there are also another four

On the 29th January 2017, Hans-Bernhard Meyer informed me that the following can be proved:

Every 6x6 L-AMS with magic sum S and horizontal centre line has the property:

x04+x11+x18+x19+x26+x33 = x03+x08+x13+x24+x29+x34 = S

and conversely, every 6x6 L-AMS with magic sum S and

x04+x11+x18+x19+x26+x33 = x03+x08+x13+x24+x29+x34 = S

has a horizontal centre line.

He stated that there are many examples of such L-AMS and strongly supposed that the example illustrated above had the lowest possible magic sum. He was able to confirm that examples with 2 horizontal lines do not exist.

On the same day, Hans-Bernhard also informed me that in the order-6, the L-AMS with the lowest possible magic sum had a magic constant of 402. He had found that there were several of these L-AMS, and kindly sent me details of an example, which I have illustrated below:

*area semi-magic parallelograms*displayed on this cylinder (with cells 44, 124, 144, or 19 in their top left corners).On the 29th January 2017, Hans-Bernhard Meyer informed me that the following can be proved:

Every 6x6 L-AMS with magic sum S and horizontal centre line has the property:

x04+x11+x18+x19+x26+x33 = x03+x08+x13+x24+x29+x34 = S

and conversely, every 6x6 L-AMS with magic sum S and

x04+x11+x18+x19+x26+x33 = x03+x08+x13+x24+x29+x34 = S

has a horizontal centre line.

He stated that there are many examples of such L-AMS and strongly supposed that the example illustrated above had the lowest possible magic sum. He was able to confirm that examples with 2 horizontal lines do not exist.

On the same day, Hans-Bernhard also informed me that in the order-6, the L-AMS with the lowest possible magic sum had a magic constant of 402. He had found that there were several of these L-AMS, and kindly sent me details of an example, which I have illustrated below:

The total areas of this square are 2,412. The last digits of the first and last numbers of the non-parallel rows are either 2 for the even number sequences, or 7 for the odd number sequences. The even number sequence of the bottom row begins with 12, which Hans-Bernhard informs me is the lowest possible vertex area for L-AMS (linear area magic squares) of order-6.

These are only the first of an infinite number of examples that remain to be discovered, and I therefore expect to be regularly publishing updates to this post!

A former post Area Magic Squares and Tori of Order-3 can be found in these pages since the 13th January 2017.

On the 25th January 2017, Hans-Bernhard Meyer published an article entitled Observations on 4x4 Area Magic Squares with vertical lines in his website: Math'-pages.

On the 3rd February 2017, Walter Trump published a chapter entitled Area Magic Squares in his website: Notes on Magic Squares. This chapter includes many analyses and examples of area magic squares of the third and fourth-orders.

Since the 8th February 2017, "Area Magic Squares of Order-4" relates the first findings of area magic squares of the fourth-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 CarrĂ©s Magiques d'Aires."

In the December 2018 issue of "Spektrum der Wissenschaft" (a Springer Nature journal, and the German edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "FLĂ„CHENMAGISCHE QUADRATE."

__Related links__

A former post Area Magic Squares and Tori of Order-3 can be found in these pages since the 13th January 2017.

On the 25th January 2017, Hans-Bernhard Meyer published an article entitled Observations on 4x4 Area Magic Squares with vertical lines in his website: Math'-pages.

On the 3rd February 2017, Walter Trump published a chapter entitled Area Magic Squares in his website: Notes on Magic Squares. This chapter includes many analyses and examples of area magic squares of the third and fourth-orders.

Since the 8th February 2017, "Area Magic Squares of Order-4" relates the first findings of area magic squares of the fourth-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 CarrĂ©s Magiques d'Aires."

In the December 2018 issue of "Spektrum der Wissenschaft" (a Springer Nature journal, and the German edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "FLĂ„CHENMAGISCHE QUADRATE."

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