The initially square cells will become regular or irregular quadrilaterals, excluding complex quadrilaterals. This third rule took into account the flattened square that we are accustomed to looking at, but it also implied that the torus would degenerate into a sphere With these new constraints, on the 6th January I was able to come up with the area magic torus illustrated below:.
These extended best wishes, expressed through the complete set of third-order magic and semi-magic squares, include messages that reflect the multiple nationalities of the members of the magic square circle. The magic square viewpoint of the torus is placed at the top left. The patterns still need adjusting in order to achieve the correct areas, but as there is more flexibility, I am fairly confident that at least one accurate solution can be found. On the 6th January Walter Trump also sent us a new area magic design that used 4 continuous straight dissection lines. Although this schema could not be adapted to the number sequence 1 to 9, it was area magic:.
Area Magic Schema by Walter Trump.
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Inder Jeet Taneja then suggested that the problem with the sequence 1 to 9 was that the total areas did not add up to a perfect square area. On the same day, following Inder's suggestion, Walter Trump amazed us all with this first third-order linear area magic square! Bravo Walter for your achievement following Inder's suggestion! I hardly dared to believe that such a simple area pattern could produce a magic square!
Having other commitments to honour, Walter Trump then requested that somebody else searched for solutions using lower sequences. No other volunteers came forward, so I decided to look for myself, and came up with this second linear area magic square of the third-order using the sequence 3 to As I am not a programmer, I used Autocad to construct this linear area magic square manually. The areas of this provisional square are therefore only accurate up to two decimal places, before computer verification and area optimisation.
In the meantime Walter Trump continued writing a computer program to handle the equations and search iteratively for solutions. This was able to give orthogonal coordinates having at least 14 decimals after the commas. At first unable to solve the non-linear equations explicitly, Walter progressively improved his approach, and on the 8th January he sent us a message stating that for some sequences there were two solutions, and that the lowest number sequence ranged from 2 to 10!
This explains the method of construction and gives the reason why two solutions exist for most, but not all of the sequences studied:. Using the coordinates generated by Walter Trump's computer program, both solutions of the 5 lowest sequences of the third-order linear area magic squares are illustrated in square form below. It will be seen that the 1 to 9 sequence fails, and the linear area magic squares are only possible from the sequence 2 to 10 upwards when the continuous straight dissection lines produce no more than 4 intersections inside the squares :.
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