Number sequence tasks are among the most frequently used numerical items in psychometric tests of intelligence or ability. In general, a sequence of about four to ten natural (less frequently rational) numbers is presented, which either contains some discrepant element that has to be detected and marked, or which has to be continued by one or two ''correct'' subsequent elements. Solving number sequence tasks requires the ability to discover a general rule or relation among specific elements and to apply it to new elements. Similar to analogy or matrix problems, and very like letter sequence tasks, solving number sequence tasks is considered to be a prime example of inductive reasoning. Inductive reasoning has been an integral part of standardized intelligence and ability tests for many decades and is generally assumed to be a central component of many cognitive activities (Holyoak, 1984; Pellegrino & Glaser, 1980). Apart from their common use in psychometric tests, number sequence tasks as measures of inductive-reasoning skills are also frequently applied in various fields of psychological research (for examples see Grabitz & Wittmann, 1986; Hackett, Betz, O'Halloran & Romac, 1990; Linde & Bergström, 1992, experiment 2).
It seems that items utilizing letter or number sequences provide appropriate instruments both within psychometrics for measuring components of intelligence and ability as well as for investigating certain theoretical and practical problems in cognitive and experimental psychology. One reason for this suitability of sequence tasks may be the fact that number sequence tasks as well as letter series tasks allow for creating a universe of items with almost arbitrary variations in difficulty levels. In particular, number sequence tasks offer the additional advantage that only knowledge about natural (sometimes rational) numbers and some elementary arithmetic operations is required.
Nevertheless, despite their obvious advantages there is an often voiced criticism directed to the careless use of sequence tasks in psychological contexts. Let us assume for the moment that the problem of the existence of a solution (the solvability problem) for some number sequence task has been settled since the number sequence is constructed according to a certain rule and is thus also solvable according to this rule. There remains, however, the criticism that, generally, aside from the implemented rule of the sequence and the keyed answer considered correct, many differing answers basing on other unintended (say accidental) regularities of the number sequence seem possible and might be judged to be correct. That is, even when the sequence task is solvable according to the implemented constructional rule, the solution may not be unique. This (presupposed) problem is referred to in this paper as the (non-)uniqueness problem of number sequence tasks.
The non-uniqueness problem of number sequence tasks, however, is usually not taken seriously. Jensen (1980), for example, rejects the criticism as ''utterly trivial'',
''... because the other correct solutions are usually possible only for a mathematician; they involve a level of mathematical sophistication far beyond that required for the most obvious solution. Even an expert mathematician who could figure out other possible solutions would not do so in a test situation, because it would take so much more time, and anyone capable of figuring out one of the more complex solutions would certainly have no difficulty arriving at the simplest solution, which in every item is the keyed answer'' (p.153).Particularly in the ''psychometric approach'' as characterized by Mayer, Larkin & Kadane (1984), there seems to be widespread carelessness about the non-uniqueness problem of number sequence items. Possibly, a test constructor primarily engaged in developing a new test of ''inductive reasoning'' will be inclined to eliminate the non-uniqueness problem by identifying inappropriate (unreliable or invalid) items on the basis of empirical data and excluding such items from the test. However, to dismiss the non-uniqueness problem by purely empirical means and arguments seems not only uneconomical but from the perspective of the ''cognitive approach'' (Mayer et al., 1984) also hardly acceptable.
Cognitively oriented research on inductive reasoning has been focused on identifying the cognitive processes involved in psychometric tasks such as analogy problems (Holzman, Pellegrino & Glaser, 1982; Sternberg, 1977) and series-completion problems (Holzman, Pellegrino & Glaser, 1983; Kotovsky & Simon, 1973; Simon & Kotovsky, 1963). Prominent models of sequence completion solution, on which a large part of research has been based, include four component processes: detection of relations, discovery of periodicity, completion of pattern description, and extrapolation, where the first three processes combine to generate a pattern description. Conclusions about performance on series-completion problems are derived mainly from a presumed relationship between pattern descriptions and working memory: more complex pattern descriptions make greater demands on working memory. Clearly, on this level of model building, the problem of non-unique solvability of sequential items may not be ignored. In any case, modeling the solution process for some type of task has to be founded on a careful task analysis where all characteristics of the specific type of task and particularly all possible alternative solution ways have to be taken into account.
Unfortunately, it just happens to be the class of number sequence tasks which seems so favorable in many respects, that creates considerable difficulties for the analysis of the solvability and uniqueness problems. In this paper, the solvability and (non-)uniqueness problems will be investigated for a specific class of number sequence tasks. The line of argument in this investigation is that the two problems of solvability and (non-)uniqueness of solutions can be handled efficiently only if task instruction, constructional principles of sequences and keyed answers for the applied type of number sequence tasks are explicitly referred to a limited class of permissible rules for the sequences.
In fact, the considerations in this paper arose from specific research motivated by Doignon & Falmagne's theory of knowledge spaces (Doignon & Falmagne, 1985; see also Falmagne, Koppen, Villano, Doignon & Johannesen, 1990). This theory has developed powerful methods for the qualitative modeling and diagnosis of knowledge based on ''surmised'' solution dependencies among items and corresponding ''knowledge structures''. However, in Doignon & Falmagne's behavioral approach the preferred methods of establishing surmise structures include (undoubtedly highly sophisticated) expert querying procedures that are not designed to uncover the theoretical basis for the surmised solution dependencies on a set of items. In a research project entitled ''Wissensstrukturen'' at the University of Heidelberg (see footnote *; for an overview see e.g. Lukas & Albert, 1993) an essential aim was to develop methods for theoretical foundation and goal-directed construction of knowledge structures. The intention was to make a contribution to reconcile Doignon & Falmagne's behavioral knowledge modeling and diagnosing approach with more traditional psychometric models and newer developments in cognitive-processes-approaches (see above). Two promising methods turned out to be such methods that derive hypotheses on behavioral knowledge states from the analysis of basic components of domain-specific tasks (e.g. Albert & Held, 1994; Held, 1993), and methods that generate hypothesized knowledge states by utilizing appropriate task-specific cognitive processing models (Ptucha, 1994; Schrepp, 1993; 1995). In one of these studies the intention was to establish a surmise-ordered network of number sequence tasks and to test the corresponding knowledge states empirically. For the tasks a modified type of recursive number sequences as discussed in Krause (1985) seemed appropriate. The surmise-ordered network of number sequence tasks was constructed on the basis of a restricted set of sequence rules varying in the level of complexity. However, the unavoidable problems of solvability and uniqueness of solution for the constructed items emerged: Structuring a set of items on the basis of their solution principles necessarily requires the complete explication of all alternative ways of solving the specific type of item used. In the case of the type of linear-recursive number sequences thematized in this paper the solvability and uniqueness problems can be treated in a relatively elementary way.
The basic concept of a linear-recursive number sequence is introduced in Section 2. Section 3 presents a general type of task utilizing linear-recursive number sequences and addresses the problems of solvability and uniqueness. Section 4 contains some useful and easily derivable results on the existence, uniqueness, and structure of solution formulae for the description of linear-recursive number sequences. The central ideas of this paper are presented in Sections 5 - 7, where an exemplary analysis of the special type of linear-recursive number sequences applied in the previously mentioned study is conducted. Section 7 takes up the decision procedure used in Section 4 for the solution analysis of number sequence tasks and applies it to the special type of task discussed in Sections 5-7. Section 8 contains a summary and general discussion of the results of our investigation.