# ORDER-4 MAGIC SQUARES GROUPED BY BASE SQUARE QUARTETS

The 880 numbered Frénicle order-4 magic squares are listed below in groups that are derived from the same base square quartet(s). (A base square [or base square set for catchup base square sets] and its complement are treated as the same base square in this discussion, i.e. C2 and C6 are interchangeable as are the catchup pairs F3:E20 and F19:E4.) There are four major divisions of the magic squares on this page. The first division of squares use only base squares with main diagonals that are predictable. The second division of magic squares have two base squares that use catchup in their main diagonals, the other two base squares are predictable. Squares in the third division have three base squares using catchup and in the fourth division all the base squares use catchup for their main diagonals.

In the first two divisions the magic squares in each column are made from the base square quartets given at the top of the column. There can be one, two, or four sets of base square quartets that yield the same group of magic squares after transformations using Frénicle's rules. The base square quartet(s) that generate the group of magic squares are given at the top of each column. Below the base square quartet listing is the number of magic squares that are derived from the base square quartet(s).

If the magic square group has any commonly recognized feature(s), they are listed below the number of magic squares. Below that are the classifications of that group according to Walkington and Dudeney.

Directly above the actual listing of magic squares are squares that show Walkington's sub-magic 2x2 square patterns or transpose of those patterns for all group members. A number 34 indicates a sub-magic 2x2 square with the 34 placed in its upper left corner. The numbers in that 2x2 square sum to 34. If a 2x2 square does not sum to the magic constant, there is no number present in its upper left corner. If all 16 sub squares sum to 34, the magic square is compact2. Numbers in the right column and bottom row require the pan property to complete the 2x2 sub square. The bottom right corner corresponds to the sum of the four corners of the magic square. The orientation of the sub square pattern in a Frénicle magic square will often be different than the orientation shown.

In the last two divisions of the table there are 28 and 24 groups of four and two magic squares respectively. Only eight columns are shown so the tables are split into 4 and 3 sub sections respectively. Each sub section has headings for the magic squares in that sub section.

## ORTHODOX MAGIC SQUARES

The first three columns of orthodox magic squares are defined by just one base square quartet given in the header (In the base square quartets a base square can be the base square listed or its complement, i.e. B3+C1+F1+F2 should be read as either B3 or B7 and either C1 or C5 and either F1 or F16 and either F2 or F17.). In the last four groups of magic squares, 2 base square quartets generate the same group of Frénicle order-4 magic squares. This is because the 2 base square quartets can be interconverted by a simple transformation of all four base squares. A more detailed description of the features of each group is given in ORTHODOX ORDER-4 MAGIC SQUARES. A discussion of those features presence in base squares is given in Order-4 Magic Square Features.

