Computational complexity of inverse word search problem

Word search is a classical puzzle to search for all given words on a given assignment of letters to a rectangular grid (matrix). This problem is clearly in P. The inverse of this problem is more difficult, which asks to assign letters in a given alphabet to a matrix of given size so that every word in a given wordset can be found horizontally, vertically, or diagonally. This problem is in NP; it admits a trivial polynomial-size certificate. We prove its NP-hardness. It turns out to be so even under the following restrictions: 1) the alphabet size is 2 (binary) and 2) all the words to be found are of length at most 2. These results are optimal in the sense that decreasing these bounds 2 to 1 makes the problem be trivially in P.