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FEATURES IN ORDER-4 MAGIC SQUARES

In an order-4 magic square, all four rows, all four columns, and the two main diagonals must add to the magic number, 34. All 880 Frénicle order-4 magic squares have these features. There are many additional features that have been described for these squares. They are present in some, but not all of the squares. These features require that the sum of the numbers in certain patterns within the magic squares add to a fixed value, usually the magic constant for 4 numbers or half of the magic constant for two numbers. The presence of these features has been used to classify them in groups with similar features.

If a feature is present in all of the base squares from which a magic square is constructed then it will also be present in the magic square.

The converse, if a feature is present in a magic square it will also be present in the base squares from which the magic square is constructed, is not true when features are made using catchup. The discussion on this page will focus on those features that are present in all the base squares from which the magic square is constructed. These are the predictable features.

The table below lists some features that are present in order-4 magic squares. Usually these are predictable features in the base squares from which the magic squares are constructed. If a square in the table is yellow then the base square of its row has the predictable feature of its column. The column codes for features that sum four numbers are A: all agonals add to 2 (the only requirement for base squares in the table), M: the main diagonals add to 2, 1: there is a broken diagonal adjacent to each main diagonal that adds to 2, 2: there is a broken diagonal two away from each main diagonal that adds to 2, C: the corners of all 3x3 sub squares add to 2 (it is compact₃), S: the number of 2x2 sub squares that add to 2 (a 16 means all sub squares add to 2 [yellow] and the base square contains the compact₂ feature), D: the number of diamond shapes that add to 2.

The base squares are shown in the table Order-4 Base Squares with Agonals that are Predictable.

The first seven features all require the addition of four numbers in the base square. The table only marks a feature valid if the sum of those four numbers is 2 for all repetitions of that feature in the base square. A combination of four base squares with that feature will have the feature as well. That base square quartet will be a magic square if it contains all numbers from 1 to 16.

Features made using four numbers can also appear in magic squares by using catchup. When catchup is used the sum of the four numbers in the catchup base squares is not 2. The feature is not predictable. None of the order-4 magic squares use catchup in their agonals, the diagonals that are two away from the main diagonals, the corners of 3x3 squares, or in any of the sub-magic 2x2 squares. Catchup is often used in the main diagonals and in the diagonals that are adjacent to the main diagonals. Some of the magic squares have a few diamond patterns that add to the magic sum. These always use catchup.

The Roman Numerals I to XII represent the twelve Groups defined by Dudeney. Dudeney grouped the Frénicle squares based on the patterns that are made by the complementary pairs within each magic square. Those patterns are shown in the column at left. The complementary pairs in each square have the same color. The patterns as shown appear in the rows(r) of the squares, they may also appear in the columns(c)(rotate the pattern 90°). Therefore a base square may contain the pattern in either r or c or both. Patterns I, II, and III are symmetrical and always appear in both r and c.

In order for a base square to exhibit one of the Dudeney features each complementary pair of the pattern must add to 1, i.e. one member of the pair must be a 0 and the other a 1. It is not possible to use catchup for a feature made from pairs of numbers so the Dudeney features are all predictable. A Dudeney feature will be present in a magic square only if it is present in all four of its base squares.

The base square codes in the "codes" columns always contain a base square with a 0 in the upper left corner (the first code) and that base square's complement. If the codes are red or orange then they are base squares that can initiate a catchup sequence in the main diagonals. If they are green or blue then the base squares can maintain the catchup sequence or terminate it. If the initiator is red then the green base squares can maintain it and the blue ones terminate it. For the orange initiators the blue base squares maintain and the green terminate.

Codes that are crossed out are not used in any order-4 magic square. If they exhibit a feature this is shown with a lighter yellow color. The unadorned codes represent base squares that have main diagonals that add to 2 and thus have predictable main diagonals.

HOW TO USE THE TABLE TO MAKE MAGIC SQUARES WITH FEATURES

The "Features Present in Order-4 Base Squares" table provides a quick way to make magic squares that have the features listed. The process is easy if the feature(s) are predictable in all their base squares, a little more complicated if catchup is involved. Fortunately most of the features in order-4 magic squares do not use catchup in any of their base squares. Some main diagonals, most broken diagonals one away from the main diagonals, and the diamond shape use catchup. Catchup for the main diagonals and broken diagonals can be addressed using the table.

