Pandiagonal magic square

A pandiagonal magic square or panmagic square (also diabolic square, diabolical square or diabolical magic square) is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.

A pandiagonal magic square remains pandiagonally magic not only under rotation or reflection, but also if a row or column is moved from one side of the square to the opposite side. As such, an ${\displaystyle n\times n}$ pandiagonal magic square can be regarded as having ${\displaystyle 8n^{2}}$ orientations.

3×3 pandiagonal magic squares

It can be shown that non-trivial pandiagonal magic squares of order 3 do not exist. Suppose the square

${\displaystyle {\begin{array}{|c|c|c|}\hline \!\!a_{11}\!\!\!&\!\!a_{12}\!\!\!&\!\!a_{13}\!\!\!\\\hline \!\!a_{21}\!\!\!&\!\!a_{22}\!\!\!&\!\!a_{23}\!\!\!\\\hline \!\!a_{31}\!\!\!&\!\!a_{32}\!\!\!&\!\!a_{33}\!\!\!\\\hline \end{array}}}$

is pandiagonally magic with magic sum ${\displaystyle s}$. Adding sums ${\displaystyle a_{11}+a_{22}+a_{33},}$ ${\displaystyle a_{12}+a_{22}+a_{32},}$ and ${\displaystyle a_{13}+a_{22}+a_{31}}$ results in ${\displaystyle 3s}$. Subtracting ${\displaystyle a_{11}+a_{12}+a_{13}}$ and ${\displaystyle a_{31}+a_{32}+a_{33},}$ we get ${\displaystyle 3a_{22}=s}$. However, if we move the third column in front and perform the same proof, we obtain ${\displaystyle 3a_{21}=s}$. In fact, using the symmetries of 3 × 3 magic squares, all cells must equal ${\displaystyle {\tfrac {1}{3}}s}$. Therefore, all 3 × 3 pandiagonal magic squares must be trivial.

However, if the magic square concept is generalized to include geometric shapes instead of numbers—the geometric magic squares discovered by Lee Sallows—a 3 × 3 pandiagonal magic square does exist.

4×4 pandiagonal magic squares

Euler diagram of requirements of some types of 4 × 4 magic squares. Cells of the same colour sum to the magic constant.

The smallest non-trivial pandiagonal magic squares are 4 × 4 squares. All 4 × 4 pandiagonal magic squares must be translationally symmetric to the form [1]

a	a + b + c + e	a + c + d	a + b + d + e
a + b + c + d	a + d + e	a + b	a + c + e
a + b + e	a + c	a + b + c + d + e	a + d
a + c + d + e	a + b + d	a + e	a + b + c

Since each 2 × 2 subsquare sums to the magic constant, 4 × 4 pandiagonal magic squares are most-perfect magic square. In addition, the two numbers at the opposite corners of any 3 × 3 square add up to half the magic sum. Consequently, all 4 × 4 pandiagonal magic squares that are associative must have duplicate cells.

All 4 × 4 pandiagonal magic squares using numbers 1-16 without duplicates are obtained by letting a equal 1; letting b, c, d, and e equal 1, 2, 4, and 8 in some order; and applying some translation. For example, with b = 1, c = 2, d = 4, and e = 8, we have the magic square

1	8	13	12
14	11	2	7
4	5	16	9
15	10	3	6

The number of 4 × 4 pandiagonal magic squares using numbers 1-16 without duplicates is 384 (16×24, where 16 accounts for the translation and 24 accounts for the 4! ways to assign 1, 2, 4, and 8 to b, c, d, and e).