Orthodox Order-4 Magic Squares
 2x2 sub-square pattern 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 1 8 10 15 12 13 3 6 7 2 16 9 14 11 5 4
 1 4 14 15 13 16 2 3 8 5 11 10 12 9 7 6
 1 8 12 13 14 11 7 2 15 10 6 3 4 5 9 16
 1 4 14 15 16 13 3 2 7 6 12 9 10 11 5 8
 1 4 16 13 14 15 3 2 7 6 10 11 12 9 5 8
 1 4 13 16 14 15 2 3 8 5 12 9 11 10 7 6
 1 4 13 16 8 14 3 9 15 5 12 2 10 11 6 7
 1 8 10 15 14 11 5 4 7 2 16 9 12 13 3 6
 1 4 14 15 13 16 2 3 12 9 7 6 8 5 11 10
 1 8 12 13 15 10 6 3 14 11 7 2 4 5 9 16
 1 4 14 15 16 13 3 2 11 10 8 5 6 7 9 12
 1 4 16 13 14 15 3 2 11 10 6 7 8 5 9 12
 1 4 13 16 14 15 2 3 12 9 8 5 7 6 11 10
 1 4 13 16 8 15 2 9 14 5 12 3 11 10 7 6
 1 8 11 14 12 13 2 7 6 3 16 9 15 10 5 4
 1 4 15 14 13 16 3 2 8 5 10 11 12 9 6 7
 1 8 14 11 12 13 7 2 15 10 4 5 6 3 9 16
 1 4 15 14 16 13 2 3 6 7 12 9 11 10 5 8
 1 4 16 13 15 14 2 3 6 7 11 10 12 9 5 8
 1 4 13 16 15 14 3 2 8 5 12 9 10 11 6 7
 1 4 13 16 12 14 3 5 15 9 8 2 6 7 10 11
 1 8 11 14 15 10 5 4 6 3 16 9 12 13 2 7
 1 4 15 14 13 16 3 2 12 9 6 7 8 5 10 11
 1 8 14 11 15 10 4 5 12 13 7 2 6 3 9 16
 1 4 15 14 16 13 2 3 10 11 8 5 7 6 9 12
 1 4 16 13 15 14 2 3 10 11 7 6 8 5 9 12
 1 4 13 16 15 14 3 2 12 9 8 5 6 7 10 11
 1 4 13 16 12 15 2 5 14 9 8 3 7 6 11 10
 1 8 13 12 14 11 2 7 4 5 16 9 15 10 3 6
 1 6 12 15 11 16 2 5 8 3 13 10 14 9 7 4
 1 8 15 10 12 13 6 3 14 11 4 5 7 2 9 16
 1 6 12 15 16 11 5 2 7 4 14 9 10 13 3 8
 1 6 16 11 12 15 5 2 7 4 10 13 14 9 3 8
 1 6 11 16 12 15 2 5 8 3 14 9 13 10 7 4
 1 6 11 16 8 12 5 9 15 3 14 2 10 13 4 7
 1 8 13 12 15 10 3 6 4 5 16 9 14 11 2 7
 1 6 12 15 11 16 2 5 14 9 7 4 8 3 13 10
 1 8 15 10 14 11 4 5 12 13 6 3 7 2 9 16
 1 6 12 15 16 11 5 2 13 10 8 3 4 7 9 14
 1 6 16 11 12 15 5 2 13 10 4 7 8 3 9 14
 1 6 11 16 12 15 2 5 14 9 8 3 7 4 13 10
 1 6 11 16 8 15 2 9 12 3 14 5 13 10 7 4
 1 12 6 15 14 7 9 4 11 2 16 5 8 13 3 10
 1 6 15 12 11 16 5 2 8 3 10 13 14 9 4 7
 1 12 8 13 14 7 11 2 15 6 10 3 4 9 5 16
 1 6 15 12 16 11 2 5 4 7 14 9 13 10 3 8
 1 6 16 11 15 12 2 5 4 7 13 10 14 9 3 8
 1 6 11 16 15 12 5 2 8 3 14 9 10 13 4 7
 1 6 11 16 14 12 5 3 15 9 8 2 4 7 10 13
 1 12 7 14 15 6 9 4 10 3 16 5 8 13 2 11
 1 6 15 12 11 16 5 2 14 9 4 7 8 3 10 13
 1 12 8 13 15 6 10 3 14 7 11 2 4 9 5 16
 1 6 15 12 16 11 2 5 10 13 8 3 7 4 9 14
 1 6 16 11 15 12 2 5 10 13 7 4 8 3 9 14
 1 6 11 16 15 12 5 2 14 9 8 3 4 7 10 13
 1 6 11 16 14 15 2 3 12 9 8 5 7 4 13 10
 1 12 13 8 14 7 2 11 4 9 16 5 15 6 3 10
 1 7 12 14 10 16 3 5 8 2 13 11 15 9 6 4
 1 12 14 7 15 6 4 9 8 13 11 2 10 3 5 16
 1 7 12 14 16 10 5 3 6 4 15 9 11 13 2 8
 1 7 16 10 12 14 5 3 6 4 11 13 15 9 2 8
 1 7 10 16 12 14 3 5 8 2 15 9 13 11 6 4
 1 7 10 16 8 12 5 9 14 2 15 3 11 13 4 6
 1 12 13 8 15 6 3 10 4 9 16 5 14 7 2 11
 1 7 12 14 10 16 3 5 15 9 6 4 8 2 13 11
 1 12 15 6 14 7 4 9 8 13 10 3 11 2 5 16
 1 7 12 14 16 10 5 3 13 11 8 2 4 6 9 15
 1 7 16 10 12 14 5 3 13 11 4 6 8 2 9 15
 1 7 10 16 12 14 3 5 15 9 8 2 6 4 13 11
 1 7 10 16 8 14 3 9 12 2 15 5 13 11 6 4
 1 14 7 12 15 4 9 6 10 5 16 3 8 11 2 13
 1 7 14 12 10 16 5 3 8 2 11 13 15 9 4 6
 1 14 8 11 15 4 10 5 12 7 13 2 6 9 3 16
 1 7 14 12 16 10 3 5 4 6 15 9 13 11 2 8
 1 7 16 10 14 12 3 5 4 6 13 11 15 9 2 8
 1 7 10 16 14 12 5 3 8 2 15 9 11 13 4 6
 1 7 10 16 15 12 5 2 14 9 8 3 4 6 11 13
 1 14 11 8 15 4 5 10 6 9 16 3 12 7 2 13
 1 7 14 12 10 16 5 3 15 9 4 6 8 2 11 13
 1 14 12 7 15 4 6 9 8 11 13 2 10 5 3 16
 1 7 14 12 16 10 3 5 11 13 8 2 6 4 9 15
 1 7 16 10 14 12 3 5 11 13 6 4 8 2 9 15
 1 7 10 16 14 12 5 3 15 9 8 2 4 6 11 13
 1 7 10 16 15 14 3 2 12 9 8 5 6 4 13 11
 2 7 9 16 11 14 4 5 8 1 15 10 13 12 6 3
 2 3 13 16 14 15 1 4 7 6 12 9 11 10 8 5
 2 7 11 14 13 12 8 1 16 9 5 4 3 6 10 15
 1 10 8 15 16 7 9 2 11 4 14 5 6 13 3 12
 1 8 10 15 11 14 4 5 16 9 7 2 6 3 13 12
 1 8 12 13 10 15 3 6 7 2 14 11 16 9 5 4
 1 10 7 16 12 8 9 5 15 3 14 2 6 13 4 11
 2 7 9 16 13 12 6 3 8 1 15 10 11 14 4 5
 2 3 13 16 14 15 1 4 11 10 8 5 7 6 12 9
 2 7 11 14 16 9 5 4 13 12 8 1 3 6 10 15
 1 10 8 15 16 7 9 2 13 6 12 3 4 11 5 14
 1 8 10 15 13 12 6 3 16 9 7 2 4 5 11 14
 1 8 12 13 11 14 2 7 6 3 15 10 16 9 5 4
 1 10 7 16 12 15 2 5 8 3 14 9 13 6 11 4
 2 7 12 13 11 14 1 8 5 4 15 10 16 9 6 3
 2 3 16 13 14 15 4 1 7 6 9 12 11 10 5 8
 2 7 13 12 11 14 8 1 16 9 3 6 5 4 10 15
 1 10 15 8 16 7 2 9 4 11 14 5 13 6 3 12
 1 8 11 14 10 15 4 5 16 9 6 3 7 2 13 12
 1 8 14 11 10 15 5 4 7 2 12 13 16 9 3 6
 1 10 7 16 14 8 9 3 15 5 12 2 4 11 6 13
 2 7 12 13 16 9 6 3 5 4 15 10 11 14 1 8
 2 3 16 13 14 15 4 1 11 10 5 8 7 6 9 12
 2 7 13 12 16 9 3 6 11 14 8 1 5 4 10 15
 1 10 15 8 16 7 2 9 6 13 12 3 11 4 5 14
 1 8 11 14 13 12 7 2 16 9 6 3 4 5 10 15
 1 8 14 11 13 12 2 7 4 5 15 10 16 9 3 6
 1 10 7 16 14 15 2 3 8 5 12 9 11 4 13 6
 2 7 14 11 13 12 1 8 3 6 15 10 16 9 4 5
 2 5 11 16 12 15 1 6 7 4 14 9 13 10 8 3
 2 7 16 9 11 14 5 4 13 12 3 6 8 1 10 15
 1 11 8 14 16 6 9 3 10 4 15 5 7 13 2 12
 1 8 13 12 10 15 6 3 16 9 4 5 7 2 11 14
 1 8 15 10 11 14 5 4 6 3 12 13 16 9 2 7
 1 11 6 16 12 8 9 5 14 2 15 3 7 13 4 10
 2 7 14 11 16 9 4 5 3 6 15 10 13 12 1 8
 2 5 11 16 12 15 1 6 13 10 8 3 7 4 14 9
 2 7 16 9 13 12 3 6 11 14 5 4 8 1 10 15
 1 11 8 14 16 6 9 3 13 7 12 2 4 10 5 15
 1 8 13 12 11 14 7 2 16 9 4 5 6 3 10 15
 1 8 15 10 13 12 3 6 4 5 14 11 16 9 2 7
 1 11 6 16 12 14 3 5 8 2 15 9 13 7 10 4
 2 11 5 16 13 8 10 3 12 1 15 6 7 14 4 9
 2 5 16 11 12 15 6 1 7 4 9 14 13 10 3 8
 2 11 7 14 13 8 12 1 16 5 9 4 3 10 6 15
 1 11 14 8 16 6 3 9 4 10 15 5 13 7 2 12