Only the B type base squares, C1/C4, C2/C5, F1/F17, and F2/F18 are predictable in their main diagonals. These are the only base squares that will generate magic squares that do not use catchup. Magic squares of this type are called orthodox magic squares on these pages.

Creation of a Dudeney Group IV square that does not use catchup, for example, first requires determining which base squares can be used. They must be either all r or all c. There are 9 base squares that have the Group IV feature in their rows, but only B2/B6, B3/B7, B4/B8, and C2/C6 are predictable in their main diagonals. Alternatively B1/B5, B3/B7, B4/B8, and C1/C5 have the feature in their columns. In the example, pick one base square from each pair (checkboxes), multiply them by 1, 2, 4, and 8 in any order and add one to each number to obtain a magic square that is Group IV. All such combinations will be magic squares but many will not be in the correct Frénicle conformation, they need to be transformed.

The A1/A2, F6/F22, F8/F24, F10/F26, and F12/F28 base squares should allow the presence of Dudeney Group IV squares that use catchup. The initiator A1 base square (red) can be combined with F22, F8, F26, or F28 (blue) to make a catchup base square pair. (see ORDER-4 MAGIC SQUARES USING TWO CATCHUP BASE SQUARES for further discussion.)The catchup pair needs to be combined with base squares obtained from two of the complementary pairs B2/B6, B3/B7, B4/B8, and C2/C6. No combination gives a magic square. All combination sum correctly but they repeat some numbers in the series 1-16 and miss others. Catchup triplets or quartets can also be made using the initiator A1 (red), the maintainers, F6, F24, F10, or F12 (green) and correctors F22, F8, F26, or F28 (blue). Again no combination gives a magic square, so there are no Dudeney Group IV squares that use catchup base squares for their main diagonals.

The Dudeney Group VI feature is present in 18 rows and 18 columns, more than any other feature except agonals which is required for all base squares. In both the rows and columns there are six base squares that are predictable in their main diagonals and have the Group VI feature. These six base squares generate two groups of 96 magic squares in the Orthodox Order-4 Magic Squares table (groups 6 and 7).

In both the rows and columns all four of the base squares that act as initiators of a catchup group have the Dudeney Group VI feature (A1/A2, C3/C7, F3/F19, and F4/F20). If A1, C3, F3, or F4 (red) is the initiator then E1, E2, E3, E4, E5, E6, E23, E24, F5, F6, F23, F24, F9, F10, F11, or F12 (green) can be maintainers. Terminators of the catchup group can be E16, E17, E18, E19, E20, E21, E7, E8, F21, F22, F7, F8, F25, F26, F27, or F28 (blue). The six base squares with predictable main diagonals discussed earlier are used as the predictable base squares that are not in the catchup group if needed.

From the above analysis there are many four base square quartets that can contain a 2, 3, or 4 catchup base square combination, but only a few of them are magic squares. None of the groups in the Order-4 Magic Squares with Two Catchup Base Squares table are Dudeney Group VI. Sixteen of the groups in the Order-4 Magic Squares with Three Catchup Base Squares table are Dudeney Group VI. And all 24 of the groups in the Order-4 Magic Squares with Four Catchup Base Squares table are Dudeney Group VI.

COMPARISON OF BASE SQUARE GROUPS WITH DUDENEY AND WALKINGTON GROUPS

In general the groups defined by Dudeney and Walkington are composed of one or more of the base square groups. The tables below show the relationship. The values in the tables are a number or set of numbers followed by a number in parentheses. The first number or set of numbers are the base square group designation(s) within the base square divisions. The number in parentheses is the number of Frénicle magic squares with the given properties.

Most Dudeney or Walkington groups are composed of one or more base square groups. There are two exceptions within the Walkington classification. The T4.03.1 and T4.05.1 magic squares that have main diagonals that are orthodox belong in the same base square group (Orthodox Group 7). The T4.04 Catchup2 magic squares and some T4.05.4 magic squares that use catchup for the main diagonals in two of their base squares combine to make two Catchup2 groups. The T4.03.1 and T4.04 magic squares in these exceptions contain broken Catchup2 diagonals that are not present in the remaining members of the base square groups.