Some Types

Adjacent Corner Paired

See terms.

There are no squares without wrap-around. With wrap-around there are just 2 squares in group 26,870,638:

Note that the squares are complementary.

Adjacent Side Paired

See terms.

Without wrap-around, there are 6216 squares in 3 groups:

With wrap-around, there are an additional 5468 squares in 4 groups:


There are 3600 squares in 37 groups. The group size is shown under the group number. If not all the squares in the group are pandiagonal, the number of pandiagonal squares is shown in parentheses. Group 7 is the associative squares:

There are an additional 77580 pandiagonal 1-way squares in 18445 groups ranging in size from 2 to 228960. The 5 groups with the most pandiagonal 1-way squares are:

Bent Diagonal 1-Way

There are 9190 squares in 2643 groups ranging in size from 2 to 6622. There are squares with all center values, (from 1 to 25). The 5 groups with the most bent diagonal 1-way squares are:

All the squares in these 5 groups are bent diagonal 1-way.

ZigZag2 2-Way

There are 53568 squares in 189 groups ranging in size from 2 to 10272. There are squares with center values from 6 to 20. The 5 groups with the most zigzag2 2-way squares are:

All the squares in the 189 groups are zigzag2 2-way.

Note that the complement of each number in row 2 is opposite in row 4. Therefore, the numbers in rows 1, 3, 5 of each column will sum to the magic constant with the numbers in rows 2 and 4 of any column.

Some Center 13 Types

Octant Paired

See terms.

There are 7 basic types:

The number of groups of each type is shown below the type name.
One group pattern of each type occurs in the Symborder groups shown below.
Types 6 and 7 include two orientations each.

There are also hybrid and other combinations of these basic types.


See terms.

There are 15 groups:

The number of squares in the group is shown below each group number.

Groups 1 to 5 are the biggest order 5 groups.

Group 1: opposite side, same side hybrid, (concentric, symlateral)
Group 2: opposite side, (concentric)
Group 3: same side, (symlateral)
Group 4: center symmetric, opposite side hybrid, (associative, concentric)
Group 5: center symmetric, same side hybrid, (associative, symlateral)

Group 7:     center symmetric, (associative)
Group 85:   cross corner, criss-cross, cross side, cross diagonal combination
Group 86:   cross corner, criss-cross, cross diagonal, cross diagonal combination
Group 123: cross diagonal
Group 124: cross corner, criss-cross hybrid

Group 134:    criss-cross, cross side hybrid
Group 135:    cross corner, cross side hybrid
Group 368:    cross side
Group 1,162: criss-cross
Group 1,163: cross corner