 1 10 16 7 15 8 2 9 4 11 13 6 14 5 3 12
 1 10 7 16 15 8 9 2 12 3 14 5 6 13 4 11
 1 11 6 16 15 8 9 2 14 5 12 3 4 10 7 13
 2 11 8 13 16 5 10 3 9 4 15 6 7 14 1 12
 2 5 16 11 12 15 6 1 13 10 3 8 7 4 9 14
 2 11 7 14 16 5 9 4 13 8 12 1 3 10 6 15
 1 11 14 8 16 6 3 9 7 13 12 2 10 4 5 15
 1 10 16 7 15 8 2 9 6 13 11 4 12 3 5 14
 1 10 7 16 15 8 9 2 14 5 12 3 4 11 6 13
 1 11 6 16 15 14 3 2 8 5 12 9 10 4 13 7
 2 11 14 7 13 8 1 12 3 10 15 6 16 5 4 9
 2 8 11 13 9 15 4 6 7 1 14 12 16 10 5 3
 2 11 13 8 16 5 3 10 7 14 12 1 9 4 6 15
 1 13 8 12 16 4 9 5 10 6 15 3 7 11 2 14
 1 11 16 6 14 8 3 9 4 10 13 7 15 5 2 12
 1 11 6 16 14 8 9 3 12 2 15 5 7 13 4 10
 1 13 4 16 14 8 9 3 12 2 15 5 7 11 6 10
 2 11 14 7 16 5 4 9 3 10 15 6 13 8 1 12
 2 8 11 13 9 15 4 6 16 10 5 3 7 1 14 12
 2 11 16 5 13 8 3 10 7 14 9 4 12 1 6 15
 1 13 8 12 16 4 9 5 11 7 14 2 6 10 3 15
 1 11 16 6 14 8 3 9 7 13 10 4 12 2 5 15
 1 11 6 16 14 8 9 3 15 5 12 2 4 10 7 13
 1 13 4 16 14 12 5 3 8 2 15 9 11 7 10 6
 2 13 8 11 16 3 10 5 9 6 15 4 7 12 1 14
 2 8 13 11 9 15 6 4 7 1 12 14 16 10 3 5
 2 13 7 12 16 3 9 6 11 8 14 1 5 10 4 15
 1 13 12 8 16 4 5 9 6 10 15 3 11 7 2 14
 1 12 6 15 13 8 10 3 16 5 11 2 4 9 7 14
 1 12 14 7 13 8 2 11 4 9 15 6 16 5 3 10
 1 13 4 16 15 8 9 2 12 3 14 5 6 10 7 11
 2 13 12 7 16 3 6 9 5 10 15 4 11 8 1 14
 2 8 13 11 9 15 6 4 16 10 3 5 7 1 12 14
 2 13 11 8 16 3 5 10 7 12 14 1 9 6 4 15
 1 13 12 8 16 4 5 9 7 11 14 2 10 6 3 15
 1 12 7 14 13 8 11 2 16 5 10 3 4 9 6 15
 1 12 15 6 13 8 3 10 4 9 14 7 16 5 2 11
 1 13 4 16 15 12 5 2 8 3 14 9 10 6 11 7
 3 6 9 16 13 12 7 2 8 1 14 11 10 15 4 5
 3 2 13 16 15 14 1 4 6 7 12 9 10 11 8 5
 3 6 13 12 10 15 8 1 16 9 2 7 5 4 11 14
 2 3 13 16 15 14 4 1 8 5 11 10 9 12 6 7
 2 3 15 14 13 16 4 1 8 5 9 12 11 10 6 7
 2 3 14 15 13 16 1 4 7 6 11 10 12 9 8 5
 2 3 14 15 7 13 4 10 16 6 11 1 9 12 5 8
 3 6 12 13 16 9 7 2 5 4 14 11 10 15 1 8
 3 2 13 16 15 14 1 4 10 11 8 5 6 7 12 9
 3 6 13 12 16 9 2 7 10 15 8 1 5 4 11 14
 2 3 13 16 15 14 4 1 12 9 7 6 5 8 10 11
 2 3 15 14 13 16 4 1 12 9 5 8 7 6 10 11
 2 3 14 15 13 16 1 4 11 10 7 6 8 5 12 9
 2 3 14 15 7 16 1 10 13 6 11 4 12 9 8 5
 3 6 15 10 13 12 1 8 2 7 14 11 16 9 4 5
 3 2 16 13 15 14 4 1 6 7 9 12 10 11 5 8
 3 6 16 9 10 15 5 4 13 12 2 7 8 1 11 14
 2 3 16 13 15 14 1 4 5 8 11 10 12 9 6 7
 2 3 15 14 16 13 1 4 5 8 12 9 11 10 6 7
 2 3 14 15 16 13 4 1 7 6 11 10 9 12 5 8
 2 3 14 15 11 13 4 6 16 10 7 1 5 8 9 12
 3 6 15 10 16 9 4 5 2 7 14 11 13 12 1 8
 3 2 16 13 15 14 4 1 10 11 5 8 6 7 9 12
 3 6 16 9 13 12 2 7 10 15 5 4 8 1 11 14
 2 3 16 13 15 14 1 4 9 12 7 6 8 5 10 11
 2 3 15 14 16 13 1 4 9 12 8 5 7 6 10 11
 2 3 14 15 16 13 4 1 11 10 7 6 5 8 9 12
 2 3 14 15 11 16 1 6 13 10 7 4 8 5 12 9
 3 10 5 16 13 8 11 2 12 1 14 7 6 15 4 9
 3 5 10 16 12 14 1 7 13 11 8 2 6 4 15 9
 3 10 13 8 16 5 2 11 6 15 12 1 9 4 7 14
 2 5 11 16 15 12 6 1 8 3 13 10 9 14 4 7
 2 5 15 12 11 16 6 1 8 3 9 14 13 10 4 7
 2 5 12 15 11 16 1 6 7 4 13 10 14 9 8 3
 2 5 12 15 7 11 6 10 16 4 13 1 9 14 3 8
 3 10 8 13 16 5 11 2 9 4 14 7 6 15 1 12
 3 5 16 10 12 14 7 1 6 4 9 15 13 11 2 8
 3 10 16 5 13 8 2 11 6 15 9 4 12 1 7 14
 2 5 11 16 15 12 6 1 14 9 7 4 3 8 10 13
 2 5 15 12 11 16 6 1 14 9 3 8 7 4 10 13
 2 5 12 15 11 16 1 6 13 10 7 4 8 3 14 9
 2 5 12 15 7 16 1 10 11 4 13 6 14 9 8 3
 3 10 15 6 13 8 1 12 2 11 14 7 16 5 4 9
 3 5 16 10 12 14 7 1 13 11 2 8 6 4 9 15
 3 13 6 12 16 2 9 7 10 8 15 1 5 11 4 14
 2 5 16 11 15 12 1 6 3 8 13 10 14 9 4 7
 2 5 15 12 16 11 1 6 3 8 14 9 13 10 4 7
 2 5 12 15 16 11 6 1 7 4 13 10 9 14 3 8
 2 5 12 15 13 11 6 4 16 10 7 1 3 8 9 14
 3 10 15 6 16 5 4 9 2 11 14 7 13 8 1 12
 3 8 10 13 9 14 4 7 16 11 5 2 6 1 15 12
 3 13 10 8 16 2 5 11 6 12 15 1 9 7 4 14
 2 5 16 11 15 12 1 6 9 14 7 4 8 3 10 13
 2 5 15 12 16 11 1 6 9 14 8 3 7 4 10 13
 2 5 12 15 16 11 6 1 13 10 7 4 3 8 9 14
 2 5 12 15 13 16 1 4 11 10 7 6 8 3 14 9
 3 13 8 10 16 2 11 5 9 7 14 4 6 12 1 15
 3 8 13 10 9 14 7 4 6 1 12 15 16 11 2 5
 4 5 14 11 9 16 7 2 15 10 1 8 6 3 12 13
 2 8 11 13 15 9 6 4 5 3 16 10 12 14 1 7
 2 7 9 16 12 13 3 6 15 10 8 1 5 4 14 11
 2 7 11 14 9 16 4 5 8 1 13 12 15 10 6 3
 2 7 11 14 8 13 1 12 9 4 16 5 15 10 6 3
 3 13 12 6 16 2 7 9 5 11 14 4 10 8 1 15
 3 8 13 10 9 14 7 4 16 11 2 5 6 1 12 15
 4 5 14 11 15 10 1 8 9 16 7 2 6 3 12 13
 2 8 11 13 15 9 6 4 14 12 7 1 3 5 10 16
 2 7 9 16 14 11 5 4 15 10 8 1 3 6 12 13
 2 7 11 14 12 13 1 8 5 4 16 9 15 10 6 3
 2 7 13 12 8 11 1 14 9 6 16 3 15 10 4 5
 4 5 10 15 14 11 8 1 7 2 13 12 9 16 3 6
 4 1 14 15 16 13 2 3 5 8 11 10 9 12 7 6
 4 5 15 10 9 16 6 3 14 11 1 8 7 2 12 13
 2 8 13 11 15 9 4 6 3 5 16 10 14 12 1 7
 2 7 12 13 9 16 3 6 15 10 5 4 8 1 14 11
 2 7 13 12 9 16 6 3 8 1 11 14 15 10 4 5
 2 8 9 15 16 11 6 1 13 10 7 4 3 5 12 14
 4 5 11 14 15 10 8 1 6 3 13 12 9 16 2 7
 4 1 14 15 16 13 2 3 9 12 7 6 5 8 11 10
 4 5 15 10 14 11 1 8 9 16 6 3 7 2 12 13
 2 8 13 11 15 9 4 6 12 14 7 1 5 3 10 16
 2 7 12 13 14 11 8 1 15 10 5 4 3 6 9 16
 2 7 13 12 14 11 1 8 3 6 16 9 15 10 4 5
 2 8 9 15 16 13 4 1 11 10 7 6 5 3 14 12
 4 5 16 9 14 11 2 7 1 8 13 12 15 10 3 6
 4 1 15 14 16 13 3 2 5 8 10 11 9 12 6 7
 4 9 14 7 15 6 1 12 5 16 11 2 10 3 8 13
 2 9 7 16 15 8 10 1 12 3 13 6 5 14 4 11
 2 7 14 11 9 16 5 4 15 10 3 6 8 1 12 13
 2 7 16 9 12 13 6 3 5 4 11 14 15 10 1 8
 2 9 8 15 11 7 10 6 16 4 13 1 5 14 3 12
 4 5 16 9 15 10 3 6 1 8 13 12 14 11 2 7
 4 1 15 14 16 13 3 2 9 12 6 7 5 8 10 11
 4 9 15 6 14 7 1 12 5 16 10 3 11 2 8 13
 2 9 7 16 15 8 10 1 14 5 11 4 3 12 6 13
