Panmagic square
Encyclopedia
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 n×n pandiagonal magic square can be regarded as having 8n2 orientations.
In 4×4 panmagic squares, the magic constant of 34 can be seen in a number of patterns in addition to the rows, columns and diagonals:
Thus of the 86 possible sums adding to 34, 52 of them form regular patterns, compared with 10 for an ordinary 4×4 magic square.
There are only three distinct 4×4 pandiagonal magic squares, namely the one above and the following:
EWLINE
In any 4×4 pandiagonal magic square, any two numbers at the opposite corners of a 3×3 square add up to 17. Consequently, no 4×4 panmagic squares are associative
.
. The following is a 5×5 associative panmagic square:
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:
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 panmagic square
with magic sum Z. To prove the quincunx sum A+E+M+U+Y = Z (corresponding to the 20+2+13+24+6 = 65 example given above), one adds together the following:
From this sum the following are subtracted:
The net result is 5A+5E+5M+5U+5Y = 5Z, which divided by 5 gives the quincunx sum. Similar linear combinations can be constructed for the other quincunx patterns H+L+M+N+R, C+K+M+O+W, and G+I+M+Q+S.
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
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
Magic constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, and diagonal of the magic square. For example, the magic square shown below has a magic constant of 15....
.
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 n×n pandiagonal magic square can be regarded as having 8n2 orientations.
4×4 panmagic squares
The smallest non-trivial pandiagonal magic squares are 4×4 squares.| 1 | 8 | 13 | 12 |
| 14 | 11 | 2 | 7 |
| 4 | 5 | 16 | 9 |
| 15 | 10 | 3 | 6 |
In 4×4 panmagic squares, the magic constant of 34 can be seen in a number of patterns in addition to the rows, columns and diagonals:
- Any of the sixteen 2×2 squares, including those that wrap around the edges of the whole square, e.g. 14+11+4+5, 1+12+15+6
- The corners of any 3×3 square, e.g. 8+12+5+9
- Any pair of horizontally or vertically adjacent numbers, together with the corresponding pair displaced by a (2, 2) vector, e.g. 1+8+16+9
Thus of the 86 possible sums adding to 34, 52 of them form regular patterns, compared with 10 for an ordinary 4×4 magic square.
There are only three distinct 4×4 pandiagonal magic squares, namely the one above and the following:
| 1" cellspacing="0" cellpadding="2" style="text-align:center"> | |||
| 1 | 12 | 7 | 14 |
| 8 | 13 | 2 | 11 |
| 10 | 3 | 16 | 5 |
| 15 | 6 | 9 | 4 |
| 1 | 8 | 11 | 14 |
| 12 | 13 | 2 | 7 |
| 6 | 3 | 16 | 9 |
| 15 | 10 | 5 | 4 |
In any 4×4 pandiagonal magic square, any two numbers at the opposite corners of a 3×3 square add up to 17. Consequently, no 4×4 panmagic squares are associative
Associative magic square
An associative magic square is a magic square for which every pair of numbers symmetrically opposite to the center sum up to the same value.-External links:*, MathWorld...
.
5×5 panmagic squares
There are many 5×5 pandiagonal magic squares. Unlike 4×4 panmagic squares, these can be associativeAssociative magic square
An associative magic square is a magic square for which every pair of numbers symmetrically opposite to the center sum up to the same value.-External links:*, MathWorld...
. The following is a 5×5 associative panmagic 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
Quincunx
A quincunx is a geometric pattern consisting of five points arranged in a cross, that is five coplanar points, four of them forming a square or rectangle and a fifth at its center...
" 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 panmagic square
| A | B | C | D | E |
| F | G | H | I | J |
| K | L | M | N | O |
| P | Q | R | S | T |
| U | V | W | X | Y |
with magic sum Z. To prove the quincunx sum A+E+M+U+Y = Z (corresponding to the 20+2+13+24+6 = 65 example given above), one adds together the following:
- 3 times each of the diagonal sums A+G+M+S+Y and E+I+M+Q+U
- The diagonal sums A+J+N+R+V, B+H+N+T+U, D+H+L+P+Y,and E+F+L+R+X
- The row sums A+B+C+D+E and U+V+W+X+Y
From this sum the following are subtracted:
- The row sums F+G+H+I+J and P+Q+R+S+T
- The column sum C+H+M+R+W
- Twice each of the column sums B+G+L+Q+V and D+I+N+S+X.
The net result is 5A+5E+5M+5U+5Y = 5Z, which divided by 5 gives the quincunx sum. Similar linear combinations can be constructed for the other quincunx patterns H+L+M+N+R, C+K+M+O+W, and G+I+M+Q+S.