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Magic Squares Glossary

Definition List

The list below provides definitions of most of the terms used on this Magic Squares website.
Diagonals "Main Diagonals" are the oblique row of cells from one corner to another. The remainder are "Broken Diagonals" in which a short diagonal is completed by imagining a duplicate adjacent magic square.
Formula Many magic squares can be constructed with a reasonably simple Formula. This is useful when making magic squares with a spreadsheet.
Graeco - Latin
Literally this implies the combination of two arrays: one a Latin square; and the second a square made up using the Greek alphabet. In practice it is more convenient, and more common, to make a "Graeco Latin square" using two "Latin squares", one using CAPITALS and the other using lower case. The ideal arrangement distributes the letters so that no pair is duplicated and every letter appears just once in each row, column, and diagonal.
How Many One of the frequently asked questions is: "How Many Magic squares are there?" The exact interpretation of this question depends on whether reflections and rotations are counted as different.
Knight's Move A technique for distributing numbers in a Prime Number Magic square, so called because the "Next Number" is placed by stepping "Two Squares Forward" and "One Step Sideways" like the Chess Knight. With larger squares this technique can be modified to employ larger steps.
Latin Square A square array of Alphabetical characters "A, B, C, etc." arranged so that each character occurs once in each column, row and, if possible, each diagonal. All Latin Squares are "Magic Carpets".
Magic Carpet Carpet A square array of cells filled with a short sequence of numbers, e.g., zeros and ones, or 0 1 and 2, arranged so that each column, each row, and all diagonals sum to the same value. The square to the right is a Magic Carpet of order 4 and uses, eight times, the range zero to one; every row, column, and diagonal adds up to 2. For Prime Number Magic squares, the Magic Carpet is also a Latin Square - each letter is used once in each row. Where smaller sequences are used repeatedly in a row, the Magic carpet is not a true "Latin Square"
Magic Cube A three dimensional array of numbers where the cells sum to a Magic Sum in every row in all three axes.
Magic Grid An array of zeros and ones which fill a square of order 4p+2. The array is orthogonal when used in conjunction with a rotated version of itself. The resulting composite is used as a component to make 4p+2 Magic squares.
Magic Square A square array of cells filled with a sequence of numbers; each column, each row, and both diagonals all sum to the same value, the magic sum.
Magic Sum The total of the numbers in each row or column of a Magic square. When the square starts at zero and the size is order "N", the magic sum can be calculated as the number of cells in a row multiplied by the average of the first and last numbers in the whole square:
        [ N x (0+N2-1)/2 ],
which can be rewritten as:
        (N3 - N)/2 which would equal 60 for an order-5 square.

For a traditional square starting at number 1:
        Sum = (N3 + N)/2.
Nasik A Magic Square in which all the broken daigonals also sum to magic sum. This name was created by a Mr. Frost after the Indian town in which he lived. See also Pan-Diagonal and Pan-Magic
Order Order is a measure of size - the length of the side of a magic square, which is the square root of the total number of cells in the square. Thus an order 5 magic square has 25 cells
Pan-Diagonal A Magic Square in which all the broken daigonals also sum to magic sum. See also Pan-Magic and Nasik
A Magic Square in which all the broken daigonals also sum to the magic sum. See also Pan-Diagnonal and Nasik
Prime Number
Magic Squares
A Magic Square whose order is a prime number. These Magic squares can conveniently be made employing a "Knight's Move".
A Pan Magic square which can be broken down into two or more magic carpets.
Starting With When the lowest number in a Magic square is zero, it is more easily analysed and understood. Traditional Magic squares start with "1".
Magic Square
Magic square where the first number is "1"
Unique Identifier A technique for providing a unique label for a Magic square to facilitate counting how many there are.
Zero Origin When the lowest number in a Magic square is zero, it is more easily analysed and understood.


Copyright © Mar 2010 Magic Squares
Mar 6, 2010