Generating puzzles for a Four-by-Four Word Game

A few years ago, I made a crossword-like game wherein users fill out a 4×4 grid of letters to spell 8 words (4 across and 4 down): http://osric.com/chris/wordgame/

Four by Four word puzzle game

However, over the course of several days, I was able to develop only 20 puzzles by hand. Trying all the various combinations is clearly a task better suited to computers than humans. Such a grid has 16 slots, each of which can contain one of 26 letters–so there are 2616 total permutations to check. That’s about 43,000,000,000,000,000,000,000–which could take a very long time, even for a computer. One key to speeding things up is to ignore permutations that don’t contain words.
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