 2 7 14 11 12 13 8 1 15 10 3 6 5 4 9 16
 2 7 16 9 14 11 4 5 3 6 13 12 15 10 1 8
 2 9 8 15 11 16 1 6 7 4 13 10 14 5 12 3
 4 9 6 15 14 7 12 1 11 2 13 8 5 16 3 10
 4 6 9 15 11 13 2 8 14 12 7 1 5 3 16 10
 4 14 5 11 15 1 10 8 9 7 16 2 6 12 3 13
 2 9 16 7 15 8 1 10 3 12 13 6 14 5 4 11
 2 8 15 9 11 13 6 4 5 3 12 14 16 10 1 7
 2 8 9 15 11 13 4 6 7 1 16 10 14 12 5 3
 2 9 8 15 13 7 10 4 16 6 11 1 3 12 5 14
 4 9 7 14 15 6 12 1 10 3 13 8 5 16 2 11
 4 6 15 9 11 13 8 2 5 3 10 16 14 12 1 7
 4 14 9 7 15 1 6 12 5 11 16 2 10 8 3 13
 2 9 16 7 15 8 1 10 5 14 11 4 12 3 6 13
 2 8 15 9 11 13 6 4 14 12 3 5 7 1 10 16
 2 8 9 15 11 13 4 6 16 10 7 1 5 3 14 12
 2 9 8 15 13 16 1 4 7 6 11 10 12 3 14 5
 4 9 16 5 14 7 2 11 1 12 13 8 15 6 3 10
 4 6 15 9 11 13 8 2 14 12 1 7 5 3 10 16
 5 4 16 9 10 15 3 6 11 14 2 7 8 1 13 12
 2 12 7 13 15 5 10 4 9 3 16 6 8 14 1 11
 2 8 15 9 13 11 4 6 3 5 14 12 16 10 1 7
 2 8 9 15 13 11 6 4 7 1 16 10 12 14 3 5
 2 11 7 14 12 13 1 8 5 4 16 9 15 6 10 3
 4 9 16 5 15 6 3 10 1 12 13 8 14 7 2 11
 4 7 9 14 10 13 3 8 15 12 6 1 5 2 16 11
 5 4 16 9 11 14 2 7 10 15 3 6 8 1 13 12
 2 12 7 13 15 5 10 4 14 8 11 1 3 9 6 16
 2 8 15 9 13 11 4 6 12 14 5 3 7 1 10 16
 2 8 9 15 13 11 6 4 16 10 7 1 3 5 12 14
 2 11 13 8 12 7 1 14 5 10 16 3 15 6 4 9
 4 14 7 9 15 1 12 6 10 8 13 3 5 11 2 16
 4 7 14 9 10 13 8 3 5 2 11 16 15 12 1 6
 5 10 11 8 16 3 2 13 4 15 14 1 9 6 7 12
 2 12 13 7 15 5 4 10 3 9 16 6 14 8 1 11
 2 9 15 8 16 7 1 10 3 12 14 5 13 6 4 11
 2 9 8 15 16 7 10 1 11 4 13 6 5 14 3 12
 2 12 5 15 16 7 10 1 13 6 11 4 3 9 8 14
 4 14 11 5 15 1 8 10 6 12 13 3 9 7 2 16
 4 7 14 9 10 13 8 3 15 12 1 6 5 2 11 16
 5 11 10 8 16 2 3 13 4 14 15 1 9 7 6 12
 2 12 13 7 15 5 4 10 8 14 11 1 9 3 6 16
 2 9 15 8 16 7 1 10 5 14 12 3 11 4 6 13
 2 9 8 15 16 7 10 1 13 6 11 4 3 12 5 14
 2 12 5 15 16 13 4 1 7 6 11 10 9 3 14 8
 5 4 14 11 16 9 7 2 3 6 12 13 10 15 1 8
 5 2 16 11 15 12 6 1 4 7 9 14 10 13 3 8
 6 3 15 10 9 16 4 5 12 13 1 8 7 2 14 11
 2 14 7 11 15 3 10 6 9 5 16 4 8 12 1 13
 2 11 5 16 14 7 9 4 15 6 12 1 3 10 8 13
 2 11 13 8 14 7 1 12 3 10 16 5 15 6 4 9
 2 13 7 12 14 11 1 8 3 6 16 9 15 4 10 5
 5 4 15 10 16 9 6 3 2 7 12 13 11 14 1 8
 5 3 16 10 14 12 7 1 4 6 9 15 11 13 2 8
 6 3 15 10 12 13 1 8 9 16 4 5 7 2 14 11
 2 14 7 11 15 3 10 6 12 8 13 1 5 9 4 16
 2 11 8 13 14 7 12 1 15 6 9 4 3 10 5 16
 2 11 16 5 14 7 4 9 3 10 13 8 15 6 1 12
 2 13 11 8 14 7 1 12 3 10 16 5 15 4 6 9
 6 3 13 12 15 10 8 1 4 5 11 14 9 16 2 7
 6 1 15 12 16 11 5 2 3 8 10 13 9 14 4 7
 6 9 12 7 15 4 1 14 3 16 13 2 10 5 8 11
 2 14 11 7 15 3 6 10 5 9 16 4 12 8 1 13
 2 12 15 5 13 7 4 10 3 9 14 8 16 6 1 11
 2 12 5 15 13 7 10 4 11 1 16 6 8 14 3 9
 2 14 3 15 16 7 10 1 11 4 13 6 5 9 8 12
 6 3 16 9 15 10 5 4 1 8 11 14 12 13 2 7
 6 4 15 9 13 11 8 2 3 5 10 16 12 14 1 7
 6 12 9 7 15 1 4 14 3 13 16 2 10 8 5 11
 2 14 11 7 15 3 6 10 8 12 13 1 9 5 4 16
 2 12 15 5 13 7 4 10 8 14 9 3 11 1 6 16
 2 12 5 15 13 7 10 4 16 6 11 1 3 9 8 14
 2 14 3 15 16 11 6 1 7 4 13 10 9 5 12 8
 3 2 13 16 14 15 4 1 8 5 10 11 9 12 7 6
 3 2 14 15 13 16 4 1 8 5 9 12 10 11 7 6
 3 2 15 14 13 16 1 4 6 7 10 11 12 9 8 5
 3 2 15 14 6 13 4 11 16 7 10 1 9 12 5 8
 3 2 13 16 14 15 4 1 12 9 6 7 5 8 11 10
 3 2 14 15 13 16 4 1 12 9 5 8 6 7 11 10
 3 2 15 14 13 16 1 4 10 11 6 7 8 5 12 9
 3 2 15 14 6 16 1 11 13 7 10 4 12 9 8 5
 3 2 16 13 14 15 1 4 5 8 10 11 12 9 7 6
 3 2 14 15 16 13 1 4 5 8 12 9 10 11 7 6
 3 2 15 14 16 13 4 1 6 7 10 11 9 12 5 8
 3 2 15 14 10 13 4 7 16 11 6 1 5 8 9 12
 3 2 16 13 14 15 1 4 9 12 6 7 8 5 11 10
 3 2 14 15 16 13 1 4 9 12 8 5 6 7 11 10
 3 2 15 14 16 13 4 1 10 11 6 7 5 8 9 12
 3 2 15 14 10 16 1 7 13 11 6 4 8 5 12 9
 3 5 10 16 14 12 7 1 8 2 13 11 9 15 4 6
 3 5 14 12 10 16 7 1 8 2 9 15 13 11 4 6
 3 5 12 14 10 16 1 7 13 11 6 4 8 2 15 9
 3 5 12 14 6 10 7 11 16 4 13 1 9 15 2 8
 3 5 16 10 14 12 1 7 2 8 13 11 15 9 4 6
 3 5 14 12 10 16 7 1 15 9 2 8 6 4 11 13
 3 5 12 14 16 10 7 1 6 4 13 11 9 15 2 8
 3 5 12 14 13 16 1 4 10 11 6 7 8 2 15 9
 3 5 16 10 14 12 1 7 9 15 6 4 8 2 11 13
 3 5 14 12 16 10 1 7 2 8 15 9 13 11 4 6
 3 6 13 12 9 16 7 2 8 1 10 15 14 11 4 5
 3 6 13 12 8 10 1 15 9 7 16 2 14 11 4 5
 3 8 10 13 14 9 7 4 5 2 16 11 12 15 1 6
 3 5 14 12 16 10 1 7 9 15 8 2 6 4 11 13
 3 6 13 12 15 10 1 8 2 7 16 9 14 11 4 5
 3 8 9 14 16 13 4 1 10 11 6 7 5 2 15 12
 3 8 13 10 14 9 4 7 2 5 16 11 15 12 1 6
 3 6 9 16 12 13 2 7 14 11 8 1 5 4 15 10
 3 6 16 9 12 13 7 2 5 4 10 15 14 11 1 8
 3 9 8 14 10 6 11 7 16 4 13 1 5 15 2 12
 3 8 13 10 14 9 4 7 12 15 6 1 5 2 11 16
 3 6 12 13 9 16 2 7 14 11 5 4 8 1 15 10
 3 6 16 9 15 10 4 5 2 7 13 12 14 11 1 8
 3 9 8 14 13 16 1 4 6 7 10 11 12 2 15 5
 3 9 6 16 14 8 11 1 12 2 13 7 5 15 4 10
 3 6 15 10 9 16 5 4 14 11 2 7 8 1 12 13
 3 8 9 14 10 13 4 7 16 11 6 1 5 2 15 12
 3 10 13 8 12 6 1 15 5 11 16 2 14 7 4 9
 3 9 16 6 14 8 1 11 2 12 13 7 15 5 4 10
 3 6 15 10 12 13 8 1 14 11 2 7 5 4 9 16
 3 8 9 14 13 10 7 4 6 1 16 11 12 15 2 5
 3 12 5 14 16 13 4 1 6 7 10 11 9 2 15 8
 3 9 16 6 14 8 1 11 5 15 10 4 12 2 7 13
 3 8 14 9 10 13 7 4 5 2 12 15 16 11 1 6
 3 9 8 14 16 6 11 1 10 4 13 7 5 15 2 12
 3 13 6 12 15 10 1 8 2 7 16 9 14 4 11 5
 3 12 6 13 14 5 11 4 9 2 16 7 8 15 1 10
 3 8 14 9 10 13 7 4 15 12 2 5 6 1 11 16
 3 10 13 8 15 6 1 12 2 11 16 5 14 7 4 9
 3 13 10 8 15 6 1 12 2 11 16 5 14 4 7 9
 3 12 13 6 14 5 4 11 2 9 16 7 15 8 1 10
 3 8 14 9 13 10 4 7 2 5 15 12 16 11 1 6
 3 10 16 5 15 6 4 9 2 11 13 8 14 7 1 12
