• For the construction of problems, the chess motives present one possible type of problem components.
• As a principle of construction, we select a small number of motives and then produce three move problems that contain combinations of them.
• Motives --symbolized by a, b, c, and d-- are elements of a single component set C.
• We assume that an antichain order is defined on C. Hence, the principle of set inclusion can be applied.
  

  The component space is as follows:

• By means of the ordering principle of set inclusion we can infer a surmise relation R on the set Q of the 15 problems that are identified with the elements of .

Figure 6: Upward drawing for the problems identified with the elements of the component space .

• Expressed in words, the hypothesis for the investigation is:
  If a problem q identified with a component set is solved by a subject, then all problems q' which are identified with a component set with will also be solved by this subject.

• The hypothetical structure corresponds to a set of 167 knowledge states. Only about 0.5% of 2₁₅ = 32768 potential solution patterns represent valid states.

Method:

• 4 motives were selected and combined as shown in Fig. 6: fork, pin, deflection and guidance.
• The combinations of these four motives form a set of 15 problems.
• Problems were presented to 13 subjects.
• Each problem printed on a single card as a diagram. Subjects had to write down the solution.
  
• Time needed for the solution was controlled by the subjects themselves with aid of a chess clock. No time limit.
  
• Subjects were asked only to answer as accurately and as quickly as possible.
  
• Problems were presented in the order of hypothesized difficulty: Problem {a,b,c,d} with four motives was the first to be presented and the one-motive problems {a},{b},{c} and {d} and were the last to be presented.

Results:

• Criteria for the goodness of fit:
  
  1. the minimal distances between each response pattern and the closest states in the quasi-ordinal knowledge space and
  2. the mean distance between the set of response patterns and the states in the quasi-ordinal knowledge space.
  

• Symmetric Set Difference

  Let be a quasi-ordinal knowledge space, and let be the set of all response patterns in the data; let the elements of as well as those of be represented in the form of subsets of problems.
  
  Further, let and . Then the distance between X and K, abbreviated by dist(X,K), is defined as the number of elements occurring in the symmetric set difference of X and K, that is:
  
  
  The minimal distance of X to , abbreviated by mdist(X,K), is defined as the distance of X to the nearest knowledge state in K, that is:
  
  
  Now, the mean distance d(S,K) of S to K is given by:
  
  where N is the number of response patterns in S.

Table 2: Chess Problems: Minimal Distances and Mean Distance Between the Solution Patterns and the Hypothetical Knowledge Space.

Distances	d
0  1  2  3  4  5  6
4  2  1  3  2  0  1	2.08

(number of subjects)

• Hypothesis holds only for the three subjects who solved all problems and for one subject who failed only in solving problem {a,b,c,d}.
  
• Two subjects out of 13 each show inconsistencies for only one problem.

Discussion:

• Results contradict our deterministic hypothesis because the response patterns of only four subjects agree with it.
  
• Reasons:
  1. difficulty of the chess problems is probably not solely influenced by the type and number of included motives. An investigation by Albert, Schrepp, and Held(1994) show that taking the sequence of motives within problems into consideration can contribute to a more adequate problem structure.
  2. work on chess problems requires great concentration over a large period of time. Thus, we suspect that the order of problem presentation (beginning with {a,b,c,d}) might not have been the best choice.
  3. In Albert, Schrepp, and Held (1994) these problems were taken into account.

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