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The 4x4 magic square is square, because it has as many rows (from left to right = horizontal) as columns
(from top to bottom = vertical).
The 4x4 magic square consists of 4 rows which multiplied by 4 columns is 16 cells.
The 4x4 magic square must contain 16 different digits. A pure magic 4x4 square contains the digits 1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16.
The magic square is magic, because the sum of the digits of each row, each column and both diagonals al-
ways give the same result. The sum can be calculated as follows, the (even) size of the square divided by 2
multiplied by the lowest plus the highest digit: 4 / 2 x (1 + 16) = 34.
What is a 4x4 panmagic square?
A 4x4 panmagic square has the above mentioned row-, column- and diagonal features (= minimum features)
plus extra magic features.
The extra magic features of the panmagic 4x4 square are:
- Each 2x2 sub-square has the same (magic) sum (= 34);
- Each 4x4 sub-square on a carpet of 2x2 (see below) is panmagic (in the carpet you can find 16 different
4x4 panmagic squares).
Each 4x4 subsquare on the 2x2 carpet is panmagic, because of the parallel running diagonals on the carpet. The
sum of the digits of four sequencing cells on a diagonal is always 34 (= the magic sum).
(pan)diagonals from left to right (pan)diagonals from right to left
Notify that in a 4x4 sub-square not only the sum of the four digits of the ordinary diagonals but also the sum of the
four digits of the (six) broken diagonals (= pandiagonals) is 34; see below.
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What is the secret behind the panmagic 4x4 square
How to get each time the same (magic) sum? Add the the lowest digit to the highest digit, the second lowest digit to
the second highest digit, the third lowest digit to the third highest digit, …. If you split up the digits of the 4x4 square
in 1 up to 8 and 9 up to 16, than you get:
1+8 = 2+7 = 3+6 = 4+5 = 9 and 9+16 = 10+15 = 11+14 = 12+13 = 25
See the panmagic 4x4 square below.
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Each row consists of 9 + 25 = the magic sum of 34.
The square is panmagic because of the following structure (= the structure for all panmagic squares that are multiples
of 4; 4x4, 8x8, 12x12, …):
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The sum of two digits of the same colour is each time (the highest digit of the magic square + 1, in this case 16+1=) 17,
that is half of the magic sum. Each time with two colours you can produce one of the (pan)diagonals, which total to (2 x
17 = ) the magic sum of 34:
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Notify that it is possible to swap row 1&3 and/or row 2&4 and/or column 1&3 and/or column 2&4 and you get another
panmagic 4x4 square!!!
How to produce 4x4 panmagic squares?
You can make a panmagic 4x4 square in three steps:
Place 1, 2, 3 and 4 Connect 5, 6, 7 and 8 2x same colour=17
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When you want to make all panmagic 4x4 squares, the ‘trick’ is to learn the 3 basic panmagic 4x4 squares by heart;
see on page: Panmagic 4x4 square
See also methods of construction on page: Sudoku method (1) and page Panmagic 4x4 square, binary and page
Transformation method
The famous Albrecht Dürer magic 4x4 square is not panmagic, but (semi panmagic and) symmetric. There is a trick
to make the Albrecht Dürer magic 4x4 square:
(1) Place 1 up to 16 (2) mirror 1 horizontally (3) mix 1&2 [take O & X]
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For another trick to make the Albrecht Dürer magic 4x4 square, see page Dürer and Franklin transformation.
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