4 X 4 solution with a(4) = 6 words by Jaime Poniachik:
+---+---+---+---+
| | | |XXX|
+---+---+---+---+
|XXX| | | |
+---+---+---+---+
|XXX| | | |
+---+---+---+---+
|XXX|XXX|XXX| |
+---+---+---+---+
.
5 X 5 solution with a(5) = 8 words by Jaime Poniachik:
+---+---+---+---+---+
| | | |XXX|XXX|
+---+---+---+---+---+
| |XXX| | | |
+---+---+---+---+---+
| | | |XXX| |
+---+---+---+---+---+
|XXX| |XXX|XXX| |
+---+---+---+---+---+
|XXX| | | |XXX|
+---+---+---+---+---+
.
6 X 6 solution with a(6) = 12 words by Jaime Poniachik:
+---+---+---+---+---+---+
| | | |XXX|XXX| |
+---+---+---+---+---+---+
|XXX| |XXX| | | |
+---+---+---+---+---+---+
|XXX| | | |XXX| |
+---+---+---+---+---+---+
| |XXX| | | |XXX|
+---+---+---+---+---+---+
| | | |XXX| |XXX|
+---+---+---+---+---+---+
| |XXX|XXX| | | |
+---+---+---+---+---+---+
.
Example of an invalid solution: a 6 X 6 solution with a(6) = 12 words, but not valid because some words have 2 letters instead of 3, and the 3-letter words are not all interconnected (black squares completely separate the set of 3-letter words at the upper right from the set of 3-letter words at the lower left):
+---+---+---+---+---+---+
| | |XXX| | | |
+---+---+---+---+---+---+
| | |XXX| | | |
+---+---+---+---+---+---+
|XXX|XXX|XXX| | | |
+---+---+---+---+---+---+
| | | |XXX|XXX|XXX|
+---+---+---+---+---+---+
| | | |XXX| | |
+---+---+---+---+---+---+
| | | |XXX| | |
+---+---+---+---+---+---+
.
|