Free Thousandstar (#4 of the Cluster series) by Piers Anthony

wasn't it. And even had that not been the case, that system would not work. As soon as enough contestants caught on to it, no one would volunteer the wrong guesses. The competition would roll to a halt, as all waited for others to eliminate themselves. There had to be some way to have many successive correct guesses.
    "Five more correct ones," Swoon jetted. "Too many are catching on; haven't you solved it yet? They'll roll out of ships!"
    Heem suppressed an irate blast. "Two thirds of the ships remain." But he was worried. Too many other contestants, able to work on the problem with full information from the outset, were fathoming the pattern and gaining their ships. Twenty more ships might be taken suddenly.
    Back to concepts: the identical ones did not repeat, but what about variants? Hard and Soft were physical properties; so was Dense. But the sequence was Hard-Soft-Joy-Dense. If Dense was right, why had another physical property, Brittle, been ruled wrong, while Joy had been accepted in its place? Followed by Tedium-Ascent-Brittle. And Brittle had been rejected before. How was it that an invalid concept had become valid?
    The key could not be in the number of rejections or in the particular concepts. It had to be in the order of the concepts, so that any concept became wrong when out of place. Now what was that order?
    "Seven more ships!" Swoon jetted despairingly.
    Heem washed her out of his perception, along with the public spray's pronouncements. He was beginning to get it; all he needed was uninterrupted thought. First, he had to analyze and classify the concepts. Then he had to formulate a theory of progression. Then he should verify it by predicting to himself the nature of several forthcoming correct guesses. Finally he had to make his own guess—before the supply of ships was exhausted.
    He worked it out, calling on the increasingly nervous Swoon for data on occasion. There were seven or eight categories of concept: physical Properties, such as Hard, Soft, Dense, Brittle, and Diffuse; Sentient Feelings, such as Joy, Tedium, Humor, Fear, and Courage; Special Motion such as Direction, Aslant, Plunge, Rotation, and Arrival; Taste Sensation, such as Sour, Sweet, Pungent, Savory, and Insipid; Fluid Matter, such as Rain, Sea, Moist, Dry (i.e., absence of fluid), and Liquid; Number, such as One, Two, Three, Four, and Five; Sapient Qualities, such as Wisdom, Stupidity, Sanity, and Craziness; and several stray concepts that could not yet be classified with assurance because there were too many examples of each. Concept categories tended to merge at the edges, as did tastes when the fluids bearing them mixed.
    Now the order: the first two were Physical Properties, the third a Sentient Feeling, the fourth another Physical Property, the fifth another Feeling, the sixth a Direction, the Seventh another Physical Property. Did he have a pattern here? It was hard to tell.
    Analyze it mathematically , he thought. Let the first class of concepts be A, the second B, the third C. Use exponents to indicate repeat concepts.
    Heem paused. Had he really thought that? That was not the way his mind ordinarily worked! He knew of the symbol-conventions of Galactic notation—A, B, C—but did not think in them. This pressure was having a strange effect on him. Nevertheless, it was a good thought.
    He made a mental list of the successful concepts, classifying each as a mathematical notation. Taste A, taste B, taste C, and so on, eliminating for the moment the actual concepts so that the pattern, unobscured by meaning, could emerge. That's it exactly.
    There he was, tasting to himself again, encouraging himself. Perhaps this atypical mannerism stemmed from the lingering disorientation of his failed transfer-hosting. He hoped it would not interfere with his performance.
    Onward: Hard-Soft-Joy-Dense-Tedium became A-A-B-A-B. He did not bother with the exponents after all; A-A 1 -B-A 2 -B 1 seemed to be superfluous refinement, so far. He