5×5 pandiagonal magic squares

There are many 5 × 5 pandiagonal magic squares. Unlike 4 × 4 pandiagonal magic squares, these can be associative. The following is a 5 × 5 associative pandiagonal magic square:

20	8	21	14	2
11	4	17	10	23
7	25	13	1	19
3	16	9	22	15
24	12	5	18	6

In addition to the rows, columns, and diagonals, a 5 × 5 pandiagonal magic square also shows its magic sum in four "quincunx" patterns, which in the above example are:

17+25+13+1+9 = 65 (center plus adjacent row and column squares)

21+7+13+19+5 = 65 (center plus the remaining row and column squares)

4+10+13+16+22 = 65 (center plus diagonally adjacent squares)

20+2+13+24+6 = 65 (center plus the remaining squares on its diagonals)

Each of these quincunxes can be translated to other positions in the square by cyclic permutation of the rows and columns (wrapping around), which in a pandiagonal magic square does not affect the equality of the magic sums. This leads to 100 quincunx sums, including broken quincunxes analogous to broken diagonals.

The quincunx sums can be proved by taking linear combinations of the row, column, and diagonal sums. Consider the pandiagonal magic square

${\displaystyle {\begin{array}{|c|c|c|c|c|}\hline \!\!a_{11}\!\!\!&\!\!a_{12}\!\!\!&\!\!a_{13}\!\!\!&\!\!a_{14}\!\!\!&\!\!a_{15}\!\!\!\\\hline \!\!a_{21}\!\!\!&\!\!a_{22}\!\!\!&\!\!a_{23}\!\!\!&\!\!a_{24}\!\!\!&\!\!a_{25}\!\!\!\\\hline \!\!a_{31}\!\!\!&\!\!a_{32}\!\!\!&\!\!a_{33}\!\!\!&\!\!a_{34}\!\!\!&\!\!a_{35}\!\!\!\\\hline \!\!a_{41}\!\!\!&\!\!a_{42}\!\!\!&\!\!a_{43}\!\!\!&\!\!a_{44}\!\!\!&\!\!a_{45}\!\!\!\\\hline \!\!a_{51}\!\!\!&\!\!a_{52}\!\!\!&\!\!a_{53}\!\!\!&\!\!a_{54}\!\!\!&\!\!a_{55}\!\!\!\\\hline \end{array}}}$

with magic sum s. To prove the quincunx sum ${\displaystyle a_{11}+a_{15}+a_{33}+a_{51}+a_{55}=s}$ (corresponding to the 20+2+13+24+6 = 65 example given above), we can add together the following:

3 times each of the diagonal sums ${\displaystyle a_{11}+a_{22}+a_{33}+a_{44}+a_{55}}$ and ${\displaystyle a_{15}+a_{24}+a_{33}+a_{42}+a_{51}}$,

The diagonal sums ${\displaystyle a_{11}+a_{25}+a_{34}+a_{43}+a_{52}}$, ${\displaystyle a_{12}+a_{23}+a_{34}+a_{45}+a_{51}}$, ${\displaystyle a_{14}+a_{23}+a_{32}+a_{41}+a_{55}}$, and ${\displaystyle a_{15}+a_{21}+a_{32}+a_{43}+a_{54}}$,

The row sums ${\displaystyle a_{11}+a_{12}+a_{13}+a_{14}+a_{15}}$ and ${\displaystyle a_{51}+a_{52}+a_{53}+a_{54}+a_{55}}$.

From this sum, subtract the following:

The row sums ${\displaystyle a_{21}+a_{22}+a_{23}+a_{24}+a_{25}}$ and ${\displaystyle a_{41}+a_{42}+a_{43}+a_{44}+a_{45}}$,

The column sum ${\displaystyle a_{13}+a_{23}+a_{33}+a_{43}+a_{53}}$,

Twice each of the column sums ${\displaystyle a_{12}+a_{22}+a_{32}+a_{42}+a_{52}}$ and ${\displaystyle a_{14}+a_{24}+a_{34}+a_{44}+a_{54}}$.

The net result is ${\displaystyle 5a_{11}+5a_{15}+5a_{33}+5a_{51}+5a_{55}=5s}$, which divided by 5 gives the quincunx sum. Similar linear combinations can be constructed for the other quincunx patterns ${\displaystyle a_{23}+a_{32}+a_{33}+a_{34}+a_{43}}$, ${\displaystyle a_{13}+a_{31}+a_{33}+a_{35}+a_{53}}$, and ${\displaystyle a_{22}+a_{24}+a_{33}+a_{42}+a_{44}}$.

