is of this form with and . Since , we are dealing with GF(11), so pick and compute its residues (mod
11), which are

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Picking the first residues and adding 0 gives: 0, 1, 2, 4, 5, 8, which should
then be colored in the matrix obtained by writing out
the residues increasing to the left and up along the
border (0 through ,
followed by ),
then adding horizontal and vertical coordinates to get the residue to place in each
square.

(13)

To construct ,
consider the representations . Only the first form can be used,
with
and .
We therefore use GF(19), and color 9 residues plus 0
white.

Now consider a more complicated case. For , the only form having is the first, so use the GF() field. Take as the modulus the irreducible
polynomial ,
written 1021. A four-digit number can always be written using only three digits,
since
and .
Now look at the moduli starting with 10, where each digit is considered separately.
Then

(14)

Taking the alternate terms gives white squares as 000, 001, 020, 021, 022, 100, 102, 110, 111, 120, 121, 202, 211, and 221.