*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 whoParticularly 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.couldfigure 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).

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.

© 1998 Pabst Science Publishers