440 Torus-Opposite Pairs of the 880 Frénicle Magic Squares of Order-4

Finding the torus-opposite pairs of magic squares in even-orders:

The following diagram, which illustrates the array of n² essentially-different square viewpoints of a basic magic torus of order-n, starting here with the 4x4 magic square that has the Frénicle index n° 1, shows how a torus-opposite magic square of order-4 can be identified:

Array of 16 essentially-different square viewpoints of a basic magic torus of order-4

This basic magic torus, now designated as type n° T4.05.1.01, can be seen to display a torus-opposite pair of Frénicle-indexed magic squares n° 1-458 (with Dudeney pattern VI), as well as 7 torus-opposite pairs of semi-magic squares that the discerning reader will easily be able to spot.

The relative positions of the numbers of torus-opposite squares can be expressed by a simple plus or minus vector. There are always two equal shortest paths towards the far side of the torus: These are, in toroidal directions, east or west along the latitudes, and in poloidal directions, north or south along the longitudes of the doubly-curved 2D surface. So, if the even-order is n, then:

v  =  ( ± n/2, ± n/2 )

"Tesseract Torus" by Tilman Piesk CC-BY-4.0
 https://commons.wikimedia.org/w/index.php?curid=101975795

Torus-opposite squares are not just limited to basic magic tori, as they also exist on pandiagonal, semi-pandiagonal, partially pandiagonal, and even semi-magic tori of even-orders. Therefore, for order-4, we can either say that there are 880 magic squares, or we can announce that there are 440 torus-opposite pairs of magic squares. Similarly, the 67,808 semi-magic squares of order-4 can be expressed as 33,904 torus-opposite pairs of semi-magic squares, etc.

Why torus-opposite pairs of magic squares cannot exist in odd-orders

A torus-opposite magic square always exists in even-orders because an even-order magic square has magic diagonals that produce a first magic intersection at a centre between numbers, and another magic intersection at a second centre between numbers on the far side of the torus (at the meeting point of the four corners of the first magic square viewpoint). However, in odd-orders, where a magic square has magic diagonals that produce a magic intersection over a number, then a sterile non-magic intersection always occurs between numbers on the far side of the torus (at the meeting point of the four corners of the initial magic square). This is why torus-opposite pairs of magic squares cannot exist in the odd-orders.

255 magic tori of order-4, listed by type, with details of the 440 torus-opposite pairs of the 880 Frénicle magic squares of order-4

The enumeration of the 255 magic tori of order-4 was first published in French on the 28th October 2011, before being translated into English on the 9th January 2012: "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus." In this previous article, the corresponding 880 fourth-order magic squares were listed by their Frénicle index numbers, but not always illustrated. The intention of the enclosed paper is therefore to facilitate the understanding of the magic tori of order-4, by portraying each case.

Here, Frénicle index numbers continue to be used, as they have the double advantage of being both well known and commonly accepted for cross-reference purposes. But please also note, that now, in order to simplify the visualisation of the magic tori, their displayed magic squares are not systematically presented in Frénicle standard form. 

In the illustrations of the 255 magic tori of order-4, listed by type with the presentation of their magic squares, the latter are labelled from left to right with their Frénicle index number, followed by, in brackets, the Frénicle index number of their torus-opposite magic square, and finally by the Roman numeral of their Dudeney complementary number pattern. Therefore, for the magic square of order-4 with Frénicle index n° 1 (that forms a torus-opposite pair of magic squares with Frénicle index n° 458; both squares having the same Dudeney pattern VI), its label is 1 (458) VI.

To find the "255 magic tori of order-4, listed by type, with details of the 440 torus-opposite pairs of the 880 Frénicle magic squares of order-4" please consult the following PDF:

Read more ...

Even and Odd Number Patterns on Magic Tori of Orders 3 and 4

Ancient references to the pattern of even and odd numbers of the 3 × 3 magic square appear in the "I Ching" or "Yih King" (Book of Changes). In the introduction to the Chou edition, the scroll of the river Loh or "Loh-Shu", which is represented by the magic square of order-3, is written with black dots (the yin symbol and emblem of earth) for the even numbers, and white dots (the yang symbol and emblem of heaven) for the odd numbers. The "Loh-Shu" is incorporated in the writings of Ts'ai Yuän-Ting who lived from 1135 to 1198:

The Scroll of Loh, according to Ts'ai Yüang-Ting

A normal magic torus (or magic square) of order-n uses consecutive numbers from 1 to n² and consequently, for the order-3, the magic constant (MC) = 15. The Loh-Shu Magic Torus of Order-3 is presented below, together with its corresponding even and odd number pattern P3:

The "Loh-Shu" Magic Torus of Order-3 and its Even and Odd Number (x mod 2) Pattern P3

The magic torus T3 of order-3 is shown here as seen from the Scroll of Loh magic square viewpoint.

The even and odd number pattern P3 has reflection symmetry (with 6 lines of symmetry); rotational symmetry (of order 4); and point symmetry (over number 5, between numbers 3 and 7, between numbers 1 and 9, and between numbers 4, 6, 2, and 8).

The Even and Odd Number Patterns of the Magic Tori of Order-4

Only consecutively numbered magic tori (with numbers from 1 to n²) are studied here, and therefore, for order-4, the magic constant MC = 34.

The "Table of Fourth-Order Magic Tori" lists and describes each magic torus of order-4 using index numbers, while "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus" gives a group categorisation of the magic tori of order-4 using type numbers. Both of these classifications give the details of the Frénicle indexed magic squares that are displayed by each torus.

In the study that follows it can be seen that there are 4 essentially different even and odd number patterns on the magic tori of order-4:

The Order-4 Even and Odd Number Pattern P4.1
(represented by the pandiagonal torus T4.198)

The Pandiagonal Torus T4.198 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.1

The pandiagonal torus of order-4 (with index n° T4.198 and type n° T4.01.2) is shown here as seen from the Frénicle n° 107 magic square viewpoint.

The even and odd number pattern P4.1 has reflection symmetry (with 6 lines of symmetry); rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The pattern P4.1 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.1 concerns 79 out of the 255 magic tori, and 404 out of the 880 magic squares of order-4. These include the 3 pandiagonal tori that display 48 pandiagonal squares; 30 semi-pandiagonal tori that display 240 semi-pandiagonal squares; 12 partially pandiagonal tori that display 48 partially pandiagonal squares; and also 34 basic magic tori that display 68 basic magic squares.

The Order-4 Even and Odd Number Pattern P4.2
(represented by the semi-pandiagonal torus T4.059)

The Semi-Pandiagonal Torus T4.059 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.2

The semi-pandiagonal torus of order-4 (with index n° T4.059 and type n° T4.02.2.01) is shown here as seen from the Frénicle n° 21 magic square viewpoint.

The even and odd number pattern P4.2 has reflection symmetry (with 8 lines of symmetry); rotational symmetry (of order 4); and multiple point symmetry when reading as a torus. The pattern P4.2 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.2 concerns 72 out of the 255 magic tori, and 256 out of the 880 magic squares of order-4. These include 18 semi-pandiagonal tori that display 144 semi-pandiagonal squares; 6 partially pandiagonal tori that display 16 partially pandiagonal squares; and also 48 basic magic tori that display 96 basic magic squares.

The Order-4 Even and Odd Number Pattern P4.3
(represented by the partially pandiagonal torus T4.186)

The Partially Pandiagonal Torus T4.186 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.3

The partially pandiagonal torus (with index n° T4.186 and type n° T4.03.1.3) is shown here as seen from the Frénicle n° 100 magic square viewpoint.

The even and odd number pattern P4.3 has reflection symmetry (with 8 lines of symmetry); rotational symmetry (of order 4); and multiple point symmetry when reading as a torus. The pattern P4.3 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.3 concerns 16 out of the 255 magic tori, and 44 out of the 880 magic squares of order-4. These include 6 partially pandiagonal tori that display 24 partially pandiagonal squares; and also 10 basic magic tori that display 20 basic magic squares.

The Order-4 Even and Odd Number Pattern P4.4
(represented by the basic magic torus T4.062)

The Basic Magic Torus T4.062 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.4

The basic magic torus (with index n° T4.062 and type n° T4.05.1.12) is shown here as seen from the Frénicle n° 37 magic square viewpoint.

The even and odd number pattern P4.4 has reflection symmetry (with 8 lines of symmetry); rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The pattern P4.4 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.4 concerns 88 out of the 255 magic tori, and 176 out of the 880 magic squares of order-4. All of these 88 magic tori and the displayed 176 magic squares are basic magic.

Observations on the Even and Odd Number Patterns of the Magic Tori and Magic Squares of Order-4

The findings of the even and odd number patterns of the magic tori of order-4 are recapitulated and analysed in comparative tables, (together with the Dudeney complementary number patterns and the Magic Torus self-complementary and paired complementary number patterns), in the file below:

The Even and Odd Number Patterns of the Semi-Magic Tori of Order-4

Only consecutively numbered semi-magic tori (with numbers from 1 to n²) are studied here, and consequently, for the order-4, the magic constant MC = 34.