(4n+2)×(4n+2) pandiagonal magic squares with nonconsecutive elements

No pandiagonal magic square exists of order ${\displaystyle 4n+2}$ if consecutive integers are used. But certain sequences of nonconsecutive integers do admit order-(${\displaystyle 4n+2}$) pandiagonal magic squares.

Consider the sum 1+2+3+5+6+7 = 24. This sum can be divided in half by taking the appropriate groups of three addends, or in thirds using groups of two addends:

1+5+6 = 2+3+7 = 12

1+7 = 2+6 = 3+5 = 8

An additional equal partitioning of the sum of squares guarantees the semibimagic property noted below:

1²+5²+6² = 2²+3²+7² = 62

Note that the consecutive integer sum 1+2+3+4+5+6 = 21, an odd sum, lacks the half-partitioning.

With both equal partitions available, the numbers 1, 2, 3, 5, 6, 7 can be arranged into 6x6 pandigonal patterns A and B, respectively given by:

1	5	6	7	3	2
5	6	1	3	2	7
6	1	5	2	7	3
1	5	6	7	3	2
5	6	1	3	2	7
6	1	5	2	7	3

6	5	1	6	5	1
1	6	5	1	6	5
5	1	6	5	1	6
2	3	7	2	3	7
7	2	3	7	2	3
3	7	2	3	7	2

Then ${\displaystyle 7A+B-7C}$ (where C is the magic square with 1 for all cells) gives the nonconsecutive pandiagonal 6x6 square:

6	33	36	48	19	8
29	41	5	15	13	47
40	1	34	12	43	20
2	31	42	44	17	14
35	37	3	21	9	45
38	7	30	10	49	16

with a maximum element of 49 and a pandiagonal magic sum of 150. This square is pandiagonal and semibimagic, that means that rows, columns, main diagonals and broken diagonals have a sum of 150 and, if we square all the numbers in the square, only the rows and the columns are magic and have a sum of 5150.

For 10th order a similar construction is possible using the equal partitionings of the sum 1+2+3+4+5+9+10+11+12+13 = 70:

1+3+9+10+12 = 2+4+5+11+13 = 35

1+13 = 2+12 = 3+11 = 4+10 = 5+9 = 14

1²+3²+9²+10²+12² = 2²+4²+5²+11²+13² = 335 (equal partitioning of squares; semibimagic property)

This leads to squares having a maximum element of 169 and a pandiagonal magic sum of 850, which are also semibimagic with each row or column sum of squares equal to 102,850.

(6n±1)×(6n±1) pandiagonal magic squares

A ${\displaystyle (6n\pm 1)\times (6n\pm 1)}$ pandiagonal magic square can be built by the following algorithm.

1. Set up the first column of the square with the first ${\displaystyle 6n\pm 1}$ natural numbers.

   1
   2
   3
   4
   5
   6
   7

2. Copy the first column into the second column but shift it ring-wise by 2 rows.

   1	6
   2	7
   3	1
   4	2
   5	3
   6	4
   7	5

3. Continue copying the current column into the next column with ring-wise shift by 2 rows until the square is filled completely.

   1	6	4	2	7	5	3
   2	7	5	3	1	6	4
   3	1	6	4	2	7	5
   4	2	7	5	3	1	6
   5	3	1	6	4	2	7
   6	4	2	7	5	3	1
   7	5	3	1	6	4	2

4. Build a second square and copy the transpose of the first square into it.

   A   1    6    4    2    7    5    3    2    7    5    3    1    6    4    3    1    6    4    2    7    5    4    2    7    5    3    1    6    5    3    1    6    4    2    7    6    4    2    7    5    3    1    7    5    3    1    6    4    2

   ${\displaystyle A^{T}}$   1    2    3    4    5    6    7    6    7    1    2    3    4    5    4    5    6    7    1    2    3    2    3    4    5    6    7    1    7    1    2    3    4    5    6    5    6    7    1    2    3    4    3    4    5    6    7    1    2

5. Build the final square by multiplying the second square by ${\displaystyle 6n\pm 1}$, adding the first square and subtract ${\displaystyle 6n\pm 1}$ in each cell of the square.

