Wednesday, 17 April 2013

Alignment and Concentricity of Sub-Magic Squares on 4th-Order Magic Tori

Once the sub-magic 2x2 squares were found I remembered having seen traces of larger sub-magic squares when studying the 4th-order panmagic torus. I therefore decided to search for these larger sub-magic squares on the different types of 4th-order magic tori. 

Since a first edition on the 17th April 2013, I have updated this article on the 21st April 2013, adding subdivisions of the tori to take into account the different Dudeney types.

I have not verified every Frénicle square, and the results illustrated below are based on simple observation and manual input. Your comments are always welcome! 

At the end of this article I propose some new hypotheses.

4th-order pandiagonal torus type T4.01


Panmagic torus type T4.01 pandiagonal sub-magic 2x2 squares


order 4 panmagic torus type T4.01 pandiagonal sub-magic 3x3 squares


order 4 panmagic torus type T4.01 pandiagonal sub-magic 4x4 squares


order 4 panmagic torus type T4.01 pandiagonal sub-squares diagram 1


order 4 panmagic torus type T4.01 pandiagonal sub-squares diagram 2


order 4 panmagic torus type T4.01 pandiagonal sub-squares diagram 3


4th-order semi-pandiagonal torus type T4.02.1

order 4 magic Torus type T4.02.1 semi-pandiagonal sub-magic 2x2 squares


order 4 magic Torus type T4.02.1 semi-pandiagonal sub-magic 3x3 squares


order 4 magic torus type T4.02.1 semi-pandiagonal sub-magic 4x4 squares


order 4 magic torus type T4.02.1 semi-pandiagonal sub-squares diagram 1


order 4 magic torus type T4.02.1 semi-pandiagonal sub-squares diagram 2


order 4 magic torus type T4.02.1 semi-pandiagonal sub-squares diagram 3


4th-order semi-pandiagonal torus type T4.02.2


order 4 magic torus type T4.02.2 semi-pandiagonal sub-magic 2x2 squares


order 4 magic torus type T4.02.2 semi-pandiagonal sub-magic 3x3 squares

order 4 magic torus type T4.02.2 semi-pandiagonal sub-magic 4x4 squares

order 4 magic torus type T4.02.2.3 semi-pandiagonal sub-squares diagram 1


order 4 magic torus type T4.02.2.3 semi-pandiagonal sub-squares diagram 2


order 4 magic torus type T4.02.2.3 semi-pandiagonal sub-squares diagram 3


For recreational water retention studies on associative magic squares, such as those displayed on the tori type T4.02.2 illustrated above, please refer to Craig Knecht's article in Wikipedia.

 