 3 15 2 14 16 6 11 1 10 4 13 7 5 9 8 12
 3 12 13 6 14 5 4 11 8 15 10 1 9 2 7 16
 3 8 14 9 13 10 4 7 12 15 5 2 6 1 11 16
 3 12 5 14 13 6 11 4 10 1 16 7 8 15 2 9
 3 15 2 14 16 10 7 1 6 4 13 11 9 5 12 8
 3 14 5 12 15 2 9 8 10 7 16 1 6 11 4 13
 3 9 14 8 16 6 1 11 2 12 15 5 13 7 4 10
 4 1 16 13 14 15 2 3 5 8 9 12 11 10 7 6
 4 1 16 13 5 14 3 12 15 8 9 2 10 11 6 7
 3 14 8 9 15 2 12 5 10 7 13 4 6 11 1 16
 3 9 14 8 16 6 1 11 5 15 12 2 10 4 7 13
 4 1 16 13 14 15 2 3 9 12 5 8 7 6 11 10
 4 1 16 13 5 15 2 12 14 8 9 3 11 10 7 6
 3 14 9 8 15 2 5 12 6 11 16 1 10 7 4 13
 3 12 14 5 13 6 4 11 2 9 15 8 16 7 1 10
 4 1 16 13 15 14 3 2 5 8 9 12 10 11 6 7
 4 1 16 13 9 14 3 8 15 12 5 2 6 7 10 11
 3 14 12 5 15 2 8 9 6 11 13 4 10 7 1 16
 3 12 14 5 13 6 4 11 8 15 9 2 10 1 7 16
 4 1 16 13 15 14 3 2 9 12 5 8 6 7 10 11
 4 1 16 13 9 15 2 8 14 12 5 3 7 6 11 10
 4 1 14 15 13 16 3 2 7 6 9 12 10 11 8 5
 4 1 13 16 14 15 3 2 7 6 10 11 9 12 8 5
 4 5 14 11 10 15 8 1 7 2 9 16 13 12 3 6
 4 5 14 11 7 9 2 16 10 8 15 1 13 12 3 6
 4 1 14 15 13 16 3 2 11 10 5 8 6 7 12 9
 4 1 13 16 14 15 3 2 11 10 6 7 5 8 12 9
 4 5 14 11 16 9 2 7 1 8 15 10 13 12 3 6
 4 5 15 10 6 9 3 16 11 8 14 1 13 12 2 7
 4 1 15 14 13 16 2 3 6 7 9 12 11 10 8 5
 4 1 13 16 15 14 2 3 6 7 11 10 9 12 8 5
 4 5 15 10 11 14 8 1 6 3 9 16 13 12 2 7
 4 6 11 13 14 15 2 3 9 12 5 8 7 1 16 10
 4 1 15 14 13 16 2 3 10 11 5 8 7 6 12 9
 4 1 13 16 15 14 2 3 10 11 7 6 5 8 12 9
 4 5 15 10 16 9 3 6 1 8 14 11 13 12 2 7
 4 7 10 13 15 14 3 2 9 12 5 8 6 1 16 11
 4 6 9 15 13 11 8 2 7 1 14 12 10 16 3 5
 4 5 10 15 11 14 1 8 13 12 7 2 6 3 16 9
 4 6 11 13 9 15 2 8 14 12 5 3 7 1 16 10
 4 9 14 7 11 5 2 16 6 12 15 1 13 8 3 10
 4 6 15 9 13 11 2 8 1 7 14 12 16 10 3 5
 4 5 11 14 10 15 1 8 13 12 6 3 7 2 16 9
 4 6 11 13 15 9 8 2 5 3 14 12 10 16 1 7
 4 9 15 6 10 5 3 16 7 12 14 1 13 8 2 11
 4 6 15 9 13 11 2 8 10 16 5 3 7 1 12 14
 4 5 16 9 10 15 6 3 13 12 1 8 7 2 11 14
 4 7 10 13 9 14 3 8 15 12 5 2 6 1 16 11
 4 10 7 13 14 15 2 3 5 8 9 12 11 1 16 6
 4 7 9 14 13 10 8 3 6 1 15 12 11 16 2 5
 4 5 16 9 11 14 7 2 13 12 1 8 6 3 10 15
 4 7 10 13 14 9 8 3 5 2 15 12 11 16 1 6
 4 11 6 13 15 14 3 2 5 8 9 12 10 1 16 7
 4 7 14 9 13 10 3 8 1 6 15 12 16 11 2 5
 4 6 13 11 9 15 8 2 7 1 10 16 14 12 3 5
 4 9 14 7 16 5 2 11 1 12 15 6 13 8 3 10
 4 14 5 11 16 9 2 7 1 8 15 10 13 3 12 6
 4 7 14 9 13 10 3 8 11 16 5 2 6 1 12 15
 4 6 13 11 9 15 8 2 16 10 1 7 5 3 12 14
 4 9 15 6 16 5 3 10 1 12 14 7 13 8 2 11
 4 14 9 7 16 5 2 11 1 12 15 6 13 3 8 10
 4 10 5 15 13 7 12 2 11 1 14 8 6 16 3 9
 4 6 13 11 15 9 2 8 1 7 16 10 14 12 3 5
 4 10 7 13 15 5 12 2 9 3 14 8 6 16 1 11
 4 15 5 10 16 9 3 6 1 8 14 11 13 2 12 7
 4 10 15 5 13 7 2 12 1 11 14 8 16 6 3 9
 4 6 13 11 15 9 2 8 10 16 7 1 5 3 12 14
 4 11 6 13 14 5 12 3 9 2 15 8 7 16 1 10
 4 15 9 6 16 5 3 10 1 12 14 7 13 2 8 11
 4 10 15 5 13 7 2 12 6 16 9 3 11 1 8 14
 4 7 13 10 9 14 8 3 6 1 11 16 15 12 2 5
 5 2 15 12 11 16 1 6 10 13 4 7 8 3 14 9
 5 2 15 12 4 11 6 13 16 7 10 1 9 14 3 8
 4 11 5 14 13 6 12 3 10 1 15 8 7 16 2 9
 4 7 13 10 9 14 8 3 16 11 1 6 5 2 12 15
 5 2 15 12 16 11 6 1 4 7 10 13 9 14 3 8
 5 2 15 12 10 16 1 7 11 13 4 6 8 3 14 9
 4 11 14 5 13 6 3 12 1 10 15 8 16 7 2 9
 4 7 13 10 14 9 3 8 1 6 16 11 15 12 2 5
 5 3 14 12 10 16 1 7 11 13 4 6 8 2 15 9
 5 3 14 12 4 10 7 13 16 6 11 1 9 15 2 8
 4 11 14 5 13 6 3 12 7 16 9 2 10 1 8 15
 4 7 13 10 14 9 3 8 11 16 6 1 5 2 12 15
 5 3 14 12 16 10 7 1 4 6 11 13 9 15 2 8
 5 3 14 12 11 16 1 6 10 13 4 7 8 2 15 9
 4 13 6 11 16 1 10 7 9 8 15 2 5 12 3 14
 4 10 13 7 15 5 2 12 1 11 16 6 14 8 3 9
 5 4 16 9 14 11 7 2 3 6 10 15 12 13 1 8
 5 10 11 8 14 4 1 15 3 13 16 2 12 7 6 9
 4 13 7 10 16 1 11 6 9 8 14 3 5 12 2 15
 4 10 13 7 15 5 2 12 6 16 11 1 9 3 8 14
 5 4 16 9 15 10 6 3 2 7 11 14 12 13 1 8
 5 11 10 8 15 4 1 14 2 13 16 3 12 6 7 9
 4 13 10 7 16 1 6 11 5 12 15 2 9 8 3 14
 4 11 13 6 14 5 3 12 1 10 16 7 15 8 2 9
 5 10 11 8 15 4 1 14 2 13 16 3 12 7 6 9
 5 14 3 12 16 11 6 1 4 7 10 13 9 2 15 8
 4 13 11 6 16 1 7 10 5 12 14 3 9 8 2 15
 4 11 13 6 14 5 3 12 7 16 10 1 9 2 8 15
 5 11 10 8 14 4 1 15 3 13 16 2 12 6 7 9
 5 15 2 12 16 10 7 1 4 6 11 13 9 3 14 8
 5 2 16 11 12 15 1 6 9 14 4 7 8 3 13 10
 5 4 14 11 9 16 2 7 12 13 3 6 8 1 15 10
 6 1 16 11 12 15 2 5 9 14 3 8 7 4 13 10
 6 1 16 11 3 12 5 14 15 8 9 2 10 13 4 7
 5 3 16 10 12 14 1 7 9 15 4 6 8 2 13 11
 5 4 15 10 9 16 3 6 12 13 2 7 8 1 14 11
 6 1 16 11 15 12 5 2 3 8 9 14 10 13 4 7
 6 1 16 11 9 15 2 8 12 14 3 5 7 4 13 10
 5 12 9 8 14 3 2 15 4 13 16 1 11 6 7 10
 5 9 12 8 16 4 1 13 2 14 15 3 11 7 6 10
 6 3 15 10 13 12 8 1 4 5 9 16 11 14 2 7
 6 3 15 10 4 9 5 16 13 8 12 1 11 14 2 7
 5 12 9 8 15 2 3 14 4 13 16 1 10 7 6 11
 5 9 12 8 16 4 1 13 3 15 14 2 10 6 7 11
 6 3 15 10 16 9 5 4 1 8 12 13 11 14 2 7
 6 4 13 11 12 15 2 5 9 14 3 8 7 1 16 10
 6 1 15 12 11 16 2 5 10 13 3 8 7 4 14 9
 6 3 13 12 10 15 1 8 11 14 4 5 7 2 16 9
 6 4 13 11 9 15 2 8 12 14 3 5 7 1 16 10
 6 9 12 7 13 3 2 16 4 14 15 1 11 8 5 10
 6 4 15 9 11 13 2 8 10 16 3 5 7 1 14 12
 6 3 16 9 10 15 4 5 11 14 1 8 7 2 13 12
 6 4 13 11 15 9 8 2 3 5 12 14 10 16 1 7
 6 12 9 7 16 3 2 13 1 14 15 4 11 5 8 10
 6 11 10 7 13 4 1 16 3 14 15 2 12 5 8 9
 6 10 11 7 15 3 2 14 1 13 16 4 12 8 5 9
 6 9 12 7 16 3 2 13 1 14 15 4 11 8 5 10
 6 13 4 11 15 12 5 2 3 8 9 14 10 1 16 7
 6 11 10 7 16 1 4 13 3 14 15 2 9 8 5 12
 6 10 11 7 15 3 2 14 4 16 13 1 9 5 8 12
 6 12 9 7 13 3 2 16 4 14 15 1 11 5 8 10
 6 15 3 10 16 9 5 4 1 8 12 13 11 2 14 7