   Example: ${\displaystyle A+(6n\pm 1)A^{T}-(6n\pm 1)B}$, where B is the magic square with all cells as 1.

   1	13	18	23	35	40	45
   37	49	5	10	15	27	32
   24	29	41	46	2	14	19
   11	16	28	33	38	43	6
   47	3	8	20	25	30	42
   34	39	44	7	12	17	22
   21	26	31	36	48	4	9

4n×4n pandiagonal magic squares

A ${\displaystyle 4n\times 4n}$ pandiagonal magic square can be built by the following algorithm.

1. Put the first ${\displaystyle 2n}$ natural numbers into the first row and the first ${\displaystyle 2n}$ columns of the square.

   1	2	3	4

2. Put the next ${\displaystyle 2n}$ natural numbers beneath the first ${\displaystyle 2n}$ natural numbers in reverse. Each vertical pair must have the same sum.

   1	2	3	4
   8	7	6	5

3. Copy that ${\displaystyle 2\times 2n}$ rectangle ${\displaystyle 2n-1}$ times beneath the first rectangle.

   1	2	3	4
   8	7	6	5
   1	2	3	4
   8	7	6	5
   1	2	3	4
   8	7	6	5
   1	2	3	4
   8	7	6	5

4. Copy the left ${\displaystyle 4n\times 2n}$ rectangle into the right ${\displaystyle 4n\times 2n}$ rectangle but shift it ring-wise by one row.

   1	2	3	4	8	7	6	5
   8	7	6	5	1	2	3	4
   1	2	3	4	8	7	6	5
   8	7	6	5	1	2	3	4
   1	2	3	4	8	7	6	5
   8	7	6	5	1	2	3	4
   1	2	3	4	8	7	6	5
   8	7	6	5	1	2	3	4

5. Build a second 4n×4n square and copy the first square into it but turn it by 90°.

   A   1    2    3    4    8    7    6    5    8    7    6    5    1    2    3    4    1    2    3    4    8    7    6    5    8    7    6    5    1    2    3    4    1    2    3    4    8    7    6    5    8    7    6    5    1    2    3    4    1    2    3    4    8    7    6    5    8    7    6    5    1    2    3    4

   B   5    4    5    4    5    4    5    4    6    3    6    3    6    3    6    3    7    2    7    2    7    2    7    2    8    1    8    1    8    1    8    1    4    5    4    5    4    5    4    5    3    6    3    6    3    6    3    6    2    7    2    7    2    7    2    7    1    8    1    8    1    8    1    8

6. Build the final square by multiplying the second square by ${\displaystyle 4n}$, adding the first square and subtract ${\displaystyle 4n}$ in each cell of the square.

   Example: ${\displaystyle A+4nB-4nC}$, where C is the magic square with all cells as 1.

   33	26	35	28	40	31	38	29
   48	23	46	21	41	18	43	20
   49	10	51	12	56	15	54	13
   64	7	62	5	57	2	59	4
   25	34	27	36	32	39	30	37
   24	47	22	45	17	42	19	44
   9	50	11	52	16	55	14	53
   8	63	6	61	1	58	3	60

If we build a ${\displaystyle 4n\times 4n}$ pandiagonal magic square with this algorithm then every ${\displaystyle 2\times 2}$ square in the ${\displaystyle 4n\times 4n}$ square will have the same sum. Therefore, many symmetric patterns of ${\displaystyle 4n}$ cells have the same sum as any row and any column of the ${\displaystyle 4n\times 4n}$ square. Especially each ${\displaystyle 2n\times 2}$ and each ${\displaystyle 2\times 2n}$ rectangle will have the same sum as any row and any column of the ${\displaystyle 4n\times 4n}$ square. The ${\displaystyle 4n\times 4n}$ square is also a Most-perfect magic square.

(6n+3)×(6n+3) pandiagonal magic squares

A ${\displaystyle (6n+3)\times (6n+3)}$ pandiagonal magic square can be built by the following algorithm.

