Letters/Numbers Grid-Maze

Each integer from 1 to 36 is associated with a letter (a, b, or c) as given below.

Place the integer/letter combinations ('ILC's) into the spaces (one ILC per space) of a 6-by-6 grid such that:

* No two ILCs of the same letter are adjacent (left/right/above/below).

* The ILC with the number m on it is adjacent (left/right/above/below) to the ILC with a (m-1) and the ILC with the (m+1) on it (for 2 <= m <= 35).

* The ILC with the 36 on it is adjacent to the ILC with the 1.

(The ILCs can be thought of a necklace of 36 multicolored beads (connected in order) instead of ILCs, if you want.)

The ILCs are numbered and lettered as follows:

1a 2b 3c 4a 5c 6a

7c 8a 9b 10a 11b 12a

13b 14a 15c 16b 17c 18a

19c 20b 21c 22b 23a 24c

25a 26b 27c 28b 29c 30b

31c 32a 33c 34b 35a 36b

In other words:

a: 1 4 6 8 10 12 14 18 23 25 32 35

b: 2 9 11 13 16 20 22 26 28 30 34 36

c: 3 5 7 15 17 19 21 24 27 29 31 33

I am wondering if someone can find a solution that is not the solution that I am thinking of. (Of course, there is more than one solution

It might just be easier to think about the ILCs as simply an ordered string of letters (a's, b's, and c's), instead of worrying about the particular integers paired with them (since the only real reason for the integers is to order the letters).

Thanks, Leroy Quet