## MAGIC SQUARES USING TWO CATCHUP BASE SQUARES

The order-4 magic squares that use two catchup base squares in order to have main diagonals that add correctly are shown in the table below. There are eight groups of squares that have this catchup property. Each group can be created from four different base square quartets. These four base square quartets generate the same group of Frénicle magic squares after rearrangement using Frénicle's rules. An example is given below. In the example the B8 and B7 base squares are not the same as the B4 and B3 given in the "Order-4 Magic Squares with Two Catchup Base Squares" table given below for 2Catchup Group 1, they are complements of predictable base squares, however, and therefore interchangeable

Interconversion of the Four Base Square Quartets of 2Catchup Group 1
The First Base Square Quartet of 2Catchup Group 1
 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1
 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1
 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1
 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1
First Base Square Quartet Transposed
 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1
 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1
 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1
 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1
First Base Square Quartet Rotated 180°
 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1
 1 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1
 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1
 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0
First Base Square Quartet Transposed and Rotated 180°
 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1
 1 1 0 0 0 1 1 0 1 0 0 1 0 0 1 1
 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1
 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0

The four base square quartets are shown at the top of each group. The two base squares within the parentheses are the catchup squares. The other two base squares have predictable main diagonals. There are 24 Frénicle squares in each group of this section. The properties of the squares within each group are the same except for groups five and six which each have four squares with an additional broken diagonal feature. A more detailed description of the features of each group is given in ORDER-4 MAGIC SQUARES USING TWO CATCHUP BASE SQUARES.

Some of the numbers within the magic squares are in blue. These are the top numbers of a diamond shaped features that adds to 34. All of the diamond shaped features require catchup. Pairs of groups have similar patterns of diamond shaped figures.