1. Create a ${\displaystyle (2n+1)\times 3}$ rectangle with the first ${\displaystyle 6n+3}$ natural numbers so that each column has the same sum. You can do this by starting with a 3 × 3 magic square and set up the rest cells of the rectangle in meander-style. You can also use the pattern shown in the following examples.

   For 9 × 9 square  1   2   3   5   6   4   9   7   8  vertical sum = 15

   For 15 × 15 square  1   2   3   5   6   4   9   7   8   10   11   12   15   14   13  vertical sum = 40

   For 21 × 21 square  1   2   3   5   6   4   9   7   8  10 11 12 15 14 13 16 17 18 21 20 19 vertical sum = 77

2. Put this rectangle in the left upper corner of the ${\displaystyle (6n+3)\times (6n+3)}$ square and two copies of the rectangle beneath it so that the first 3 columns of the square are filled completely.

   1	2	3
   5	6	4
   9	7	8
   1	2	3
   5	6	4
   9	7	8
   1	2	3
   5	6	4
   9	7	8

3. Copy the left 3 columns into the next 3 columns, but shift it ring-wise by 1 row.

   1	2	3	9	7	8
   5	6	4	1	2	3
   9	7	8	5	6	4
   1	2	3	9	7	8
   5	6	4	1	2	3
   9	7	8	5	6	4
   1	2	3	9	7	8
   5	6	4	1	2	3
   9	7	8	5	6	4

4. Continue copying the current 3 columns into the next 3 columns, shifted ring-wise by 1 row, until the square is filled completely.

   1	2	3	9	7	8	5	6	4
   5	6	4	1	2	3	9	7	8
   9	7	8	5	6	4	1	2	3
   1	2	3	9	7	8	5	6	4
   5	6	4	1	2	3	9	7	8
   9	7	8	5	6	4	1	2	3
   1	2	3	9	7	8	5	6	4
   5	6	4	1	2	3	9	7	8
   9	7	8	5	6	4	1	2	3

5. Build a second square and copy the transpose of the first square into it.

   A   1    2    3    9    7    8    5    6    4   5   6   4   1   2   3   9   7   8   9   7   8   5   6   4   1   2   3   1   2   3   9   7   8   5   6   4   5   6   4   1   2   3   9   7   8   9   7   8   5   6   4   1   2   3   1   2   3   9   7   8   5   6   4   5   6   4   1   2   3   9   7   8   9   7   8   5   6   4   1   2   3

   ${\displaystyle A^{T}}$   1    5    9    1    5    9    1    5    9   2   6   7   2   6   7   2   6   7   3   4   8   3   4   8   3   4   8   9   1   5   9   1   5   9   1   5   7   2   6   7   2   6   7   2   6   8   3   4   8   3   4   8   3   4   5   9   1   5   9   1   5   9   1   6   7   2   6   7   2   6   7   2   4   8   3   4   8   3   4   8   3

6. Build the final square by multiplying the second square by ${\displaystyle 6n+3}$, adding the first square and subtract ${\displaystyle 6n+3}$ in each cell of the square.

   Example: ${\displaystyle A+(6n+3)A^{T}-(6n+3)B}$, where B is the magic square with all cells as 1.

   1	38	75	9	43	80	5	42	76
   14	51	58	10	47	57	18	52	62
   27	34	71	23	33	67	19	29	66
   73	2	39	81	7	44	77	6	40
   59	15	49	55	11	48	63	16	53
   72	25	35	68	24	31	64	20	30
   37	74	3	45	79	8	41	78	4
   50	60	13	46	56	12	54	61	17
   36	70	26	32	69	22	28	65	21

References

1. Ng, Louis (May 13, 2018). "Magic Counting with Inside-Out Polytopes" (PDF).

• W. S. Andrews, Magic Squares and Cubes. New York: Dover, 1960. Originally printed in 1917. See especially Chapter X.

External links

• Panmagic Square at MathWorld
• http://www.azspcs.net/Contest/PandiagonalMagicSquares