Bordered Magic Squares

Terminology

There are differences in the usage of the term Bordered Magic Squares . Some sites use it to mean the same as Concentric Magic Squares while others restrict it to mean only the Consecutively Concentric Magic Squares.

Most of the squares presented here are of the restricted consecutively concentric variety. These have the property that each nested square consists of consecutive integers. A nested square of order m may be converted to a normal magic square by subtracting the (n²+1)/2 of the border order n square and adding (m²+1)/2, which is equivalent to subtracting (n²-m²)/2.

Background

The algorithms here were first programmed on the pocket calculators HP-67 and HP-41CV. The bones numbers are computed and, optionally, (n²+1)/2 added to each number to produce the actual square.

The programs are designed to compute one cell number at a time, given the row and column. A (row, column) is entered to calculate any cell number or, optionally, the square can be stepped through, computing one cell at a time, by repeatedly pressing the R/S button.

Because the squares are not stored in the calculator, the biggest square that can be made is determined by the computational and display capability of the calculator, and, time and patience .

Odd Order Squares

Example Squares: Order 3, 5, 7

To see the normal magic square of the order 3 square at the center of the order 7 square, subtract (7² - 3²)/2 = 20 from the nested order 3 square numbers. Similarly, subtract (7² - 5²)/2 = 12 for the nested order 5 square. Similarly, subtract (5² - 3²)/2 = 8 for the nested order 3 square of the order 5 square.

Bones of Above Squares

Imagine an X-axis through the center row and a Y-axis through the center column of each bones. A couple of observations:

• bones are nested: order 7 encloses order 5, which encloses order 3, (which encloses order 1)
• the symmetry is apparent:
  • main diagonal numbers are reflected in the center
  • between main diagonals, row numbers are reflected in the X-axis
  • between main diagonals, column numbers are reflected in the Y-axis
  • the sign of the number changes with each reflection

Even Order Squares

Example Squares: Order 4, 6, 8

To see the normal magic square of the order 4 square at the center of the order 8 square, subtract (8² - 4²)/2 = 24 from the nested order 4 square numbers. Similarly, subtract (8² - 6²)/2 = 14 for the nested order 6 square. Similarly, subtract (6² - 4²)/2 = 10 for the nested order 4 square of the order 6 square.

Bones of Above Squares

Note that ½ has been ommitted from all the bones numbers displayed here.
Therefore, to convert these displayed bones numbers to the actual square numbers:

• add n²/2 + 1 to the positive numbers
• add n²/2 to the negative numbers

Again, imagine a horizontal X-axis and a vertical Y-axis through the center of each bones. Some observations:

• bones are nested: order 8 encloses order 6, which encloses order 4
• the symmetry is apparent:
  • main diagonal numbers are reflected in the center
  • for the order 4, the other numbers are also reflected in the center
  • except for order 4:
    • between main diagonals, row numbers are reflected in the X-axis
    • between main diagonals, column numbers are reflected in the Y-axis
  • the sign of the number changes with each reflection

The center 4x4 square can be any of the 880 order 4 squares, (or their rotations and reflections).

Make Squares

Instructions

Click START.

To make a square:

• enter the order
• click OK

To make a bones:

• enter the order
• click the Make Bones box
• click OK

Note: In the bones displayed here, the minus sign is dropped from negative numbers. Positive and negative numbers are distinguished by different colors, (red, blue). Also, ½ is dropped from all numbers for even order bones.

Freeware Disclaimer

The software on this site may be used freely.
You should use only if you agree to the freeware disclaimer.

Please send errata or comments to Harry White   sharrywhite@eastlink.ca

REFERENCES

Bordered Magic Squares:

Heinz, Harvey "Glossary" http://www.magic-squares.net/glossary.htm

Nakamura, Mitsutoshi "Terms" http://homepage2.nifty.com/googol/magcube/en/terms.htm

Weisstein, Eric W. "Border Square" http://mathworld.wolfram.com/BorderSquare.html

See also: Magic Squares - REFERENCES

Calculators:

Furlow, Warren "The HP-41 Archive Website" http://www.hp41.org/Intro.cfm

Hicks, David G. "The Museum of HP Calculators" http://www.hpmuseum.org/