4th-order semi-pandiagonal torus type T4.02.3

order 4 magic torus type T4.02.3 semi-pandiagonal sub-magic 2x2 squares


order 4 magic torus type T4.02.3 semi-pandiagonal sub-magic 3x3 squares


order 4 magic torus type T4.02.3 semi-pandiagonal sub-magic 4x4 squares


4th-order partially panmagic torus type T4.03.1

order 4 magic torus type T4.03.1 partially pandiagonal sub-magic 2x2 squares


order 4 magic torus type T4.03.1 partially pandiagonal sub-magic 4x4 squares

order 4 magic torus type T4.03.1.2 partially pandiagonal sub squares diagram 1


order 4 magic torus type T4.03.1.2 partially pandiagonal sub squares diagram 2


order 4 magic torus type T4.03.1.2 partially pandiagonal sub squares diagram 3



4th-order partially panmagic torus type T4.03.2

order 4 magic torus type T4.03.2 partially pandiagonal sub-magic 2x2 squares


order 4 magic torus type T4.03.2 partially pandiagonal sub-magic 4x4 squares


4th-order partially panmagic torus type T4.03.3

order 4 magic torus type T4.03.3 partially pandiagonal sub-magic 2x2 squares


order 4 magic torus type T4.03.3 partially pandiagonal sub-magic 4x4 squares


order 4 magic torus type T4.03.3 partially pandiagonal sub squares diagram 1


order 4 magic torus type T4.03.3 partially pandiagonal sub squares diagram 2


order 4 magic torus type T4.03.3 partially pandiagonal sub squares diagram 3


4th-order partially panmagic torus type T4.04

order 4 magic torus type T4.04 partially pandiagonal sub-magic 2x2 squares


order 4 magic torus type T4.04 partially pandiagonal sub-magic 4x4 squares


order 4 magic torus type T4.04 partially pandiagonal sub squares diagram 1



order 4 magic torus type T4.04 partially pandiagonal sub squares diagram 2


order 4 magic torus type T4.04 partially pandiagonal sub squares diagram 3


4th-order basic magic torus type T4.05.1

order 4 magic torus type T4.05.1 basic magic sub-magic 2x2 squares


order 4 magic torus type T4.05.1 basic magic sub-magic 4x4 squares

order 4 magic torus type T4.05.1.2 basic magic sub squares diagram 1


order 4 magic torus type T4.05.1.2 basic magic sub squares diagram 2


Magic torus type T4.05.1.2 basic magic sub squares diagram 3


  4th-order basic magic torus type T4.05.2

order 4 magic torus type T4.05.2 basic magic sub-magic 2x2 squares


order 4 magic torus type T4.05.2 basic magic sub-magic 4x4 squares


4th-order basic magic torus type T4.05.3

order 4 magic torus type T4.05.3 basic magic sub-magic 2x2 squares


order 4 magic torus type T4.05.3 basic magic sub-magic 4x4 squares

order 4 magic torus type T4.05.3.4 basic magic sub squares diagram_1


order 4 magic torus type T4.05.3.4 basic magic sub squares diagram 2


order 4 magic torus type T4.05.3.4 basic magic sub squares diagram 3



  4th-order basic magic torus type T4.05.04

order 4 magic torus type T4.05.4 basic magic sub-magic 2x2 squares


order 4 magic torus type T4.05.4 basic magic sub-magic 4x4 squares



4th-order magic torus and magic tori summary order 4

Observations


Although magic rows, columns and diagonals are important, it seems that the sub-magic squares are a driving force of the 4th-order magic tori.

The number patterns and quantities of 2x2 and 4x4 sub-magic squares are identical. On 4th-order tori the corner numbers of a 2x2 sub-magic square always coincide with the corner numbers of a 4x4 sub-magic square and vice versa.

The T4.04 and the T5.03.4 tori are similarly constructed. The partially panmagic tori type T4.04 deserve a separate classification although their third diagonals do not produce magic intersections.

Apparently the sub-magic 3x3 squares are only displayed on panmagic and semi-panmagic tori. There seems to be a relationship between the spacing of magic diagonals and the presence of sub-magic 3x3 squares, and a one in two spacing of magic diagonals only occurs on the panmagic and semi-panmagic tori.

Whilst examining only magic tori when I first wrote this article, I thought that only the semi-magic tori with similarly spaced (but non-intersecting) diagonals such as the semi-magic tori type T4.06 and T4.08 would display sub-magic 3x3 squares. Dwane Campell has since demonstrated that sub-magic 3x3 squares are also displayed on the semi-magic tori type T4.10 without magic diagonals. Here are his examples:

order 4 semi-magic torus type T4.05.10 semi-magic sub-magic 3x3 squares

The sub-magic 2x2 diamond shapes illustrated above were unknown to me until Dwane Campbell brought them to my attention. In his interesting web site article on 4th-order magic square classification he identifies the different cases of 2x2 diamond shapes on traditional Frénicle 4x4 squares.

Hypotheses


On 4th-order magic tori:
1/ Sub-magic 3x3 squares are only displayed on panmagic and semi-panmagic tori (or squares).
2/ Sub-magic 3x3 squares can be reduced to 4 essentially different subsquares per panmagic or per semi-panmagic torus. The addition of the central numbers of these essentially different sub-magic 3x3 squares totals 34.
3/ Sub-magic 2x2, 3x3, and 4x4 squares always display two even numbers and two odd numbers at their corners.

Developments


The discovery of the sub-magic 2x2 squares has inspired Mr Kanji Setsuda to make new studies of 4x4 squares, and a classification of standard magic squares by the 'composite conditions' of 2x2 squares within.

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