1. Fast, Solvable Crosswords James Connor, John Duchi, Bruce Lo

2. Symmetry Heavily constrained puzzles often exhibit symmetry All five by five and six by six puzzles had totally symmetric solutions Many of the provided crosswords had symmetric solutions

3. Symmetry Examples The important one

4. Bit Arrays Allow for compact data representation in spite of empty space (20,000 words ~2 MB) Bit-wise manipulations are very fast We feel cool for using things at such a low level

5. Bit Arrays Call all the words of length 6 a class Each class is represented by a collection of bit arrays. Each letter ch at each position p gets a bit array These bit arrays represent the words with letter ch in position p.

6. Bit Array Example ‘a’ in position 1: 001011 The third, fifth and sixth words in the lexicon have an ‘a’ in position 1. ‘c’ in position 4: 001000 The third word in the lexicon has a ‘c’ in position 4.

7. Bit Array Example (cont’d) ‘a’: 001011 ‘c’: 001000 *a**c*: 001000 Result: The third word in the lexicon has an ‘a’ in the first position, a ‘c’ in the fourth

8. Timing Results Solved in an average of one-tenth of a second 13 x 13 crossword with 60 variables

9. Easy vs. Hard Constraint Satisfaction is NP-Complete Solve in around 6 seconds median Never Solve!

10. New York Times Wednesday Proves very difficult to solve Solution rate of 60% Average solution time (when successful) 180 seconds

11. New York Times Thursday Significantly Easier 100% solution rate Average solution time 10 seconds

12. Why Easier? Long words intersect with relatively few long words in the Thursday puzzle. Forward checking with conflict-directed backjumping can focus on a few words. Allows, basically, local search for the big words. Humans just are not good at big words.

13. The Hardest Puzzle