In "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus" the 4,038 semi-magic tori of order-4 have been sorted, and type numbers have been attributed taking into account the arrangements of their magic diagonals (if any). 

Each of the four even and odd number patterns already found above on the magic tori of order-4, also occur on semi-magic tori of order-4. For example, from left to right below, we find a semi-magic torus type T4.08.0W with pattern P4.1; a semi-magic torus type T4.08.0X with pattern P4.2; a semi-magic torus type T4.07.0Y with pattern P4.3; and a semi-magic torus type T4.09.00Z with pattern P4.4:

Semi-Magic Tori of Order-4 with Even and Odd Number (x mod 2) Patterns P4.1, P4.2, P4.3 and P4.4

The even and odd number patterns P4.1 to P4.4 are not the only ones that can be found, but the patterns which are specific to the semi-magic tori of order-4 have neither been fully investigated nor precisely counted, and the examples that follow only represent some of the different varieties.

The Order-4 Even and Odd Number Pattern P4.5
(represented by the semi-magic torus type T4.06.0A)

The Semi-Magic Torus type T4.06.0A of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.5

This semi-magic torus type n° T4.06.0A was found by Walter Trump, and has been previously illustrated in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The even and odd number pattern P4.5 has diagonal reflection symmetry (with 4 lines of symmetry); and negative (evens to odds and vice versa) rotational symmetry (of order 2). The blocks of even and odd numbers are translations of each other.

The Order-4 Even and Odd Number Pattern P4.6
(represented by the semi-magic torus type T4.07.0A)

The Semi-Magic Torus type T4.07.0A of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.6

This semi-magic torus type n° T4.07.0A was found by Walter Trump, and has been previously illustrated in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The even and odd number pattern P4.6 has reflection symmetry (with 4 lines of symmetry); rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The blocks of even and odd numbers are translations of each other.

The Order-4 Even and Odd Number Pattern P4.7
(represented by the semi-magic torus type T4.09.00M)

The Semi-Magic Torus type T4.09.00M of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.7

This semi-magic torus type n° T4.09.00M was found by Walter Trump, and has been previously illustrated in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The even and odd number pattern P4.7 has both positive and negative reflection symmetry; negative rotational symmetry (of order 2); and multiple negative point symmetry when reading as a torus. The blocks of even and odd numbers are translations of each other.

The Order-4 Even and Odd Number Pattern P4.8
(represented by the semi-magic torus type T4.10.000X)

The Semi-Magic Torus type T4.10.000X of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.8

3,726 semi-magic tori of the type 10 have been found by Walter Trump, and his findings are detailed in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The T4.10.000X's even and odd number pattern P4.8 has negative reflection symmetry (2 lines of symmetry); negative rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The blocks of even and odd numbers are translations of each other.

General Observations

The "Table of Fourth-Order Magic Tori" has been revised to include the even and odd number pattern references for each of the magic tori of order-4 (listed together with the Frénicle indexed magic squares that each magic torus displays).

Other interesting websites that examine, but do not enumerate, the different even and odd number patterns on magic squares of order-4, include "4x4 Magic Squares" by Dan Rhett and "Pattern in Magic Squares" by Vipul Chaskar.

A brief look at higher-orders suggests that the order-5 has many irregular patterns; the order-6 has many patterns with negative reflection symmetry (evens to odds and vice versa); the order-7 has many patterns with rotational symmetry; and the order-8 has patterns very like those of the order-4 examined above. All of these orders merit an in-depth analysis of their even and odd number patterns.

In conclusion, it is interesting to note that the Arab mathematicians of the tenth century constructed odd-order magic squares with striking even and odd number patterns (as for example the magic squares in figures b44 and b49 on pages 243 and 248 of Magic Squares in the Tenth Century: Two Arabic treatises by Anṭākī and Būzjānī, translated by Jacques Sesiano). And in the same context, Paul Michelet has recently brought to my attention a magnificent 15 x 15 magic square by Ali b. Ahmad al-Anṭākī (d. 987). Al-Anṭākī's bordered magic square is solid evidence of the Middle Eastern mathematicians' mastery of even and odd number patterns!

Read more ...

Perimeter Magic Squares

After the first discoveries of area magic squares, I decided to do some more exploring and search for examples of perimeter magic squares. I found that the latter do indeed exist, and that although these appear to be similar to linear area magic squares, their construction is quite different for two reasons: Depending on the slopes (and lengths) of the slanting dissection lines of a perimeter magic square, the hinge points of these lines are offset when compared with those of an area magic square. The total perimeter of a perimeter magic square results from the slopes (and lengths) of the slanting dissection lines, and not from the addition of the individual cell perimeters.

Unless stated otherwise, the perimeter magic squares illustrated here have been created by the author of this post. Although perimeter magic squares are shown to be different from area magic squares, they were deduced using Walter Trump's area magic square construction techniques.

For want of a better word, please note that these perimeter magic squares are measured geometric perimeter magic squares, and that they are therefore not to be confused with the square examples of perimeter-magic polygons, (first created by Terrel Trotter Jr. and further discussed on the website of Harvey Heinz).

Linear Perimeter Magic Square of Order-3

On the 25th February 2017, the first Linear Perimeter Magic Square (L-PMS) of Order-3 was constructed using AutoCAD:

This L-PMS is drawn to a precision of 2 decimal places for each of the 9 cell perimeters, and the resulting total perimeter of the square is therefore approximately 47.372 (before computer validation and optimisation). The coordinates of this square are available upon request.

Maybe the square perimeters of such L-PMS of order-3 are proportional to the regular consecutive number cell perimeter sequences that they display, and those amongst you with programming skills will be able to generate different similarly-constructed examples, and thus identify the relationship?

Linear Perimeter Magic Squares of Order-4

The following Semi-Orthogonal Linear Perimeter Magic Squares (L-PMS) of order-4 have cell perimeter entries rounded to 6 digits. Because their central slanted lines are not derived from Pythagorean triangles, the lengths of the adjoining cell perimeters are not finite integers, but irrational "infinite" numbers. The precision of the cell perimeters can be further increased, but with each increase in precision the cell entries and the corresponding magic sums become higher, and the dimensions have to be modified accordingly.

Constructed on the 23rd February 2017, the following L-PMS of order-4 has a rounded magic sum of S = 1936532:

The upper slanted line is derived from a Pythagorean triangle with long leg 24, short leg 7, and hypotenuse 25. The central slanted line is derived from a triangle with long leg 120, short leg 8, and hypotenuse 120.26637102698... The lower slanted line is derived from a Pythagorean triangle with long leg 40, short leg 9, and hypotenuse 41. The total sum of the 16 cell perimeters is 1936532 x 4 = 7746128. The average cell perimeter is 7746128 / 16 = 484133. The perimeter of the magic square is 120000 x 4 x 4 = 1920000. The coordinates of this square are available upon request. This perimeter magic square's cell values are voluntarily rounded to 6 digits. Greater accuracy is possible, but this will inevitably lead to very high entries. If more or less digits are used for the cell perimeter values, then the slanted lines will need vertical translations to take into account the revised magic sums, and this will imply new dimensions. There are some interesting relationships between the broken diagonals of this linear perimeter magic square (L-PMS), as shown in the following diagram: 

Constructed on the 24th February 2017, the following L-PMS of order-4 has a rounded magic sum of S = 1930156:

The upper slanted line is derived from a Pythagorean triangle with long leg 1200, short leg 49, and hypotenuse 1201. The central slanted line is derived from a triangle with long leg 120, short leg 26.9, and hypotenuse 122.978087479... The lower slanted line is derived from a Pythagorean triangle with long leg 60, short leg 11, and hypotenuse 61. The total sum of the 16 cell perimeters is 1930156 x 4 = 7720624. The average cell perimeter is 7720624 / 16 = 482539. The perimeter of the magic square is 120000 x 4 x 4 = 1920000. The coordinates of this square are available upon request. This perimeter magic square's cell values are voluntarily rounded to 6 digits. Greater accuracy is possible, but this will inevitably lead to very high entries. If more or less digits are used for the cell perimeter values, then the slanted lines will need vertical translations to take into account the revised magic sums, and this will imply new dimensions. There are some interesting relationships between the broken diagonals of this linear perimeter magic square (L-PMS), as shown in the following diagram: 

Perhaps a reader with programming skills will be tempted to find the first semi-orthogonal linear perimeter magic square of order-4 with all slanted diagonals derived from Pythagorean triangles, and therefore endowed with finite integer values for its cell entries?

Linear Perimeter Semi-Magic Squares of Order-4

For comparison with the above examples of linear perimeter magic squares with rounded cell entries, the following linear perimeter semi-magic squares of order-4 are each constructed with all three slanting lines derived from Pythagorean triangles. Their individual cell perimeters can thus be defined as integers, and the resulting cell entries and magic sums of these perimeter semi-magic squares are therefore rational and finite.

