Every 4 X 4 board with 3 filled squares either cannot be completed, or can be completed in two or more ways. But with 4 filled squares it is possible:
+-----+-----+
| . 1 | 2 . |
| . . | . . |
+-----+-----+
| . . | 1 . |
| . . | . 3 |
+-----+-----+
Thus a(2) = 4.
The following 16 X 16 puzzle with 55 clues has a unique solution:
+------------+------------+------------+------------+
| . . . 9 | . . . . | . 3 . . | . . . 2 |
| . . . . |15 . . 12 |16 . . . | . 10 . 8 |
| . 4 . 5 | . . . . | . 9 . . | . . . . |
| . . . . | . . . 10 | . . 13 . | . . . 15 |
+------------+------------+------------+------------+
| . . 8 . | . . . . | . . . . | . . . 16 |
| . . . . | . 5 . . | . . . . | . . . . |
|10 . 15 . | . . . . | . . . . | . . . 12 |
| . . . . | . 13 9 . | . 4 . . | . . 7 . |
+------------+------------+------------+------------+
| . . . . |16 . . 14 | . . . . | . . . . |
| . 5 . 4 | . . . . | . 7 . 11 | 1 13 9 . |
| . . . 3 | . . . . | . 1 . . | 5 . 4 . |
| . . . . |10 . . 15 | . . . . | . . . . |
+------------+------------+------------+------------+
|15 . 16 . | . . . . | 8 . 10 . | . . . 14 |
| . . . . | . 1 4 . | . . . . | 2 . 5 . |
| 8 . . . | . . . . |12 . 16 . | . . . . |
| . . . . | . 9 7 3 | . . . . | . . 1 . |
+------------+------------+------------+------------+
Thus a(4) <= 55.
|