Sunday, 22 April 2012

A New Census of Fourth-Order Magic Squares

Since Mr Henry Ernest Dudeney's first article in "The Queen" on the 15th January 1910, (the findings of which he confirmed on the page 120 of his book "Amusements in Mathematics," in 1917), the present generally accepted census of fourth-order magic squares includes three main categories as follows:

"Nasik" (Pandiagonal) : 48 magic squares
"Semi-Nasik" (Semi-pandiagonal) : 384 magic squares
"Simple" (Basic magic) : 448 magic squares
All Types : 880 = the total number of essentially different 4th-order magic squares in Frénicle standard form.

After defining magic squares as: "in their simple form of consecutive whole numbers arranged in a square so that every column, every row, and each of the two long diagonals shall add up alike," Dudeney then characterised his three main categories of fourth-order magic squares as follows:

A "Simple" square is a "square that fulfils the simple conditions and no more."
A "Semi-Nasik" (or semi-pandiagonal) square "has the additional property that the opposite short diagonals of two cells each together sum to 34."
A "Nasik" (or pandiagonal) square is a magic square on which "all the broken diagonals sum to 34."

This census was misleading, as Dudeney implied that for all his "simple" magic squares, only the main diagonals were magic. Because this was not the case, he had erroneously categorised certain partially pandiagonal squares as "simple" magic squares: There are 88 fourth-order partially pandiagonal squares that have been neglected until now, perhaps because their magic diagonals are only singly symmetric and did not satisfy Dudeney's initial conditions for semi-pandiagonal classification...

Considering the commencement of pandiagonality to take place when at least one of a magic square's broken diagonals is magic, and wishing to clarify the classification of these squares, I propose that 88 partially pandiagonal squares (including six Dudeney types VI, VII, VIII, IX, X, and XI) should be taken into account. These squares are identified in a previous article: "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus." For example the Frénicle index squares n° 46, 50, 337 and 545, classed "simple" by Dudeney, all share the same partially pandiagonal characteristics, displayed on the magic torus T4.108 (type n° T4.03.1.1). It is clear that these partially pandiagonal squares, with two singly symmetric broken diagonals, are essentially different when compared with their basic, or "simple," fourth-order magic square cousins:

order 4 partially pandiagonal magic torus T4.108

Additionally, the Frénicle index squares n° 40 and 552, classed "simple" by Dudeney, both share the same partially pandiagonal characteristics, displayed on the magic torus T4.096 (type n° T4.04.01). It is clear that these partially pandiagonal squares, with one singly symmetric broken diagonal, are essentially different when compared with their basic, or "simple," fourth-order magic square cousins:

order 4 partially pandiagonal magic torus T4.096

Since highlighting the 255 magic tori that display the 880 Frénicle magic squares, I now propose five main categories of 4th-order magic squares as follows:

Pandiagonal ("Nasik") : 48 magic squares
Semi-pandiagonal ("Semi-Nasik") : 384 magic squares with 2 four times symmetric broken magic diagonals
Partially pandiagonal : 80 magic squares with 2 singly symmetric broken magic diagonals
Partially pandiagonal : 8 magic squares with 1 singly symmetric broken magic diagonal
Basic ("Simple") : 360 magic squares
All Types : 880 = the total number of essentially different 4th-order magic squares in Frénicle standard form.

Nevertheless, these 4th-order magic squares only offer partial glimpses of the convex or concave 2D number systems that they represent. Instead of counting the magic squares we could be reasoning otherwise, and I therefore also propose an alternative census of the 255 essentially different magic tori that display the 880 4th-order magic squares:

