A Statistical Study of Failures in Solving Crossword Puzzles
S. Naranan · Journal of Quantitative Linguistics · 2010
Crossword puzzles are the most popular form of linguistic puzzles; for the solver they are intellectually challenging and entertaining as well. An interesting exercise for this author, a keen solver of the British-style ‘cryptic’ crossword puzzles, has been the statistical distribution of the number of unsolved clues (x) in a puzzle. Data are cumulated over a decade (total number of puzzles 3404). The large sample size makes it possible to examine the tail of the distribution at large x, up to 12. It is found that the Poisson distribution with one free parameter (λ) is inadequate, but the negative binomial distribution (NBD) with two free parameters (p, k) fits the distribution well as vouched by a χ 2-test. The NBD can be interpreted as a “mixture” of Poisson and Gamma distributions. It is suggested that this is an appropriate model for the distribution of x. Surprisingly, a 3-parameter lognormal distribution (LND2) also fits the observed distribution of x equally well. The popular model for LND – theory of proportional effect – does have some relevance for the crossword puzzle solving. It appears that the dichotomy (NBD and LND2) exists only for a limited range of (p, k) of the NBD. Both the NBD and LND2 have wide application in many branches of science. It is conjectured that NBD may apply to all crossword puzzles and all solvers. The present work has relevance to linguistics, especially the co-existence of random and orderly features as reflected in the many statistical regularities of language texts.