Constructed on the 24th February 2017, the following linear perimeter semi-magic square (L-PSMS) has a magic sum of S = 15492:

The upper slanted line is derived from a Pythagorean triangle with long leg 24, short leg 7, and hypotenuse 25. The central slanted line is derived from a Pythagorean triangle with long leg 480, short leg 31, and hypotenuse 481. The lower slanted line is derived from a Pythagorean triangle with long leg 40, short leg 9, and hypotenuse 41. The total sum of the 16 cell perimeters is 15492 x 4 = 61968. The average cell perimeter is 61968 / 16 = 3873. The perimeter of the semi-magic square is 960 x 4 x 4 = 15360. The coordinates of this square are available upon request. There are some interesting relationships between the broken diagonals of this linear perimeter semi-magic square (L-PSMS), as shown in the following diagram: 

Also constructed on the 24th February 2017, the following linear perimeter semi-magic square (L-PSMS) has a magic sum of S = 970:

The upper slanted line is derived from a Pythagorean triangle with long leg 40, short leg 9, and hypotenuse 41. The central slanted line is derived from a Pythagorean triangle with long leg 60, short leg 11, and hypotenuse 61. The lower slanted line is derived from a Pythagorean triangle with long leg 24, short leg 7, and hypotenuse 25. The total sum of the 16 cell perimeters is 970 x 4 = 3880. The average cell perimeter is 3880 / 16 = 242.5. The perimeter of the semi-magic square is 60 x 4 x 4 = 960. The coordinates of this square are available upon request. There are some interesting relationships between the broken diagonals of this linear perimeter semi-magic square (L-PSMS), as shown in the following diagram: 

Further Developments

If anyone wishes to contribute linear or strict geometrical constructions of other perimeter magic squares, then please send me the x and y coordinates of the cell intersections that define the perimeters correctly up to two or more decimal places. No prizes can be given, but the authors of pertinent solutions will, if they wish, have their solutions published here.

A new post on "Polyomino Area Magic Tori" (PAMT) can be found in this blog since the 7th June 2022. Perhaps these PAMT will suggest new possibilities for the exploration of polyomino perimeter magic squares (PPMS) and polyomino perimeter magic tori (PPMT)...

Read more ...

251,449,712 Fifth-Order Magic Tori

After finding the number of 3rd and 4th-order magic tori it was tempting to look further and Walter Trump kindly offered his computing skills to determine the number of the 5th-order magic tori.

Please note that the squares used to illustrate the magic tori are not necessarily "magic," except for the pandiagonal examples, as partially pandiagonal and basic magic tori display not only magic squares but also "semi-magic" cousins. To see the magic squares you sometimes need to move the viewpoint round the torus until a magic intersection of magic diagonals coincides with the centre of the grid, as explained in a previous article "From the Magic Square to the Magic Torus." With odd-order tori a magic intersection of magic diagonals occurs over a number and not over a space between numbers.

Provisionally, the different types of fifth-order tori are listed in descending magic order (taking into account the number of magic squares per torus). The results of Walter Trump's computing are thus as follows:

T 5.01 Pandiagonal or Panmagic Tori Type 1
with 10 crossed magic diagonals producing 25 magic intersections

Total   : 144 pandiagonal or panmagic tori type 1 that display 3,600 pandiagonal or panmagic squares

T 5.02 Partially Pandiagonal Tori Type 2
with 5 magic diagonals crossed by 3 other magic diagonals producing 15 magic intersections

Total   : 402 partially pandiagonal tori type 2 that display 6,030 partially pandiagonal squares and 4,020 semi-magic squares

T 5.03 Partially Pandiagonal Tori Type 3
with 5 magic diagonals crossed by 2 other magic diagonals producing 10 magic intersections

Total   : 136 partially pandiagonal tori type 3 that display 1,360 partially pandiagonal squares and 2,040 semi-magic squares

T 5.04 Partially Pandiagonal Tori Type 4
with 3 magic diagonals crossed by 3 other magic diagonals producing 9 magic intersections

Total   : 2,856 partially pandiagonal tori type 4 that display 25,704 partially pandiagonal squares and 45,696 semi-magic squares

T 5.05 Partially Pandiagonal Tori Type 5
with 3 magic diagonals crossed by 2 other magic diagonals producing 6 magic intersections

Total   : 21,348 partially pandiagonal tori type 5 that display 128,088 partially pandiagonal squares and 405,612 semi-magic squares

T 5.06 Partially Pandiagonal Tori Type 6
with 5 magic diagonals crossed by 1 other magic diagonal producing 5 magic intersections

Total   : 14,038 partially pandiagonal tori type 6 that display 70,190 partially pandiagonal squares and 280,760 semi-magic squares

T 5.07 Partially Pandiagonal Tori Type 7
with 2 magic diagonals crossed by 2 other magic diagonals producing 4 magic intersections

Total   : 518,536 partially pandiagonal tori type 7 that display 2,074,144 partially pandiagonal squares and 10,889,256 semi-magic squares

T 5.08 Partially Pandiagonal Tori Type 8
with 3 magic diagonals crossed by 1 other magic diagonal producing 3 magic intersections

Total   : 523,036 partially pandiagonal tori type 8 that display 1,569,108 partially pandiagonal squares and 11,506,792 semi-magic squares

T 5.09 Partially Pandiagonal Tori Type 9
with 2 magic diagonals crossed by 1 other magic diagonal producing 2 magic intersections

Total   : 21,057,784 partially pandiagonal tori type 9 that display 42,115,568 partially pandiagonal squares and 484,329,032 semi-magic squares

T 5.10 Basic Magic Tori Type 10
with 2 crossed magic diagonals producing 1 magic intersection (over a number) and one non-magic intersection (between numbers)

Total   : 229,311,432 basic magic tori type 10 that display 229,311,432 basic magic squares and 5,503,474,368 semi-magic squares

GRAND TOTALS OF THE MAGIC TORI :
251,449,712 FIFTH-ORDER MAGIC TORI THAT DISPLAY 275,305,224 MAGIC SQUARES AND 6,010,937,576 SEMI-MAGIC SQUARES

T 5.11 Semi-Magic Tori Type 11
with 5 parallel magic diagonals producing no magic intersections

Total   : 64,108 semi-magic tori type 11 that display 1,602,700 semi-magic squares


T 5.12 Semi-Magic Tori Type 12
with 3 parallel magic diagonals producing no magic intersections

Total   : 3,823,164 semi-magic tori type 12 that display 95,579,100 semi-magic squares


T 5.13 Semi-Magic Tori Type 13
with 2 parallel magic diagonals producing no magic intersections

Total   : 187,679,364 semi-magic tori type 13 that display 4,691,984,100 semi-magic squares

T 5.14 Semi-Magic Tori Type 14
with 1 magic diagonal producing no magic intersection

Total   : 4,138,786,720 semi-magic tori type 14 that display 103,469,668,000 semi-magic squares

T 5.15 Semi-Magic Tori Type 15
with no magic diagonals

Total   : 18,579,918,980 semi-magic tori type 15 that display 464,497,974,500 semi-magic squares

GRAND TOTALS OF THE SEMI-MAGIC TORI :
22,910,272,336 FIFTH-ORDER SEMI-MAGIC TORI THAT DISPLAY

572,756,808,400 SEMI-MAGIC SQUARES

ALL TOGETHER:
23,161,722,048 FIFTH-ORDER MAGIC AND SEMI-MAGIC TORI THAT DISPLAY 275,305,224 MAGIC SQUARES
AND 6,010,937,576 + 572,756,808,400 = 578,767,745,976 SEMI-MAGIC SQUARES

THE TOTAL NUMBER OF FIFTH-ORDER MAGIC AND SEMI-MAGIC SQUARES IS
579,043,051,200

CONCLUSIONS:

The study reveals the existence of 251,449,712 fifth-order magic tori and 22,910,272,336 fifth-order semi-magic tori.

The overall total of the fifth-order magic and semi-magic tori is 23,161,722,048 which is equal to the number of fifth-order magic and semi-magic squares (see Walter Trump's previous enumeration) divided by 25, (5² being the number of squares per torus for 5^th-order tori).

251,449,712 is now the fifth number of the sequence A270876 "Number of magic tori of order n composed of the numbers from 1 to n^2," published by the On-Line Encyclopedia of Integer Sequences (OEIS).

23,161,722,048 (251,449,712 + 22,910,272,336) is now the fifth number of the sequence A271104 "Number of magic and semi-magic tori of order n composed of the numbers from 1 to n^2," published by the On-Line Encyclopedia of Integer Sequences (OEIS).

Read more ...

From the Magic Square to the Magic Torus

This post is a translation of the article "Passage du Carré Magique au Tore Magique," first published in French on the 11th November 2011.