Order-4 Magic Squares with Two Catchup Base Squares
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34
 2x2 sub-square pattern 34 34 34 34
 2x2 sub-square pattern 34 34 34 34
 2x2 sub-square pattern 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2 5 16 11 13 12 1 8 4 7 14 9 15 10 3 6
 1 8 10 15 16 13 3 2 5 4 14 11 12 9 7 6
 1 3 16 14 13 15 2 4 8 6 11 9 12 10 5 7
 2 4 13 15 14 16 3 1 7 5 10 12 11 9 8 6
 1 3 16 14 8 15 2 9 13 6 11 4 12 10 5 7
 2 5 12 15 14 16 3 1 11 9 6 8 7 4 13 10
 2 5 16 11 8 12 1 13 9 7 14 4 15 10 3 6
 1 4 14 15 16 11 5 2 9 6 12 7 8 13 3 10
 2 7 14 11 13 10 3 8 4 5 16 9 15 12 1 6
 1 8 13 12 16 11 2 5 3 6 15 10 14 9 4 7
 1 3 16 14 13 15 2 4 12 10 7 5 8 6 9 11
 2 4 13 15 14 16 3 1 11 9 6 8 7 5 12 10
 1 3 16 14 12 15 2 5 13 10 7 4 8 6 9 11
 2 5 12 15 16 14 1 3 9 11 8 6 7 4 13 10
 2 7 14 11 8 10 3 13 9 5 16 4 15 12 1 6
 1 4 15 14 16 10 5 3 9 7 12 6 8 13 2 11
 2 9 16 7 13 8 1 12 4 11 14 5 15 6 3 10
 1 12 13 8 16 7 2 9 3 10 15 6 14 5 4 11
 1 4 14 15 9 12 6 7 16 5 11 2 8 13 3 10
 2 4 13 15 16 14 1 3 5 7 12 10 11 9 8 6
 1 5 16 12 8 14 3 9 10 4 13 7 15 11 2 6
 2 9 8 15 14 16 3 1 7 5 10 12 11 4 13 6
 2 9 16 7 12 8 1 13 5 11 14 4 15 6 3 10
 1 10 8 15 16 13 3 2 5 4 14 11 12 7 9 6
 2 11 14 7 13 6 3 12 4 9 16 5 15 8 1 10
 1 14 4 15 16 11 5 2 9 6 12 7 8 3 13 10
 1 4 15 14 9 12 7 6 16 5 10 3 8 13 2 11
 2 4 13 15 16 14 1 3 9 11 8 6 7 5 12 10
 1 5 16 12 15 14 3 2 10 11 6 7 8 4 9 13
 2 9 8 15 16 14 1 3 5 7 12 10 11 4 13 6
 2 11 14 7 12 6 3 13 5 9 16 4 15 8 1 10
 1 10 15 8 16 6 3 9 5 11 14 4 12 7 2 13
 3 2 16 13 10 15 1 8 7 6 12 9 14 11 5 4
 1 15 4 14 16 10 5 3 9 7 12 6 8 2 13 11
 1 5 16 12 10 14 3 7 8 4 13 9 15 11 2 6
 3 1 16 14 13 15 4 2 8 6 9 11 10 12 5 7
 1 9 16 8 14 12 5 3 4 6 11 13 15 7 2 10
 3 2 15 14 12 16 5 1 13 9 4 8 6 7 10 11
 3 2 16 13 8 15 1 10 9 6 12 7 14 11 5 4
 1 13 8 12 16 11 2 5 3 6 15 10 14 4 9 7
 3 6 12 13 10 11 5 8 7 2 16 9 14 15 1 4
 1 15 10 8 16 6 3 9 5 11 14 4 12 2 7 13
 1 5 16 12 10 14 3 7 15 11 6 2 8 4 9 13
 3 1 16 14 13 15 4 2 12 10 5 7 6 8 9 11
 1 9 16 8 15 12 5 2 4 7 10 13 14 6 3 11
 3 2 15 14 16 12 1 5 9 13 8 4 6 7 10 11
 3 5 12 14 6 11 8 9 15 2 13 4 10 16 1 7
 1 13 12 8 16 7 2 9 3 10 15 6 14 4 5 11
 3 6 15 10 16 11 2 5 1 8 13 12 14 9 4 7
 2 5 16 11 15 10 3 6 4 7 14 9 13 12 1 8
 2 3 13 16 10 11 5 8 15 6 12 1 7 14 4 9
 3 1 16 14 15 13 2 4 6 8 11 9 10 12 5 7
 2 4 15 13 5 14 3 12 16 7 10 1 11 9 6 8
 3 8 9 14 13 15 4 2 12 10 5 7 6 1 16 11
 3 5 12 14 8 9 6 11 13 4 15 2 10 16 1 7
 2 3 13 16 15 12 6 1 10 5 11 8 7 14 4 9
 3 8 13 10 16 9 4 5 1 6 15 12 14 11 2 7
 2 7 9 16 15 14 4 1 6 3 13 12 11 10 8 5
 2 3 16 13 10 11 8 5 15 6 9 4 7 14 1 12
 3 1 16 14 15 13 2 4 10 12 7 5 6 8 9 11
 2 4 15 13 9 14 3 8 16 11 6 1 7 5 10 12
 3 8 9 14 15 13 2 4 10 12 7 5 6 1 16 11
 3 6 12 13 8 11 5 10 9 2 16 7 14 15 1 4
 2 3 16 13 15 9 6 4 10 8 11 5 7 14 1 12
 3 9 16 6 10 8 1 15 7 13 12 2 14 4 5 11
 2 9 16 7 15 6 3 10 4 11 14 5 13 8 1 12
 2 4 15 13 16 14 3 1 5 7 10 12 11 9 6 8
 3 7 10 14 12 16 5 1 6 2 11 15 13 9 8 4
 2 6 15 11 7 13 4 10 9 3 14 8 16 12 1 5
 3 9 8 14 12 16 5 1 6 2 11 15 13 7 10 4
 3 9 8 14 10 7 12 5 15 2 13 4 6 16 1 11
 2 9 7 16 15 14 4 1 6 3 13 12 11 8 10 5
 3 10 7 14 13 4 9 8 12 5 16 1 6 15 2 11
 2 13 3 16 15 12 6 1 10 5 11 8 7 4 14 9
 2 4 15 13 16 14 3 1 9 11 6 8 7 5 10 12
 3 7 10 14 12 16 5 1 13 9 4 8 6 2 15 11
 2 6 15 11 16 13 4 1 9 12 5 8 7 3 10 14
 3 9 8 14 16 12 1 5 2 6 15 11 13 7 10 4
 3 9 8 14 12 5 10 7 13 4 15 2 6 16 1 11
 2 9 16 7 15 5 4 10 6 12 13 3 11 8 1 14
 3 10 15 6 16 7 2 9 1 12 13 8 14 5 4 11
 2 15 6 11 16 5 12 1 9 4 13 8 7 10 3 14
 2 6 15 11 9 13 4 8 7 3 14 10 16 12 1 5
 3 7 10 14 16 12 1 5 2 6 15 11 13 9 8 4
 2 10 15 7 13 11 6 4 3 5 12 14 16 8 1 9
 3 12 5 14 13 15 4 2 8 6 9 11 10 1 16 7
 3 9 16 6 15 8 1 10 2 13 12 7 14 4 5 11
 2 15 4 13 16 6 11 1 9 3 14 8 7 10 5 12
 3 12 13 6 16 5 4 9 1 10 15 8 14 7 2 11
 2 15 10 7 16 9 8 1 3 6 11 14 13 4 5 12
 2 6 15 11 9 13 4 8 16 12 5 1 7 3 10 14
 3 7 10 14 16 12 1 5 9 13 8 4 6 2 15 11
 2 10 15 7 16 11 6 1 3 8 9 14 13 5 4 12
 3 12 5 14 15 13 2 4 6 8 11 9 10 1 16 7
 3 13 12 6 15 4 5 10 2 9 16 7 14 8 1 11
 2 15 4 13 16 10 7 1 5 3 14 12 11 6 9 8
 4 9 8 13 10 7 14 3 15 2 11 6 5 16 1 12
 3 2 16 13 14 11 5 4 7 6 12 9 10 15 1 8
 3 1 14 16 13 15 2 4 8 6 11 9 10 12 7 5
 4 8 9 13 11 15 6 2 14 10 3 7 5 1 16 12
 3 1 14 16 8 15 2 9 13 6 11 4 10 12 7 5
 4 1 16 13 11 15 6 2 14 10 3 7 5 8 9 12
 4 10 15 5 16 7 2 9 1 14 11 8 13 3 6 12
 3 2 13 16 14 12 7 1 11 5 10 8 6 15 4 9
 4 9 8 13 14 3 10 7 11 6 15 2 5 16 1 12
 3 8 13 10 14 11 2 7 1 6 15 12 16 9 4 5
 3 1 14 16 13 15 2 4 12 10 7 5 6 8 11 9
 4 8 9 13 15 11 2 6 10 14 7 3 5 1 16 12
 3 1 14 16 12 15 2 5 13 10 7 4 6 8 11 9
 4 1 16 13 15 11 2 6 10 14 7 3 5 8 9 12
 4 14 11 5 16 3 6 9 1 10 15 8 13 7 2 12
 3 2 16 13 14 9 7 4 11 8 10 5 6 15 1 12
 5 2 16 11 4 15 1 14 13 10 8 3 12 7 9 6
 3 9 16 6 14 4 5 11 7 13 12 2 10 8 1 15
 3 2 13 16 11 10 5 8 14 7 12 1 6 15 4 9
 5 1 16 12 10 14 7 3 8 4 9 13 11 15 2 6
 3 2 16 13 7 12 6 9 14 5 11 4 10 15 1 8
 5 2 15 12 8 16 9 1 11 3 6 14 10 13 4 7
 5 2 15 12 4 13 8 9 14 3 10 7 11 16 1 6
 3 13 8 10 14 11 2 7 1 6 15 12 16 4 9 5
 5 3 16 10 4 14 1 15 13 11 8 2 12 6 9 7
 3 12 13 6 14 7 2 11 1 10 15 8 16 5 4 9
 3 2 16 13 11 10 8 5 14 7 9 4 6 15 1 12
 5 1 16 12 14 10 3 7 4 8 13 9 11 15 2 6
 3 7 14 10 9 12 5 8 16 13 4 1 6 2 11 15
 5 2 15 12 16 8 1 9 3 11 14 6 10 13 4 7
 5 2 15 12 8 9 4 13 10 7 14 3 11 16 1 6
 3 13 12 6 14 7 2 11 1 10 15 8 16 4 5 9
 5 4 13 12 10 7 2 15 8 9 16 1 11 14 3 6
 3 13 2 16 14 12 7 1 11 5 10 8 6 4 15 9
 3 7 14 10 16 12 5 1 2 6 11 15 13 9 4 8
 5 8 10 11 13 16 2 3 4 9 7 14 12 1 15 6
 3 11 14 6 13 10 7 4 2 5 12 15 16 8 1 9
 5 3 14 12 8 16 9 1 10 2 7 15 11 13 4 6
 5 2 16 11 14 15 1 4 3 10 8 13 12 7 9 6
 3 14 7 10 16 4 13 1 9 5 12 8 6 11 2 15
 5 4 13 12 11 6 3 14 8 9 16 1 10 15 2 7
 3 14 11 6 16 9 8 1 2 7 10 15 13 4 5 12
 3 7 14 10 16 12 5 1 9 13 4 8 6 2 11 15
 5 8 11 10 13 16 3 2 4 9 6 15 12 1 14 7
 3 11 14 6 16 10 7 1 2 8 9 15 13 5 4 12
 5 3 14 12 16 8 1 9 2 10 15 7 11 13 4 6
 5 3 16 10 15 14 1 4 2 11 8 13 12 6 9 7
 3 14 7 10 16 11 6 1 2 5 12 15 13 4 9 8
 5 4 14 11 16 13 3 2 1 8 10 15 12 9 7 6
 4 1 15 14 13 12 6 3 8 5 11 10 9 16 2 7
 4 1 14 15 12 9 6 7 13 8 11 2 5 16 3 10
 5 13 4 12 16 8 1 9 2 10 15 7 11 3 14 6
 4 1 15 14 8 11 5 10 13 6 12 3 9 16 2 7
 5 10 8 11 15 14 4 1 2 7 9 16 12 3 13 6
 5 10 8 11 14 7 9 4 3 2 16 13 12 15 1 6
 4 1 14 15 13 11 8 2 12 6 9 7 5 16 3 10
 5 8 10 11 16 9 7 2 1 4 14 15 12 13 3 6
 4 5 16 9 13 10 3 8 2 7 14 11 15 12 1 6
 4 1 15 14 12 9 7 6 13 8 10 3 5 16 2 11
 5 13 4 12 16 8 1 9 3 11 14 6 10 2 15 7
 4 2 13 15 5 14 3 12 16 7 10 1 9 11 8 6
 5 14 4 11 15 10 8 1 2 3 13 16 12 7 9 6
 5 11 8 10 15 6 9 4 2 3 16 13 12 14 1 7
 4 1 15 14 13 10 8 3 12 7 9 6 5 16 2 11
 6 3 14 11 4 13 12 5 15 2 7 10 9 16 1 8
 4 9 16 5 13 6 3 12 2 11 14 7 15 8 1 10
 4 2 13 15 16 14 3 1 5 7 10 12 9 11 8 6
 6 7 12 9 14 15 4 1 3 10 5 16 11 2 13 8
 4 2 13 15 9 14 3 8 16 11 6 1 5 7 12 10
 6 1 16 11 7 15 10 2 12 4 5 13 9 14 3 8
 6 4 15 9 16 13 2 3 1 12 7 14 11 5 10 8
 4 13 2 15 16 6 11 1 9 3 14 8 5 12 7 10
 6 3 14 11 12 5 4 13 7 10 15 2 9 16 1 8
 4 10 15 5 13 3 6 12 8 14 11 1 9 7 2 16
 4 2 13 15 16 14 3 1 9 11 6 8 5 7 12 10
 6 14 3 11 15 7 2 10 4 12 13 5 9 1 16 8
 4 8 13 9 10 11 6 7 15 14 3 2 5 1 12 16
 6 1 16 11 15 7 2 10 4 12 13 5 9 14 3 8
 6 12 7 9 16 5 10 3 1 4 15 14 11 13 2 8
 4 13 2 15 16 10 7 1 5 3 14 12 9 8 11 6
 7 2 15 10 4 13 12 5 14 3 6 11 9 16 1 8
 4 13 12 5 14 11 6 3 1 8 9 16 15 2 7 10
 4 8 13 9 15 11 6 2 1 5 12 16 14 10 3 7
 7 6 12 9 15 14 4 1 2 11 5 16 10 3 13 8
 4 12 13 5 14 9 8 3 1 6 11 16 15 7 2 10
 7 1 16 10 6 14 11 3 12 4 5 13 9 15 2 8
 7 4 14 9 16 13 3 2 1 12 6 15 10 5 11 8
 4 13 8 9 15 3 14 2 10 6 11 7 5 12 1 16
 7 2 15 10 12 5 4 13 6 11 14 3 9 16 1 8
 4 13 12 5 15 10 7 2 1 8 9 16 14 3 6 11
 4 8 13 9 15 11 6 2 10 14 3 7 5 1 12 16
 7 14 4 9 15 6 12 1 2 3 13 16 10 11 5 8
 4 12 13 5 15 9 8 2 1 7 10 16 14 6 3 11
 7 1 16 10 14 6 3 11 4 12 13 5 9 15 2 8
 7 12 6 9 16 5 11 2 1 4 14 15 10 13 3 8
 4 13 8 9 15 12 5 2 1 6 11 16 14 3 10 7

