An hypothesis approach to the solution of anagrams

GERALD A. MENDELSOHN 0 0 Institute of Personality Assessment and Research, University of California , Berkeley, California 91 ,720 The attempts of subjects to reorganize the letters of an anagram were construed as a series of hypotheses about the correct letter order. It was predicted, consequently, that variables which reduce the number of tenable hypotheses or influence the order in which hypotheses are generated. will aff~ct problem difficulty. Five such variables, plus solution word frequency, were used to pr~dlct solu~lOn probabilities in two studies. The multiple Rs obtained were .92 and .82 and the two regression equations were effectively interchangeable. The process of anagram solution was described as entailing the retrieval of words from memory storage on the basis of letter order cues generated by the subject or, less usually, present in the anagram itself. - The process of anagram solution can be analyzed into two phases consisting, respectively, of attempts to reorganize the letters of the anagram and to retrieve the solution word from memory. In a recent paper, Mendelsohn and O'Brien (I974) argued that in the reorganization phase subjects formulate hypotheses about the correct letter order on the basis of the em pirical probabilities of letter events in the language. Like previous investigators following the lead of Mayzner and Tresselt (I962; see also, Mayzner, Tresselt, & Helbock, 1964), they assumed that the bigram is the basic unit with which the subject works in reorganizing the anagram, but in their formulation, the pool of bi grams that can be formed from the anagram constitutes a limited set of hypotheses from which the subject samples. The order in which hypotheses are formed corresponds roughly to the relative transition letter probabilities (TPs) of the bigrams in the pool, the most probable appearing first, and so on. The hypotheses are then tested by attempting to retrieve from memory a whole word which matches the partial reorganization of the anagram or by attempting to arrange the remain ing letters about the hypothesized pair. It follows from this formulation that, when a solution word consists of bigrams which are probable relative to the other (incorrect) bigrams which can be formed from the anagram, the likelihood of solution should be high and solution latency should be low. Conversely, when the bigrams of a solution word are relatively improbable, high latencies and infrequent solutions should be the case. The measure developed by Mendelsohn and O'Brien (I974) to index the relative frequencies of correct and incorrect bigrams consisted of the sum, across the correct bigrams, of the number of incorrect bigrams having higher TPs. It should be noted that the TPs were obtained from Mayzner and Tresselt's (1965) frequency counts and, thus, take word length and letter position into account. Mendelsohn and O'Brien reported a correlation of about .75 between this measure and solution scores, thus confirming their prediction. It was found, further, that this relationship was uninfluenced by solution word frequency. The present paper seeks to extend the hypothesis approach to anagram solution by examining character istics of letters and words other than bigram TPs which may either constrain the number of tenable hypotheses the subject can formulate or affect the sequence in which hypotheses are likely to be formulated. Four such variables were identified as follows: 1. Total number of bigrams with frequency greater than zero. It should, in general, be the case that the fewer the tenable hypotheses (i.e., bigrams with fre quency > 0 which can be formed from the letters of the anagram), the less the problem difficulty. This variable is conceptually related to Ronning's (I965) "ruleout" factor, but is based on the bigram unit rather than on all letters of the anagram considered at once, 2. Number of vowels in the solution word. With very few exceptions, words of five or more letters containing a single vowel do not begin or end with the vowel. Consequently, in this case, otherwise tenable bigram hypotheses involving the vowel can be eliminated and only those in which the vowel occupies an interior position require consideration. The effect should be to reduce the number of tenable hypotheses and, thus, problem difficulty for one-vowel words. 3. First letter of the solution word, vowel or con sonant. Although there are many words which begin with vowels, it is far more common for the first letter to be a consonant. In Mayzner and Tresselt's (I965) sample of 3,422 five-letter words, for example, 85% begin with a consonant; likewise, a vowel is less likely to occupy the first position than any other in five-letter words. Consequently, hypotheses involving a VC com bination in Positions 1-2 should occur relatively late in the sequence of reorganization attempts. Anagrams formed from solution words beginning with a vowel should thus be more difficult to solve. 4. Key letters. Cohen (1968) found that anagrams containing one of the six most infrequent letters in the language (1, K, Q, V, X, Z) were easier to solve. He reasoned that "uncommon letters, if present, reduced the number of letter groupings which were plausible" and thereby "remove more uncertainty than common letters" (p.80). The letter "V," for example, the most frequent of the 6 letters above, can be preceded by 12 different letters and followed by only 6, all vowels, according to the Underwood and Schulz (1960) count. In contrast, "B," the next most infrequent letter, can be preceded by 19 and followed by 17 letters. It is not infrequency per se which is critical but the limited number of combinatorial possibilities the letters possess. Thus, the presence of an infrequent letter provides the possibility of a maximizing strategy, i.e., if the reorgani zation effort begins with a bigram including an infre quent letter, the number of plausible hypotheses can be rapidly exhausted and in some cases reduced to one or two. Consequently, independent of the total number of plausible hypotheses or the relative probabilities of hypotheses, problem difficulty should be reduced in the case of a key letter. Six solution word variables [bigram rank (BR), word frequency (WF), and the four just described] were used as predictors of anagram difficulty in two studies. In addition to zero-order correlations, multiple correlations were obtained and the results for each sample cross-validated on the other. Study 1 The data of the first study were drawn from Mendelsohn and O'Brien (1974). Since that paper includes a detailed descrip tion of the selection of solution words, construction of ana grams and anagram lists, subjects, and procedure, a brief sum mary can suffice, Thirty five-letter solution words were used, half of them low frequency (6 to 10) in the Kucera and Francis (1967) count and half high frequency (35 to 100). There were five low-BR (5-30), five middle-BR (50-70), and five high-B (...truncated)