To visualise a magic torus let's begin with the Frénicle index square n° 91 (Bernard Frénicle de Bessy's classification of 880 fourth order magic squares - published posthumously in 1693 in his book "Des Quarrez Magiques" - see the listing on Harvey Heinz's site - Frénicle n°s 1 - 200, Frénicle n°s 201 - 400, Frénicle n°s 401 - 600, Frénicle n°s 601 - 880):

Extending this magic square we can construct a matrix of possibilities sized (2N-1)² to discover the other squares that are displayed on the magic torus:

This way we can find another 15 squares. We can see that there are 4 partially pandiagonal squares and 12 semi-magic squares displayed on the magic torus:

Then we can convert the squares that we have found into Frénicle standard form:

We can see that the magic torus displays not only the partially pandiagonal square Frénicle n°91, but also 3 other partially pandiagonal squares (Frénicle n°150, 320, and 394), as well as 12 semi-magic squares.

We are looking at the partially pandiagonal torus type n° T4.03.1.2. (index n° T4.174):

Ironically, although the procedure is invaluable in eliminating doubles, conversion into Frénicle standard form (by rotation, transposition, and / or reflection) has until today concealed the toroidal continuity of magic squares.

In the end, magic squares are to be observed in only two fundamental ways: either from outside the torus or from within, and it does not really matter which, as the torus can always be turned inside out...

Inside-out torus by Surot [Public domain], via Wikimedia Commons

Read more ...

255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus

This post is a translation of the article "255 Tores Magiques d'Ordre 4, et 1 Tore Magique d'Ordre 3," first published in French on the 28th October 2011.

By rolling a square and then connecting the two ends of the cylinder, the limited surface of the square is transformed into the limitless surface of a torus. The torus permits a better visualisation of the scrolling numbers of a magic square, but with the inconvenient that certain numbers are hidden on the rear external surface and inside the ring.

"Torus from rectangle" by Lucas Vieira [Public domain], via Wikimedia Commons

In order to optimise the visualisation of the scrolling effect we can take a new flattened look. When extending a repetition of the lines, columns and diagonals of a pandiagonal magic square we obtain a grid of numbers. On this grid, starting from the initial pandiagonal square (coloured black) it is possible to generate several other different squares that are all pandiagonal:

All the pandiagonal squares that are created this way can exist in a zone of seven by seven squares, (here indicated by the dotted enclosure), beyond which the squares become redundant because of the scrolling phenomenon. Beginning with an N-order pandiagonal square the number of different squares that exist in the system is N².

Convinced by the interest of this approach I chose to study the « Table générale des quarrez magiques de quatre » established by Bernard Frénicle de Bessy and published posthumously in 1693 in his book « Des quarrez magiques ». The census of these 880 4x4 squares is given in the appendix to the book « New Recreations With Magic Squares » by William H. Benson and Oswald Jacoby (Edition 1976) and also on the website of the late Harvey D. Heinz:  Frénicle n° 1 à 200, Frénicle n° 201 à 400, Frénicle n° 401 à 600, Frénicle n° 601 à 880.

Thanks to the computing skills of Walter Trump the amended and extended results of this research now reveal the existence of 255 fourth-order magic tori and 4 038 fourth-order semi-magic tori:

Fourth-order Magic Tori 

 

T4.01 Pandiagonal or Panmagic Tori Type 1

with 8 crossed magic diagonals producing 16 magic intersections.

Each torus type 1 is entirely covered by 16 sub-magic 2x2 squares.

All of the displayed pandiagonal squares are Dudeney type I. 

T4.01.1 (index n° T4.194):

16 pandiagonal squares Frénicle N°102, 104, 174, 201, 279, 281, 365, 393, 473, 530, 565, 623, 690, 748, 785 and 828. The torus is entirely covered by 16 sub-magic 2x2 squares.

The illustration below shows a way to spot the similar number sequences in different Frenicle squares showing that they come from a same torus:

For more details of how to find a magic torus, please refer to the article "From the Magic Square to the Magic Torus".  Also, the "Table of Fourth-Order Magic Tori" shows how to convert any fourth-order magic square to normalised torus form, and thus discover the index number and complete details of the corresponding magic torus.

T4.01.2 (index n° T4.198):

16 pandiagonal squares Frénicle N°107, 109, 171, 204, 292, 294, 355, 396, 469, 532, 560, 621, 691, 744, 788, and 839. The torus is entirely covered by 16 sub-magic 2x2 squares.

T4.01.3 (index n° T4.213):

16 pandiagonal squares Frénicle N°116, 117, 177, 178, 304, 305, 372, 375, 483, 485, 536, 537, 646, 647, 702 and 704. The torus is entirely covered by 16 sub-magic 2x2 squares.

Total   : 3 pandiagonal or panmagic tori type 1 that display 48 pandiagonal or panmagic squares, and 48 sub-magic 2x2 squares.

All of the displayed pandiagonal squares are Dudeney type I.

T4.02 Semi-pandiagonal Tori Type 2

with 4 crossed magic diagonals producing 8 magic intersections.

Each torus type 2.1 is entirely covered by 12 sub-magic 2x2 squares

Each torus type 2.2 is entirely covered by 8 sub-magic 2x2 squares

Each torus type 2.3 is entirely covered by 8 sub-magic 2x2 squares 

T4.02.1 Semi-pandiagonal Tori Type 2.1

with 4 crossed magic diagonals producing 8 magic intersections. 

Each torus type 2.1 is entirely covered by 12 sub-magic 2x2 squares.

Half of the displayed semi-pandiagonal squares are Dudeney type IV and the other half are Dudeney type VI.

T4.02.1.01 (index n° T4.038):

8 semi-pandiagonal squares Frénicle N° 16, 160, 224, 385, 435, 603, 810, and 855, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.02 (index n° T4.027):

8 semi-pandiagonal squares Frénicle N° 17, 85, 225, 322, 436, 604, 778, and 843, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.03 (index n° T4.036):

8 semi-pandiagonal squares Frénicle N° 18, 141, 226, 334, 432, 599, 809, and 854, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.04 (index n° T4.023):

8 semi-pandiagonal squares Frénicle N° 19, 61, 227, 251, 433, 600, 766, and 817, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.05 (index n° T4.053):

8 semi-pandiagonal squares Frénicle N° 24, 137, 216, 342 443, 589, 806, and 859, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.06 (index n° T4.044):

8 semi-pandiagonal squares Frénicle N° 25, 55, 217, 262, 444, 590, 761, and 822, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.07 (index n° T4.073):

8 semi-pandiagonal squares Frénicle N° 30, 155, 236, 380, 419, 581, 802, and 851, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.08 (index n° T4.064):

8 semi-pandiagonal squares Frénicle N° 31, 77, 237, 313, 420, 582, 771, and 840, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.09 (index n° T4.115):

8 semi-pandiagonal squares Frénicle N° 48, 192, 255, 400, 570, 734, 763, and 824, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.10 (index n° T4.101):

8 semi-pandiagonal squares Frénicle N° 49, 93, 256, 326, 467, 498, 664, and 670, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.11 (index n° T4.111):

8 semi-pandiagonal squares Frénicle N° 54, 140, 261, 333, 499, 548, 671, and 715, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.12 (index n° T4.125):

8 semi-pandiagonal squares Frénicle N° 60, 136, 250, 341, 504, 543, 669, and 716, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.13 (index n° T4.138):

8 semi-pandiagonal squares Frénicle N° 64, 184, 273, 378, 507, 534, 681, and 696, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.14 (index n° T4.130):

8 semi-pandiagonal squares Frénicle N° 65, 74, 274, 311, 456, 508, 651, and 682, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.15 (index n° T4.155):

8 semi-pandiagonal squares Frénicle N° 73, 191, 310, 399, 572, 735, 774, and 842, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.16 (index n° T4.151):

8 semi-pandiagonal squares Frénicle N° 76, 159, 312, 384, 459, 515, 654, and 705, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.17 (index n° T4.165):

8 semi-pandiagonal squares Frénicle N° 84, 154, 321, 379, 451, 521, 649, and 708, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.18 (index n° T4.175):

8 semi-pandiagonal squares Frénicle N° 92, 182, 325, 367, 466, 540, 663, and 699, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.19 (index n° T4.201):

8 semi-pandiagonal squares Frénicle N° 110, 196, 287, 402, 566, 730, 791, and 836, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.20 (index n° T4.204):

8 semi-pandiagonal squares Frénicle N° 111, 195, 288, 401, 569, 731, 792, and 837, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.21 (index n° T4.220):

8 semi-pandiagonal squares Frénicle N° 119, 163, 296, 387, 488, 528, 636, and 713, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.22 (index n° T4.224):

8 semi-pandiagonal squares Frénicle N° 121, 162, 298, 386, 490, 527, 638, and 712, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.23 (index n° T4.226):

8 semi-pandiagonal squares Frénicle N° 123, 147, 307, 351, 475, 550, 630, and 723, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

T4.02.1.24 (index n° T4.231):

8 semi-pandiagonal squares Frénicle N° 125, 146, 309, 350, 477, 549, 633, and 722, and also 8 semi-magic squares. The torus is entirely covered by 12 sub-magic 2x2 squares.

