How to produce bordered squares?
Introduction
There are 3 similar types of squares: inlaid squares, bordered squares and concentric squares. See for
the exact descriptions on the website of Harvey Heinz: www.magic-squares.net/glossary.htm
A bordered square is an (impure) magic square inside a bigger (pure) magic square. A bordered square
consist of the center magic square with a singular border around the center square. The center square
consist of the middle numbers of the magic square and the border contains the lowest numbers and their
complements (the highest numbers). See below a method of construction to produce even bordered
squares, an example of an odd bordered square and a reference to a website with (more complicated)
inlaid squares (= more squares inside a bigger magic square).
Even bordered squares
The smallest even bordered square is an (impure) 4x4 square inside a (pure) 6x6 square. A pure 6x6
square consist of all the digits from 1 up to 36.
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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11
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12
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13
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14
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15
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16
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17
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18
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19
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20
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21
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22
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23
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24
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25
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26
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27
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28
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29
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30
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31
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32
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33
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34
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35
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36
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-10
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-9
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-8
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-7
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-6
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-5
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-4
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-3
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-2
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-1
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Use the middle (yellow marked) 16 digits to produce the 4x4 inlaid square. Take for example the first basic
panmagic 4x4 square (see page ‘panmagic 4x4’) and add 10 to each digit.
Pure 4x4 +10 = (impure) inlaid 4x4
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1
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8
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13
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12
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11
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18
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23
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22
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15
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10
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3
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6
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25
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20
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13
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16
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4
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5
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16
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9
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14
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15
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26
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19
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14
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11
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2
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7
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24
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21
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12
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17
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To produce the border there are the following three boundary conditions:
● On the opposite of a possitive digit in the row, the column or the diagonal must be filled in the same
negative digit (= translation of the digits from 27 up to 36);
● In the top row, the bottom row, the left column and the right column must be filled in each time 6
different (3 positive and 3 negative) digits;
● The digits of the top row, the bottom row, the left column and the right column must each time total
to 0.
Produce the top row and the right column and fill in the digit 1 at the left corner on the top row. Because
the digit at the left corner on the top row is 1, at the right corner on the bottom row the digit -1 must be
filled in. The digit (?) to be filled in at the right corner on the top row is in the top row as well as in the right
column.
In the top row and the right column must be filled in 4x 3 digits from (+/-) 1 up to 10. Because the digits from
the top row and the right column must total to 0, it will be wise to use 4x the same sum (2x positive and 2x
negative is 0). It will be also wise to choose a sum, which gives the maximum number of possibilities to get
that sum with three digits out of 1 up to10. This sum can be calculated by multiplying the sum of 1 up to10,
\that is 55, with 3/10 (the choice of 3 digits out of 10); 55 x 3/10 = 16,5. Choose 16 or 17. If you choose 16,
than there are the following 10 possibilities to get 16 with three digits out of 1 up to 10:
1+5+10
1+6+ 9
1+7+ 8
2+4+10
2+5+ 9
2+6+ 8
3+4+ 9
3+5+ 8
3+6+ 7
4+5+ 7
Choose 4x 3 possibilities. Establish that in the four possibilities the digit 1 and another (?) are double (and the
other digits are single). See the four marked rows and notify that only the 1 and the 8 are double (so +/- 8
must be filled in at the right corner an the top row). The digits can be filled in as follows:
Fill in 4 possibilities Fill in opposite digits
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1
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6
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9
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-3
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-5
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-8
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1
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6
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9
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-3
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-5
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-8
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2
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-2
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2
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4
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-4
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4
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10
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-10
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10
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-7
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7
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-7
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-1
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8
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-6
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-9
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3
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5
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-1
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The final result is:
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1
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6
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9
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34
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32
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29
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35
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11
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18
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23
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22
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2
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33
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25
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20
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13
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16
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4
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27
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14
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15
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26
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19
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10
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7
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24
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21
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12
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17
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30
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8
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31
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28
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3
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5
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36
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You can use this method of construction for all even bordered magic squares (6x6 inside 8x8, 8x8 inside 10x10,
10x10 inside 12x12, …).
Odd bordered squares
The smallest bordered square is an (impure) 3x3 square inside the (pure) 5x5 square. It is possible to use the 9
(sequencing) middle digits out of 1 up to 25 to produce the 3x3 center square; see the result below.
3x3 inlaid inside 5x5 square
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1
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22
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20
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19
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3
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2
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10
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17
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12
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24
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18
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15
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13
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11
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8
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21
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14
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9
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16
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5
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23
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4
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6
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7
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25
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It is not possible to produce bigger odd bordered squares in the same way. I will try to find a method of construc-
tion to produce (bigger) odd bordered squares.
(more complicated) inlaid squares
You can find (more complicated) inlaid squares on the website of Harvey Heinz:
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