Pandiagional : 3 Tori Type 4.01 with 8 crossed magic diagonals producing 16 magic intersections. Each torus T4.01 is entirely covered by 16 sub-magic 2x2 squares.
The 48 pandiagonal squares that are displayed are Dudeney type I.
Semi-pandiagonal : 24 Tori Type 4.02.1 with 4 crossed magic diagonals producing 8 magic intersections. Each torus T4.02.1 is entirely covered by 12 sub-magic 2x2 squares.
The 192 semi-pandiagonal squares that are displayed are Dudeney types IV and VI
Semi-pandiagonal : 12 Tori Type 4.02.2 with 4 crossed magic diagonals producing 8 magic intersections. Each torus T4.2.2 is entirely covered by 8 sub-magic 2x2 squares.
The 96 semi-pandiagonal squares that are displayed are Dudeney type II and III.
Semi-pandiagonal : 12 Tori Type 4.02.3 with 4 crossed magic diagonals producing 8 magic intersections. Each torus T4.02.3 is entirely covered by 8 sub-magic 2x2 squares.
The 96 semi-pandiagonal squares that are displayed are Dudeney type V.
Partially pandiagonal : 6 Tori Type 4.03.1 with 4 crossed magic diagonals producing 4 magic intersections. Each torus T4.03.1 is entirely covered by 8 sub-magic 2x2 squares.
The 24 partially panpandiagonal squares that are displayed are Dudeney type VI.
Partially pandiagonal : 12 Tori Type 4.03.2 with 4 crossed magic diagonals producing 4 magic intersections. Each torus T4.03.2 is entirely covered by 8 sub-magic 2x2 squares.
The 48 partially pandiagonal squares that are displayed are Dudeney types VII and IX.
Partially pandiagonal : 2 Tori Type 4.03.3 with 4 crossed magic diagonals producing 4 magic intersections. Each torus T4.03.3 is entirely covered by 8 sub-magic 2x2 squares.
The 8 partially pandiagonal squares that are displayed are Dudeney type XI.
Partially pandiagonal : 4 Tori Type 4.04 with 3 crossed magic diagonals producing 2 magic intersection. Each torus T4.04 is entirely covered by 4 sub-magic 2x2 squares.
The 8 partially pandiagonal squares that are displayed are Dudeney types VIII and X.
Basic : 92 Tori Type 4.05.1 with 2 crossed magic diagonals producing 2 magic intersections. Each torus T4.05.1 is entirely covered by 8 sub-magic 2x2 squares.
The 184 basic magic squares that are displayed are Dudeney type VI.
Basic : 32 Tori Type 4.05.2 with 2 crossed magic diagonals producing 2 magic intersections. Each torus T4.05.2 is entirely covered by 8 sub-magic 2x2 squares.
The 64 basic magic squares that are displayed are Dudeney types VII and IX.
Basic : 4 Tori Type 4.05.3 with 2 crossed magic diagonals producing 2 magic intersections. Each torus T4.05.3 is entirely covered by 4 sub-magic 2x2 squares.
The 8 basic magic squares that are displayed are Dudeney type XII.
Basic : 52 Tori Type 4.05.4 with 2 crossed magic diagonals producing 2 magic intersections. Each torus T4.05.4 is entirely covered by 4 sub-magic 2x2 squares.
The 104 basic magic squares that are displayed are Dudeney types VIII and X.
All Types : 255 = the total number of essentially different fourth-order magic tori.

The first table below compares the magic square enumerations using the 12 Dudeney types and the 12 Magic torus types. If you click on this table, a full-size version will open in a new window, which will allow for easier reading:

order 4 new census of 4th order magic squares

The second table below recapitulates the different characteristics of the 255 fourth-order magic tori and the magic squares that they display:

4th order magic tori recap order 4

For convenient reference, the 255 magic tori that display the 880 fourth-order magic squares are now indexed and listed in normalised square form, in the "Table of Fourth-Order Magic Tori."

For further analysis of this new fourth-order magic square census, please refer to Dwane Campbell's web site, in which he has made interesting studies of 4th-order Frénicle squares constructed from base squares, and also a comparison of the Dudeney and Walkington classifications.
I wish to express my gratitude to Walter Trump, not only for his valuable advice concerning the choice of terminology, but also for his computing skills that identified the fourth-order magic torus type 4.04, and enabled a precise count.
My thanks also go out to Miguel Angel Amela who ran a computer program that confirmed the different counts of magic diagonals on 4th-order magic squares.
I also wish to express my thanks to Harry White who kindly ran a programme to check and validate my hypothesis that sub-magic 2x2 squares entirely cover each 4th-order magic torus.

New development!


Posted on the 21st January 2018, an article entitled "Multiplicative Magic Tori" shows that the 255 magic tori of order-4 are different multiplied states of 82 Multiplicative Magic Tori (MMT) of order-4!

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