Total    : 24 semi-pandiagonal tori type 2.1 that display 192 semi-pandiagonal squares, 192 semi-magic squares, and 288 sub-magic 2x2 squares

Half of the displayed semi-pandiagonal squares are Dudeney type IV and the other half are Dudeney type VI.

T4.02.2 Semi-pandiagonal Tori Type 2.2

with 4 crossed magic diagonals producing 8 magic intersections.
Each torus type 2.2 is entirely covered by 8 sub-magic 2x2 squares.
Half of the displayed semi-pandiagonal squares are Dudeney type II and the other half are Dudeney type III (associated or self-complementary squares). 

T4.02.2.01 (index n° T4.059):
8 semi-pandiagonal squares Frénicle N° 21, 176, 213, 361, 445, 591, 808, and 860, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.2.02 (index n° T4.048):
8 semi-pandiagonal squares Frénicle N° 22, 113, 214, 290, 446, 592, 790, and 835, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.2.03 (index n° T4.077):

8 semi-pandiagonal squares Frénicle N° 27, 175 (the Dürer square), 233, 360, 421, 583, 803, and 850, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.2.04 (index n° T4.067):
8 semi-pandiagonal squares Frénicle N° 28, 112, 234, 289, 422, 584, 789, and 834, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.2.05 (index n° T4.127):
8 semi-pandiagonal squares Frénicle N° 56, 203, 246, 392, 562, 746, 768, and 818, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.2.06 (index n° T4.121):
8 semi-pandiagonal squares Frénicle N° 57, 122, 247, 299, 489, 503, 637, and 668, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.2.07 (index n° T4.137):
8 semi-pandiagonal squares Frénicle N° 62, 185, 269, 377, 505, 535, 678, and 695, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.2.08 (index n° T4.133):
8 semi-pandiagonal squares Frénicle N° 63, 120, 270, 297, 487, 506, 635, and 679, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.2.09 (index n° T4.164):
8 semi-pandiagonal squares Frénicle N° 82, 206, 316, 395, 558, 741, 779, and 844, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.2.10 (index n° T4.158):
8 semi-pandiagonal squares Frénicle N° 83, 126, 308, 317, 450, 478, 632, and 648, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.2.11 (index n° T4.173):
8 semi-pandiagonal squares Frénicle N° 89, 183, 323, 368, 464, 539, 661, and 698, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.2.12 (index n° T4.169):
8 semi-pandiagonal squares Frénicle N° 90, 124, 306, 324, 465, 476, 628, and 662, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

Total    : 12 semi-pandiagonal tori type 2.2 that display 96 semi-pandiagonal squares, 96 semi-magic squares, and 96 sub-magic 2x2 squares. 

Half of the displayed semi-pandiagonal squares are Dudeney type II and the other half are Dudeney type III (associated or self-complementary squares).

T4.02.3 Semi-pandiagonal Tori Type 2.3

with 4 crossed magic diagonals producing 8 magic intersections.

Each torus type 2.3 is entirely covered by 8 sub-magic 2x2 squares.

All of the displayed semi-pandiagonal squares are Dudeney type V.

T4.02.3.01 (index n° T4.085):

8 semi-pandiagonal squares Frénicle N° 32, 173, 228, 362, 425, 577, 798, and 853, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.3.02 (index n° T4.080):

8 semi-pandiagonal squares Frénicle N° 33, 108, 229, 293, 426, 578, 787,and 838, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.3.03 (index n° T4.084):

8 semi-pandiagonal squares Frénicle N° 34, 169, 230, 356, 423, 575, 797, and 852, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.3.04 (index n° T4.079):

8 semi-pandiagonal squares Frénicle N° 35, 103, 231, 282, 424, 576, 784, and 827, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.3.05 (index n° T4.145):

8 semi-pandiagonal squares Frénicle N° 66, 165, 264, 389, 511, 524, 674, and 710, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.3.06 (index n° T4.144):

8 semi-pandiagonal squares Frénicle N° 67, 115, 265, 302, 479, 512, 640, and 675, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.3.07 (index n° T4.146):

8 semi-pandiagonal squares Frénicle N° 68, 164, 266, 388, 513, 523, 676, and 709, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.3.08 (index n° T4.143):

8 semi-pandiagonal squares Frénicle N° 69, 101, 267, 280, 471, 514, 622, and 677, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.3.09 (index n° T4.183):

8 semi-pandiagonal squares Frénicle N° 95, 149, 328, 345, 460, 554, 657, and 721, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.3.10 (index n° T4.181):

8 semi-pandiagonal squares Frénicle N° 96, 114, 301, 329, 461, 481, 642, and 658, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.3.11 (index n° T4.184):

8 semi-pandiagonal squares Frénicle N° 97, 148, 330, 344, 462, 553, 659, and 720, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.02.3.12 (index n° T4.178):

8 semi-pandiagonal squares Frénicle N° 98, 106, 291, 331, 463, 468, 620, and 660, and also 8 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

Total    : 12 semi-pandiagonal tori type 2.3 that display 96 semi-pandiagonal squares, 96 semi-magic squares, and 96 sub-magic 2x2 squares.  

All of the displayed semi-pandiagonal squares are Dudeney type V.

T4.03 Partially pandiagonal Tori Type 3

with 4 crossed magic diagonals producing 4 magic intersections and 4 non magic intersections 

Each torus type 3 is entirely covered by 8 sub-magic 2x2 squares.

Each torus type 3.1 displays partially pandiagonal squares Dudeney type VI.

Each torus type 3.2 displays partially pandiagonal squares Dudeney types VII and IX.

Each torus type 3.3 displays partially pandiagonal squares Dudeney type XI.

T4.03.1 Partially pandiagonal Tori Type 3.1

with 4 crossed magic diagonals producing 4 magic intersections and 4 non magic intersections.

Each torus type 3.1 is entirely covered by 8 sub-magic 2x2 squares.

All of the displayed partially pandiagonal squares are Dudeney type VI.

T4.03.1.1 (index n° T4.108): 

4 partially pandiagonal squares Frénicle N° 46, 50, 337 and 545, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.1.2 (index n° T4.174):
4 partially pandiagonal squares Frénicle N° 91, 150, 320 and 394, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares. (see this torus here).

T4.03.1.3 (index n° T4.186): 

4 partially pandiagonal squares Frénicle N° 100, 285, 617 and 793, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.1.4 (index n° T4.246): 

4 partially pandiagonal squares Frénicle N° 132, 253, 259 and 719, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.1.5 (index n° T4.216): 

4 partially pandiagonal squares Frénicle N° 179, 376, 482 and 645, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.1.6 (index n° T4.168):
4 partially pandiagonal squares Frénicle N° 187, 366, 667 and 780, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

Total    : 6 partially pandiagonal tori type 3.1 that display 24 partially pandiagonal squares, 72 semi-magic squares, and 48 sub-magic 2x2 squares.  

All of the displayed partially pandiagonal squares are Dudeney type VI.

T4.03.2 Partially pandiagonal Tori Type 3.2

with 4 crossed magic diagonals producing 4 magic intersections and 4 non magic intersections.

Each torus type 3.2 is entirely covered by 8 sub-magic 2x2 squares.

Half of the displayed partially pandiagonal squares are Dudeney type VII and the other half are Dudeney type IX.

T4.03.2.01 (index n° T4.193):
4 partially pandiagonal squares Frénicle N° 105, 439, 442 and 786, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.2.02 (index n° T4.212):
4 partially pandiagonal squares Frénicle N° 118, 271, 272 and 486, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.2.03 (index n° T4.214): 

4 partially pandiagonal squares Frénicle N° 180, 347, 349 and 538, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.2.04 (index n° T4.188): 

4 partially pandiagonal squares Frénicle N° 200, 737, 765 and 868, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.2.05 (index n° T4.189): 

4 partially pandiagonal squares Frénicle N° 207, 736, 775 and 831, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.2.06 (index n° T4.195):
4 partially pandiagonal squares Frénicle N° 208, 525, 526 and 693, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.2.07 (index n° T4.060):
4 partially pandiagonal squares Frénicle N° 283, 472, 588 and 795, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.2.08 (index n° T4.141):
4 partially pandiagonal squares Frénicle N° 303, 509, 510 and 644, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.2.09 (index n° T4.234):
4 partially pandiagonal squares Frénicle N° 371, 551, 552 and 701, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.2.10 (index n° T4.123):
4 partially pandiagonal squares Frénicle N° 390, 571, 782 and 867, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.2.11 (index n° T4.221):
4 partially pandiagonal squares Frénicle N° 407, 531, 692 and 711, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.2.12 (index n° T4.161): 

4 partially pandiagonal squares Frénicle N° 408, 555, 783 and 829, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

Total    : 12 partially pandiagonal tori type 3.2 that display 48 partially pandiagonal squares, 144 semi-magic squares, and 96 sub-magic 2x2 squares.  