## MAGIC SQUARES USING THREE CATCHUP BASE SQUARES

The order-4 magic squares that use three catchup base squares in order to have main diagonals that add correctly are shown in the table below. There are 28 groups of squares that have this catchup property. Each group can be created from four different base square quartets. These four base square quartets generate the same group of Frénicle magic squares because they represent different orientations of the four base squares.

This table is split into four sub sections with 8 groups in each of the first three sub sections and 4 groups in the last sub section. Four base square quartets are shown at the top of each group. The three base squares within the parentheses are the catchup squares. The other base square is predictable. There are 4 squares in each group of this section. The properties of the group members within each group are the same. A more detailed description of the features of each group is given in ORDER-4 MAGIC SQUARES USING THREE CATCHUP BASE SQUARES.

Order-4 Magic Squares with Three Catchup Base Squares
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 1 8 9 16 14 13 4 3 7 2 15 10 12 11 6 5
 2 7 10 15 11 12 5 6 8 1 16 9 13 14 3 4
 1 12 13 8 16 9 4 5 2 7 14 11 15 6 3 10
 2 11 8 13 15 4 9 6 10 5 16 3 7 14 1 12
 1 6 11 16 14 15 2 3 7 4 13 10 12 9 8 5
 2 3 14 15 11 16 1 6 8 5 12 9 13 10 7 4
 4 9 16 5 11 8 1 14 6 15 10 3 13 2 7 12
 1 12 6 15 16 9 7 2 13 8 10 3 4 5 11 14
 1 12 13 8 15 10 3 6 2 7 14 11 16 5 4 9
 3 6 15 10 13 8 1 12 4 9 16 5 14 11 2 7
 1 14 7 12 16 5 10 3 9 4 15 6 8 11 2 13
 3 6 15 10 14 7 2 11 4 9 16 5 13 12 1 8
 1 11 6 16 12 14 3 5 13 7 10 4 8 2 15 9
 3 5 12 14 6 16 1 11 15 9 8 2 10 4 13 7
 4 11 6 13 15 2 9 8 10 7 16 1 5 14 3 12
 1 14 11 8 16 5 4 9 7 12 13 2 10 3 6 15
 2 11 14 7 16 9 4 5 1 8 13 12 15 6 3 10
 4 5 16 9 14 7 2 11 3 10 15 6 13 12 1 8
 2 11 14 7 15 10 3 6 1 8 13 12 16 5 4 9
 4 5 16 9 13 8 1 12 3 10 15 6 14 11 2 7
 2 11 14 7 12 13 8 1 5 4 9 16 15 6 3 10
 4 5 16 9 6 15 10 3 11 2 7 14 13 12 1 8
 7 2 16 9 6 15 1 12 11 14 4 5 10 3 13 8
 2 11 5 16 15 10 8 1 14 7 9 4 3 6 12 13
 4 3 14 13 15 10 7 2 6 5 12 11 9 16 1 8
 5 6 11 12 16 9 8 1 3 4 13 14 10 15 2 7
 4 11 14 5 15 2 7 10 6 13 12 3 9 8 1 16
 4 9 6 15 11 8 13 2 14 1 12 7 5 16 3 10
 4 1 16 13 15 12 5 2 6 7 10 11 9 14 3 8
 5 2 15 12 16 13 4 1 3 8 9 14 10 11 6 7
 7 6 11 10 14 3 2 15 4 13 16 1 9 12 5 8
 2 11 14 7 15 4 5 10 8 13 12 1 9 6 3 16
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34
 2x2 sub-square pattern 34 34 34 34
 2x2 sub-square pattern 34 34 34 34
 2x2 sub-square pattern 34 34 34 34
 1 2 15 16 13 14 3 4 12 7 10 5 8 11 6 9
 2 8 11 13 10 16 5 3 7 1 12 14 15 9 6 4
 1 6 11 16 13 10 7 4 12 3 14 5 8 15 2 9
 2 8 11 13 14 12 1 7 3 5 16 10 15 9 6 4
 1 2 16 15 13 14 4 3 12 7 9 6 8 11 5 10
 2 8 11 13 10 16 5 3 15 9 4 6 7 1 14 12
 4 6 11 13 10 16 7 1 15 9 2 8 5 3 14 12
 1 6 12 15 13 10 8 3 16 7 9 2 4 11 5 14
 1 7 14 12 9 15 4 6 8 2 13 11 16 10 3 5
 3 4 13 14 15 16 1 2 6 9 8 11 10 5 12 7
 1 7 14 12 11 13 2 8 6 4 15 9 16 10 3 5
 3 12 5 14 15 8 9 2 6 1 16 11 10 13 4 7
 1 7 14 12 9 15 4 6 16 10 5 3 8 2 11 13
 3 4 14 13 15 16 2 1 6 9 7 12 10 5 11 8
 4 6 11 13 16 10 1 7 9 15 8 2 5 3 14 12
 1 7 16 10 11 13 4 6 14 12 5 3 8 2 9 15
 2 1 16 15 14 13 4 3 11 8 9 6 7 12 5 10
 4 3 14 13 16 15 2 1 5 10 7 12 9 6 11 8
 2 5 12 15 14 9 8 3 11 4 13 6 7 16 1 10
 4 11 6 13 16 7 10 1 5 2 15 12 9 14 3 8
 2 1 15 16 14 13 3 4 11 8 10 5 7 12 6 9
 4 3 13 14 16 15 1 2 5 10 8 11 9 6 12 7
 7 4 14 9 11 16 2 5 6 13 3 12 10 1 15 8
 2 5 11 16 14 9 7 4 15 8 10 1 3 12 6 13
 4 6 15 9 14 12 7 1 3 5 10 16 13 11 2 8
 5 3 16 10 11 13 8 2 6 4 9 15 12 14 1 7
 4 6 15 9 16 10 5 3 1 7 12 14 13 11 2 8
 5 3 16 10 15 9 4 6 2 8 13 11 12 14 1 7
 4 6 15 9 14 12 7 1 11 13 2 8 5 3 10 16
 4 6 9 15 14 12 1 7 11 13 8 2 5 3 16 10
 7 11 6 10 16 4 1 13 2 14 15 3 9 5 12 8
 2 8 15 9 14 12 5 3 11 13 4 6 7 1 10 16
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 1 2 15 16 12 14 3 5 13 7 10 4 8 11 6 9
 2 3 14 15 8 16 1 9 11 5 12 6 13 10 7 4
 1 6 11 16 12 10 7 5 13 3 14 4 8 15 2 9
 2 7 10 15 8 12 5 9 11 1 16 6 13 14 3 4
 1 6 11 16 7 15 2 10 14 4 13 3 12 9 8 5
 2 10 7 15 11 16 1 6 8 5 12 9 13 3 14 4
 1 7 14 12 8 13 2 11 9 4 15 6 16 10 3 5
 2 11 8 13 14 12 1 7 3 5 16 10 15 6 9 4
 1 9 8 16 14 15 2 3 7 4 13 10 12 6 11 5
 3 5 12 14 15 16 1 2 6 9 8 11 10 4 13 7
 1 11 6 16 14 13 4 3 7 2 15 10 12 8 9 5
 3 13 4 14 15 8 9 2 6 1 16 11 10 12 5 7
 1 11 6 16 13 14 3 4 12 7 10 5 8 2 15 9
 3 4 13 14 6 16 1 11 15 9 8 2 10 5 12 7
 1 13 12 8 15 10 3 6 2 7 14 11 16 4 5 9
 3 6 15 10 12 8 1 13 5 9 16 4 14 11 2 7
 2 1 16 15 11 13 4 6 14 8 9 3 7 12 5 10
 4 6 11 13 16 15 2 1 5 10 7 12 9 3 14 8
 2 5 12 15 11 9 8 6 14 4 13 3 7 16 1 10
 4 14 3 13 16 7 10 1 5 2 15 12 9 11 6 8
 2 12 5 15 14 13 4 3 11 8 9 6 7 1 16 10
 4 3 14 13 5 15 2 12 16 10 7 1 9 6 11 8
 2 14 11 7 16 9 4 5 1 8 13 12 15 3 6 10
 4 5 16 9 11 7 2 14 6 10 15 3 13 12 1 8
 4 14 3 13 15 12 5 2 6 7 10 11 9 1 16 8
 5 2 15 12 3 13 4 14 16 8 9 1 10 11 6 7
 4 15 6 9 16 10 5 3 1 7 12 14 13 2 11 8
 5 3 16 10 6 9 4 15 11 8 13 2 12 14 1 7
 4 1 16 13 6 12 5 11 15 7 10 2 9 14 3 8
 5 11 6 12 16 13 4 1 3 8 9 14 10 2 15 7
 4 3 14 13 6 10 7 11 15 5 12 2 9 16 1 8
 5 15 2 12 16 9 8 1 3 4 13 14 10 6 11 7
 2x2 sub-square pattern 34 34 34 34
 2x2 sub-square pattern 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 4 3 14 13 10 16 7 1 15 9 2 8 5 6 11 12
 1 7 16 10 14 13 4 3 11 12 5 6 8 2 9 15
 4 9 16 5 14 8 1 11 3 15 10 6 13 2 7 12
 1 6 12 15 16 9 7 2 13 8 10 3 4 11 5 14
 4 3 14 13 16 10 1 7 9 15 8 2 5 6 11 12
 1 12 6 15 13 10 8 3 16 7 9 2 4 5 11 14
 4 14 3 13 15 2 9 8 10 7 16 1 5 11 6 12
 1 11 14 8 16 5 4 9 7 12 13 2 10 6 3 15
 7 4 14 9 5 16 2 11 12 13 3 6 10 1 15 8
 2 8 15 9 11 12 5 6 14 13 4 3 7 1 10 16
 7 2 16 9 12 15 1 6 5 14 4 11 10 3 13 8
 2 5 11 16 15 10 8 1 14 7 9 4 3 12 6 13
 7 5 12 10 16 4 1 13 2 14 15 3 9 11 6 8
 2 11 5 16 14 9 7 4 15 8 10 1 3 6 12 13
 7 12 5 10 14 3 2 15 4 13 16 1 9 6 11 8
 2 14 11 7 15 4 5 10 8 13 12 1 9 3 6 16

## MAGIC SQUARES USING FOUR CATCHUP BASE SQUARES

The order-4 magic squares that require four catchup base squares in order to have main diagonals that add correctly are shown in the table below. There are 24 groups of squares that have this catchup property. Each group can be created from four different base square quartets. These four base square quartets generate the same group of Frénicle magic squares because they represent different orientations of the same base squares.

The four base square quartets are shown at the top of each group. In this table there are three sub sections of eight groups each. All four base squares are in the parentheses because they are all catchup squares. The two middle base squares are in brackets because their multipliers can be exchanged. There are only 2 squares in each group of this section. The properties of the group members within each group are the same. A more detailed description of the features of each group is given in ORDER-4 MAGIC SQUARES USING FOUR CATCHUP BASE SQUARES.

Order-4 Magic Squares with Four Catchup Base Squares
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 1 10 7 16 12 13 4 5 15 8 9 2 6 3 14 11
 4 5 12 13 7 16 1 10 14 11 6 3 9 2 15 8
 1 12 5 16 14 9 8 3 15 6 11 2 4 7 10 13
 4 7 14 9 15 6 1 12 2 11 16 5 13 10 3 8
 1 10 15 8 12 13 6 3 5 4 11 14 16 7 2 9
 4 5 14 11 7 16 9 2 10 1 8 15 13 12 3 6
 1 3 14 16 10 13 4 7 15 6 11 2 8 12 5 9
 4 7 10 13 14 16 1 3 5 9 8 12 11 2 15 6
 1 10 7 16 14 11 6 3 15 8 9 2 4 5 12 13
 6 3 14 11 7 16 1 10 12 13 4 5 9 2 15 8
 1 12 15 6 14 9 4 7 3 8 13 10 16 5 2 11
 6 7 12 9 15 4 1 14 2 13 16 3 11 10 5 8
 1 10 15 8 14 11 4 5 3 6 13 12 16 7 2 9
 4 5 14 11 15 8 1 10 2 9 16 7 13 12 3 6
 1 5 12 16 10 11 6 7 15 4 13 2 8 14 3 9
 4 14 5 11 15 8 1 10 2 9 16 7 13 3 12 6
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 1 3 14 16 15 13 4 2 12 8 9 5 6 10 7 11
 4 2 15 13 14 16 1 3 7 11 6 10 9 5 12 8
 4 12 5 13 14 6 11 3 7 1 16 10 9 15 2 8
 1 7 10 16 15 9 8 2 12 4 13 5 6 14 3 11
 1 3 14 16 15 13 4 2 10 6 11 7 8 12 5 9
 4 2 15 13 14 16 1 3 5 9 8 12 11 7 10 6
 1 12 5 16 15 13 4 2 10 6 11 7 8 3 14 9
 4 2 15 13 5 16 1 12 14 9 8 3 11 7 10 6
 1 5 12 16 15 11 6 2 14 8 9 3 4 10 7 13
 6 2 15 11 12 16 1 5 7 13 4 10 9 3 14 8
 6 12 7 9 14 4 1 15 3 13 16 2 11 5 10 8
 1 7 10 16 15 9 8 2 14 6 11 3 4 12 5 13
 1 5 12 16 15 11 6 2 10 4 13 7 8 14 3 9
 4 10 7 13 14 8 9 3 5 1 16 12 11 15 2 6
 1 14 3 16 15 11 6 2 10 4 13 7 8 5 12 9
 4 5 14 11 10 8 1 15 7 9 16 2 13 12 3 6
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 2x2 sub-square pattern 34 34 34 34 34 34 34 34
 4 2 15 13 7 16 1 10 14 11 6 3 9 5 12 8
 1 10 7 16 15 11 6 2 14 8 9 3 4 5 12 13
 4 7 14 9 12 6 1 15 5 11 16 2 13 10 3 8
 1 12 5 16 15 9 8 2 14 6 11 3 4 7 10 13
 1 3 14 16 12 13 4 5 15 8 9 2 6 10 7 11
 4 5 12 13 14 16 1 3 7 11 6 10 9 2 15 8
 1 7 10 16 12 9 8 5 15 4 13 2 6 14 3 11
 4 14 7 9 15 6 1 12 2 11 16 5 13 3 10 8
 6 2 15 11 7 16 1 10 12 13 4 5 9 3 14 8
 1 10 7 16 15 13 4 2 12 8 9 5 6 3 14 11
 6 7 12 9 14 4 1 15 3 13 16 2 11 10 5 8
 1 14 3 16 15 9 8 2 12 4 13 5 6 7 10 11
 1 5 12 16 14 11 6 3 15 8 9 2 4 10 7 13
 6 3 14 11 12 16 1 5 7 13 4 10 9 2 15 8
 1 7 10 16 14 9 8 3 15 6 11 2 4 12 5 13
 6 12 7 9 15 4 1 14 2 13 16 3 11 5 10 8