Half of the displayed partially pandiagonal squares are Dudeney type VII and the other half are Dudeney type IX.

T4.03.3 Partially pandiagonal Tori Type 3.3

with 4 crossed magic diagonals producing 4 magic intersections and 4 non magic intersections.

Each torus type 3.3 is entirely covered by 8 sub-magic 2x2 squares.

All of the displayed partially pandiagonal squares are Dudeney type XI.

T4.03.3.1 (index n° T4.217):
4 partially pandiagonal squares Frénicle N° 181, 374, 484 and 643, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.03.3.2 (index n° T4.187):
4 partially pandiagonal squares Frénicle N° 202, 364, 689 and 724, and also 12 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

Total    : 2 partially pandiagonal tori type 3.3 that display 8 partially pandiagonal squares, 24 semi-magic squares, and 16 sub-magic 2x2 squares.  

All of the displayed partially pandiagonal squares are Dudeney type XI.

T4.04 Partially pandiagonal Tori Type 4

with 3 crossed magic diagonals producing 2 magic intersections and 2 non magic intersections

Each torus type 4 is entirely covered by 4 sub-magic 2x2 squares. Half of the displayed partially pandiagonal squares are Dudeney type VIII and the other half are Dudeney type X.
(tori identified thanks to Walter Trump's computing skills)

T4.04.1 (index n° T4.096): 

2 partially pandiagonal squares Frénicle N° 40 and 522, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.04.2 (index n° T4.098): 

2 partially pandiagonal squares Frénicle N° 275 and 519, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.04.3 (index n° T4.120): 

2 partially pandiagonal squares Frénicle N° 542 and 866, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.04.4 (index n° T4.157): 

2 partially pandiagonal squares Frénicle N° 727 and 865 and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

Total    : 4 partially pandiagonal tori type 4 that display 8 partially pandiagonal squares, 56 semi-magic squares, and 16 sub-magic 2x2 squares
Half of the displayed partially pandiagonal squares are Dudeney type VIII and the other half are Dudeney type X.

T4.05 Basic Magic Tori Type 5

with 2 crossed magic diagonals producing 2 magic intersections

Each torus type 5.1 and 5.2 is entirely covered by 8 sub-magic 2x2 squares.

Each torus type 5.3 and 5.4 is entirely covered by 4 sub-magic 2x2 squares.

T4.05.1 Basic Magic Tori Type 5.1

with 2 crossed magic diagonals producing 2 magic intersections. 

Each torus type 5.1 is entirely covered by 8 sub-magic 2x2 squares.

All of the displayed magic squares are Dudeney type VI.

T4.05.1.01 (index n° T4.002): 

2 magic squares Frénicle N° 1 and 458, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.02 (index n° T4.001): 

2 magic squares Frénicle N° 2 and 448, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.03 (index n° T4.012): 

2 magic squares Frénicle N° 4 and 672, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.04 (index n° T4.008): 

2 magic squares Frénicle N° 5 and 625, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.05 (index n° T4.011): 

2 magic squares Frénicle N° 6 and 611, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.06 (index n° T4.007): 

2 magic squares Frénicle N° 7 and 612, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.07 (index n° T4.034): 

2 magic squares Frénicle N° 12 and 520, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.08 (index n° T4.035): 

2 magic squares Frénicle N° 13 and 339, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.09 (index n° T4.025): 

2 magic squares Frénicle N° 14 and 457, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.10 (index n° T4.029): 

2 magic squares Frénicle N° 15 and 258, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.11 (index n° T4.089): 

2 magic squares Frénicle N° 36 and 742, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.12 (index n° T4.062): 

2 magic squares Frénicle N° 37 and 832, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.13 (index n° T4.088): 

2 magic squares Frénicle N° 38 and 706, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.14 (index n° T4.043): 

2 magic squares Frénicle N° 39 and 826, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

  

T4.05.1.15 (index n° T4.103):  

2 magic squares Frénicle N° 44 and 352, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.16 (index n° T4.107):  

2 magic squares Frénicle N° 45 and 557, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.17 (index n° T4.109):  

2 magic squares Frénicle N° 47 and 556, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.18 (index n° T4.063):  

2 magic squares Frénicle N° 51 and 772, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.19 (index n° T4.114):  

2 magic squares Frénicle N° 52 and 222, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.20 (index n° T4.100):  

2 magic squares Frénicle N° 53 and 223, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.21 (index n° T4.148):  

2 magic squares Frénicle N° 70 and 391, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.22 (index n° T4.149):  

2 magic squares Frénicle N° 71 and 359, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.23 (index n° T4.132):  

2 magic squares Frénicle N° 72 and 745, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.24 (index n° T4.066):  

2 magic squares Frénicle N° 75 and 857, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.25 (index n° T4.119):  

2 magic squares Frénicle N° 78 and 725, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.26 (index n° T4.047):  

2 magic squares Frénicle N° 79 and 856, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.27 (index n° T4.046):  

2 magic squares Frénicle N° 80 and 760, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.28 (index n° T4.118):  

2 magic squares Frénicle N° 81 and 430, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.29 (index n° T4.171):  

2 magic squares Frénicle N° 86 and 363, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.30 (index n° T4.172):  

2 magic squares Frénicle N° 87 and 318, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.31 (index n° T4.243): 

2 magic squares Frénicle N° 127 and 220, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

  

T4.05.1.32 (index n° T4.136):  

2 magic squares Frénicle N° 130 and 561, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.33 (index n° T4.134):  

2 magic squares Frénicle N° 131 and 624, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.34 (index n° T4.072):  

2 magic squares Frénicle N° 133 and 807, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.35 (index n° T4.071): 

2 magic squares Frénicle N° 134 and 830, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T 4.05.1.36 (index n° T4.244):  

2 magic squares Frénicle N° 135 and 219, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.37 (index n° T4.052):  

2 magic squares Frénicle N° 138 and 825, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.38 (index n° T4.124):  

2 magic squares Frénicle N° 139 and 610, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.39 (index n° T4.227):  

2 magic squares Frénicle N° 143 and 627, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.40 (index n° T4.232):  

2 magic squares Frénicle N° 144 and 631, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.41 (index n° T4.253):  

2 magic squares Frénicle N° 151 and 286, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.42 (index n° T4.211):  

2 magic squares Frénicle N° 152 and 454, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.43 (index n° T4.202):  

2 magic squares Frénicle N° 153 and 447, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.44 (index n° T4.252): 

2 magic squares Frénicle N° 156 and 284, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.45 (index n° T4.057):  

2 magic squares Frénicle N° 157 and 801, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.46 (index n° T4.248):  

2 magic squares Frénicle N° 158 and 428, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.47 (index n° T4.076):  

2 magic squares Frénicle N° 166 and 848, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.48 (index n° T4.058):  

2 magic squares Frénicle N° 167 and 846, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.49 (index n° T4.192):  

2 magic squares Frénicle N° 168 and 609, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.50 (index n° T4.140):  

2 magic squares Frénicle N° 186 and 683, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.51 (index n° T4.254):  

2 magic squares Frénicle N° 188 and 295, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.52 (index n° T4.128):  

2 magic squares Frénicle N° 189 and 533, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.53 (index n° T4.250):  

2 magic squares Frénicle N° 190 and 474, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.54 (index n° T4.207):  

2 magic squares Frénicle N° 194 and 480, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.55 (index n° T4.129):  

2 magic squares Frénicle N° 198 and 680, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.56 (index n° T4.196):  

2 magic squares Frénicle N° 199 and 629, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.57 (index n° T4.004):  

2 magic squares Frénicle N° 210 and 656, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.58 (index n° T4.003):  

2 magic squares Frénicle N° 211 and 619, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.59 (index n° T4.225):  

2 magic squares Frénicle N° 218 and 707, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.60 (index n° T4.131):  

2 magic squares Frénicle N° 221 and 653, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.61 (index n° T4.229):  

2 magic squares Frénicle N° 252 and 743, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.62 (index n° T4.247):  

2 magic squares Frénicle N° 254 and 740, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.63 (index n° T4.065): 

2 magic squares Frénicle N° 257 and 841, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.64 (index n° T4.045):  

2 magic squares Frénicle N° 314 and 821, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.65 (index n° T4.024):  

2 magic squares Frénicle N° 315 and 597, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.66 (index n° T4.139):  

2 magic squares Frénicle N° 336 and 747, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.67 (index n° T4.074):  

2 magic squares Frénicle N° 338 and 861, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.68 (index n° T4.150):  

2 magic squares Frénicle N° 370 and 639, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.69 (index n° T4.147):  

2 magic squares Frénicle N° 381 and 614, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.70 (index n° T4.054):  

2 magic squares Frénicle N° 382 and 849, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.71 (index n° T4.037):  

2 magic squares Frénicle N° 383 and 594, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.72 (index n° T4.126):  

2 magic squares Frénicle N° 397 and 694, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.73 (index n° T4.112):  

2 magic squares Frénicle N° 398 and 626, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.74 (index n° T4.215):  

2 magic squares Frénicle N° 404 and 641, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.75 (index n° T4.218):  

2 magic squares Frénicle N° 427 and 718, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.76 (index n° T4.102):  

2 magic squares Frénicle N° 429 and 673, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.77 (index n° T4.223):  

2 magic squares Frénicle N° 452 and 750, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.78 (index n° T4.028):  

2 magic squares Frénicle N° 502 and 596, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.79 (index n° T4.176):  

2 magic squares Frénicle N° 516 and 752, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.80 (index n° T4.039):  

2 magic squares Frénicle N° 547 and 593, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.81 (index n° T4.166):  

2 magic squares Frénicle N° 573 and 697, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.82 (index n° T4.152):  

2 magic squares Frénicle N° 574 and 634, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.83 (index n° T4.041):  

2 magic squares Frénicle N° 595 and 804, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.84 (index n° T4.042):  

2 magic squares Frénicle N° 602 and 764, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.85 (index n° T4.185):  

2 magic squares Frénicle N° 615 and 815, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.86 (index n° T4.163): 

2 magic squares Frénicle N° 665 and 777, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.87 (index n° T4.162):  

2 magic squares Frénicle N° 739 and 755, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.88 (index n° T4.167):  

2 magic squares Frénicle N° 751 and 776, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.89 (index n° T4.200):  

2 magic squares Frénicle N° 756 and 862, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.90 (index n° T4.203):  

2 magic squares Frénicle N° 769 and 864, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.91 (index n° T4.113):  

2 magic squares Frénicle N° 813 and 819, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.1.92 (index n° T4.153):  

2 magic squares Frénicle N° 816 and 833, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

Total    : 92 basic magic tori type 5.1 that display 184 basic magic squares, 1,288 semi-magic squares, and 736 sub-magic 2x2 squares
All of the displayed basic magic squares are Dudeney type VI.

T4.05.2 Basic Magic Tori Type 5.2

with 2 crossed magic diagonals producing 2 magic intersections. 

Each torus type 5.2 is entirely covered by 8 sub-magic 2x2 squares.

Half of the displayed magic squares are Dudeney type VII and the other half are Dudeney type IX.

T4.05.2.01 (index n° T4.049): 

2 magic squares Frénicle N° 23 and 767, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.02 (index n° T4.068): 

2 magic squares Frénicle N° 29 and 781, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.03 (index n° T4.026):  

2 magic squares Frénicle N° 59 and 870, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.04 (index n° T4.249):  

2 magic squares Frénicle N° 142 and 438, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.05 (index n° T4.228):  

2 magic squares Frénicle N° 145 and 529, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.06 (index n° T4.219):  

2 magic squares Frénicle N° 161 and 703, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.07 (index n° T4.040):  

2 magic squares Frénicle N° 172 and 869, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.08 (index n° T4.255):  

2 magic squares Frénicle N° 193 and 268, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.09 (index n° T4.206):  

2 magic squares Frénicle N° 197 and 346, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.10 (index n° T4.197):  

2 magic squares Frénicle N° 205 and 700, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.11 (index n° T4.122):  

2 magic squares Frénicle N° 215 and 799, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.12 (index n° T4.251):  

2 magic squares Frénicle N° 235 and 845, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

  

T4.05.2.13 (index n° T4.032):  

2 magic squares Frénicle N° 249 and 878, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.14 (index n° T4.142): 

2 magic squares Frénicle N° 300 and 559, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.15 (index n° T4.061):  

2 magic squares Frénicle N° 335 and 470, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.16 (index n° T4.199):  

2 magic squares Frénicle N° 348 and 714, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.17 (index n° T4.033):  

2 magic squares Frénicle N° 358 and 875, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

 
T4.05.2.18 (index n° T4.233):  

2 magic squares Frénicle N° 369 and 563, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.19 (index n° T4.209):  

2 magic squares Frénicle N° 373 and 717, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.20 (index n° T4.210):  

2 magic squares Frénicle N° 403 and 738, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.21 (index n° T4.205):  

2 magic squares Frénicle N° 405 and 517, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.22 (index n° T4.156):  

2 magic squares Frénicle N° 406 and 453, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.23 (index n° T4.160):  

2 magic squares Frénicle N° 418 and 805, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.24 (index n° T4.245):  

2 magic squares Frénicle N° 441 and 873, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.25 (index n° T4.154):  

2 magic squares Frénicle N° 455 and 729, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.26 (index n° T4.191):  

2 magic squares Frénicle N° 518 and 728, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.27 (index n° T4.222):  

2 magic squares Frénicle N° 564 and 732, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.28 (index n° T4.230):  

2 magic squares Frénicle N° 567 and 749, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.29 (index n° T4.110):  

2 magic squares Frénicle N° 568 and 757, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.30 (index n° T4.075):  

2 magic squares Frénicle N° 580 and 858, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.31 (index n° T4.055):  

2 magic squares Frénicle N° 587 and 879, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

T4.05.2.32 (index n° T4.116):  

2 magic squares Frénicle N° 733 and 758, and also 14 semi-magic squares. The torus is entirely covered by 8 sub-magic 2x2 squares.

Total    : 32 basic magic tori type 5.2 that display 64 basic magic squares, 448 semi-magic squares, and 256 sub-magic 2x2 squares

Half of the displayed basic magic squares are Dudeney type VII and the other half are Dudeney type IX.

T4.05.3 Basic Magic Tori Type 5.3

with 2 crossed magic diagonals producing 2 magic intersections. 

Each torus type 5.3 is entirely covered by 4 sub-magic 2x2 squares.

All of the displayed magic squares are Dudeney type XII.

T4.05.3.1 (index n° T4.005): 

2 magic squares Frénicle N° 3 and 613, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.3.2 (index n° T4.170):  

2 magic squares Frénicle N° 88 and 650, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.3.3 (index n° T4.006):  

2 magic squares Frénicle N° 209 and 449, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.3.4 (index n° T4.159):  

2 magic squares Frénicle N° 319 and 666, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

Total    : 4 basic magic tori type 5.3 that display 8 basic magic squares, 56 semi-magic squares, and 16 sub-magic 2x2 squares

All of the displayed basic magic squares are Dudeney type XII

T4.05.4 Basic Magic Tori Type 5.4

with 2 crossed magic diagonals producing 2 magic intersections. 

Each torus type 5.4 is entirely covered by 4 sub-magic 2x2 squares.

Half of the displayed magic squares are Dudeney type VIII and the other half are Dudeney type X.

T4.05.4.01 (index n° T4.019): 

2 magic squares Frénicle N° 8 and 343, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.02 (index n° T4.018): 

2 magic squares Frénicle N° 9 and 263, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.03 (index n° T4.022): 

2 magic squares Frénicle N° 10 and 240, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.04 (index n° T4.017): 

2 magic squares Frénicle N° 11 and 241, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.05 (index n° T4.050): 

2 magic squares Frénicle N° 20 and 811, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.06 (index n° T4.069): 

2 magic squares Frénicle N° 26 and 812, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.07 (index n° T4.097): 

2 magic squares Frénicle N° 41 and 493, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.08 (index n° T4.095): 

2 magic squares Frénicle N° 42 and 494, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.09 (index n° T4.092):  

2 magic squares Frénicle N° 43 and 434, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.10 (index n° T4.078):  

2 magic squares Frénicle N° 58 and 877, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.11 (index n° T4.182): 

2 magic squares Frénicle N° 94 and 655, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.12 (index n° T4.177): 

2 magic squares Frénicle N° 99 and 618, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.13 (index n° T4.239):  

2 magic squares Frénicle N° 128 and 773, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.14 (index n° T4.240):  

2 magic squares Frénicle N° 129 and 762, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.15 (index n° T4.083):  

2 magic squares Frénicle N° 170 and 874, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.16 (index n° T4.237): 

2 magic squares Frénicle N° 212 and 794, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.17 (index n° T4.242):  

2 magic squares Frénicle N° 232 and 863, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.18 (index n° T4.020):  

2 magic squares Frénicle N° 238 and 411, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.19 (index n° T4.015):  

2 magic squares Frénicle N° 239 and 412, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.20 (index n° T4.208):  

2 magic squares Frénicle N° 242 and 546, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.21 (index n° T4.105):  

2 magic squares Frénicle N° 243 and 501, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.22 (index n° T4.014):  

2 magic squares Frénicle N° 244 and 415, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.23 (index n° T4.010): 

2 magic squares Frénicle N° 245 and 416, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.24 (index n° T4.082):  

2 magic squares Frénicle N° 248 and 872, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.25 (index n° T4.016): 

2 magic squares Frénicle N° 260 and 410, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.26 (index n° T4.099):  

2 magic squares Frénicle N° 276 and 491, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.27 (index n° T4.086):  

2 magic squares Frénicle N° 277 and 685, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.28 (index n° T4.031):  

2 magic squares Frénicle N° 278 and 601, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.29 (index n° T4.179):  

2 magic squares Frénicle N° 327 and 616, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.30 (index n° T4.180):  

2 magic squares Frénicle N° 332 and 652, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.31 (index n° T4.021): 

2 magic squares Frénicle N° 340 and 409, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.32 (index n° T4.236):  

2 magic squares Frénicle N° 353 and 770, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.33 (index n° T4.104):  

2 magic squares Frénicle N° 354 and 823, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.34 (index n° T4.081):  

2 magic squares Frénicle N° 357 and 871, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.35 (index n° T4.013):  

2 magic squares Frénicle N° 413 and 607, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.36 (index n° T4.009):  

2 magic squares Frénicle N° 414 and 608, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.37 (index n° T4.238):  

2 magic squares Frénicle N° 417 and 796, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.38 (index n° T4.093):  

2 magic squares Frénicle N° 431 and 686, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.39 (index n° T4.190):  

2 magic squares Frénicle N° 437 and 814, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.40 (index n° T4.241):  

2 magic squares Frénicle N° 440 and 880, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.41 (index n° T4.094):  

2 magic squares Frénicle N° 492 and 688, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.42 (index n° T4.030):  

2 magic squares Frénicle N° 495 and 598, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.43 (index n° T4.090):  

2 magic squares Frénicle N° 496 and 754, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

 
T4.05.4.44 (index n° T4.087):  

2 magic squares Frénicle N° 497 and 684, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.45 (index n° T4.106):  

2 magic squares Frénicle N° 500 and 606, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.46 (index n° T4.235):  

2 magic squares Frénicle N° 541 and 759, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.47 (index n° T4.135):  

2 magic squares Frénicle N° 544 and 605, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.48 (index n° T4.070):  

2 magic squares Frénicle N° 579 and 847, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.49 (index n° T4.056):  

2 magic squares Frénicle N° 585 and 800, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.50 (index n° T4.051):  

2 magic squares Frénicle N° 586 and 876, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.51 (index n° T4.091):  

2 magic squares Frénicle N° 687 and 753, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

T4.05.4.52 (index n° T4.117):  

2 magic squares Frénicle N° 726 and 820, and also 14 semi-magic squares. The torus is entirely covered by 4 sub-magic 2x2 squares.

Total   : 52 basic magic tori type 5.4 that display 104 basic magic squares, 728 semi-magic squares, and 208 sub-magic 2x2 squares.

Half of the displayed basic magic squares are Dudeney type VIII and the other half are Dudeney type X.

Grand Totals of the 4th-Order Magic Tori :
255 FOURTH-ORDER MAGIC TORI THAT DISPLAY 880 MAGIC SQUARES,
3 200 SEMI-MAGIC SQUARES, AND 1 920 SUB-MAGIC 2x2 SQUARES

Fourth-order Semi-Magic Tori

A semi-magic square is a square that contains arithmetical magic in all of its rows and columns, but not in both of its main diagonals.

According to Walter Trump's first computer calculations the number of semi-magic and magic squares is 68,688. 68,688 - 880 magic squares = 67,808 semi-magic squares. 3,200 semi-magic squares are already displayed on the 255 magic tori. We can deduce that there are also 67,808 – 3,200 = 64,608 4th-order semi-magic squares displayed on another 4,038 semi-magic tori (64,608 / 16). The semi-magic tori could have crossed magic diagonals that do not produce magic intersections, parallel magic diagonals (without magic intersections), or no magic diagonals at all. Walter Trump's latest computer programme confirms that the fourth-order semi-magic tori either have parallel magic diagonals that produce no magic intersections, or no magic diagonals at all. Some examples of the different types (provisionally represented here by a single square for each torus) are shown below:

T4.06 Semi-Magic Tori Type 6

with 4 parallel magic diagonals producing no magic intersections

Total   : 12 semi-magic tori type 6 that display 192 semi-magic squares

(semi-magic tori counted by Walter Trump's computer calculations)

T4.07 Semi-Magic Tori Type 7

with 2 unequally spaced parallel magic diagonals producing no magic intersections

Total   : 12 semi-magic tori type 7 that display 192 semi-magic squares

(semi-magic tori counted by Walter Trump's computer calculations)

T4.08 Semi-Magic Tori Type 8

with 2 equally spaced parallel magic diagonals producing no magic intersections

Total   : 88 semi-magic tori type 8 that display 1,408 semi-magic squares 

(semi-magic tori counted by Walter Trump's computer calculations)

T4.09 Semi-Magic Tori Type 9

with a single magic diagonal producing no magic intersection

Total   : 200 semi-magic tori type 9 that display 3,200 semi-magic squares

(semi-magic tori counted by Walter Trump's computer calculations)

T4.10 Semi-Magic Tori Type 10

with no magic diagonals

Total   : 3,726 semi-magic tori type 10 that display 59,616 semi-magic squares

(semi-magic tori counted by Walter Trump's computer calculations)

The study remains to be completed by listing all the semi-magic tori in numerical order (using Frénicle standard form for example).

Grand Totals of the 4th-Order Semi-Magic Tori :
4,038 FOURTH-ORDER SEMI-MAGIC TORI THAT DISPLAY 64,608 SEMI-MAGIC SQUARES

 

Magic and Semi-Magic 4th-Order Tori : Conclusions

The study of the table of 880 4th-order magic squares reveals the existence of 255 magic tori, that not only display 880 magic squares, but also 3,200 semi-magic squares. All of the 255 magic tori are entirely covered by 2x2 sub-magic squares. There is a total of 1,920 sub-magic 2x2 squares on the 255 magic tori.

The number of N-order squares (magic and semi-magic squares) that are displayed on each N-order magic torus = N². 

4,038 semi-magic 4th-order tori have also been found, and these display a further 64,608 semi-magic squares. The sub-magic 2x2 squares of the 4th-order semi-magic tori have yet to be enumerated.

255 is now the fourth number of the sequence A270876 "Number of magic tori of order n composed of the numbers from 1 to n^2," published by the On-Line Encyclopedia of Integer Sequences (OEIS).

4,293 (255 + 4,038) is now the fourth number of the sequence A271104 "Number of magic and semi-magic tori of order n composed of the numbers from 1 to n^2," published by the On-Line Encyclopedia of Integer Sequences (OEIS).

T3 Third-Order Magic Torus

The study would not be complete without examining the unique 3rd-order magic torus. Beginning with the 3x3 magic square in Frénicle standard form (at the top left of the following illustration), and taking into account the scrolling effect on the 3rd order magic torus we obtain another eight 3rd-order semi-magic squares:

After transformation of these semi-magic squares into Frénicle standard form we obtain:

The 3rd-order magic torus displays just one magic square. Although there are two crossed magic diagonals on the torus, they only produce a single magic intersection over the number 5, whilst the second intersection occurs in the interval between the numbers 6, 4, 8 and 2. It is interesting to note that the mean number of 6+4+8+2 is 5, and that the mean numbers of the couples 6+4 and 8+2 are also 5.

We can deduce that basic (not pandiagonal, semi-pandiagonal or partially pandiagonal) odd number N-order tori  display not only 1 basic N-order magic square, but also N²-1 semi-magic N-order squares. For example, the smallest 1st-order magic torus displays 1 basic 1st-order magic square and 1²-1=0 semi-magic 1st-order squares. It is interesting to note that the first-order magic torus can be visualised either as a basic magic torus, or even as a pandiagonal torus!

1, (the number of third-order magic tori), is now the third number of the sequences A270876 "Number of magic tori of order n composed of the numbers from 1 to n^2" and A271104 "Number of magic and semi-magic tori of order n composed of the numbers from 1 to n^2," published by the On-Line Encyclopedia of Integer Sequences (OEIS).

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!

Acknowledgements

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 magic torus type 4.04, and enabled a precise count.

I wish to thank Miguel Angel Amela who very kindly programmed a count of 4x4 broken diagonal solutions of the fourth-order magic squares, the results of which confirm the findings of the above study.

Since publishing this study I have become aware of Aale de Winkel's "Magic Encyclopedia" database Order 04 square group theory. I wish to belatedly acknowledge Aale de Winkel's work, which, although quite different to mine, (we neither use the same reasoning nor come to the same conclusion), is complementary. Please note that in his study Aale de Winkel uses analytic numbers ranging from 0 to 15 instead of the regular numbers 1 to 16. This does not change the magical properties of the squares but you need to add 1 to each number to see the traditional Frénicle versions.

I wish to thank Kanji Setsuda for pointing out the importance of associated or self-complementary magic squares. This has lead me to create a specific subdivision of the semi-pandiagonal tori.

My thanks also go to Dwane Campbell, who has published a very interesting comparative study of the Dudeney and Walkington Order-4 magic square classifications in the three pages that follow:
"Analysis of Order-4 Magic Squares,"
"Order-4 Magic Squares Grouped by Base Square Quartets,"
"Features in Order-4 Magic Squares."

Read more ...